diff --git a/research/p104-cross-call-paid-reuse/README.md b/research/p104-cross-call-paid-reuse/README.md
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+# Earlier P104 answers do not fund these eight separately delivered centre replacements
+
+**Local-only checked result; no new κ.** This binds the internal paid outputs
+of PR5718's first two actual LOCAL invocations, at caller events **0 and 3**,
+instead of testing only their input rows. All 3,426 local paid outputs have
+exact coefficient rows, instruction versions, frames and owned epochs. The
+two invocations use **536 different independent helper columns each**. Their
+main inputs are expanded into all 32 original raw caller columns, with the
+actual stage-1 routes and native operators.
+
+The named initial target is call-event3, local-event13609:
+`MOVE owned744, frame559 -> frame593`, of rank seven. Its last logical write
+is SCALE13607, following COPY13606 from the current version of173. A difference
+in **call3 helper173 alone** leaves every call0 answer zero and every available
+destination-frame observation zero, but changes this answer by
+`(1/2) S1 Phi593 phi`. This is a nonzero full native field. It cannot be replaced
+by combinations of old-call answers and those destination-frame values.
+
+The adjacent bounded domain is the other seven rank-seven owned queries in
+the same invocation. A complete optimistic cost screen additionally allows
+**any subset of earlier paid outputs in call3**, provided each retained value
+is separately transported to the requested output chart before recombination.
+No such subset improves the H7 charge, for any tariff `Hr=r^t`, `0<t<1`.
+This is a restriction on that explicit delivery protocol, not on joint
+transport, arbitrary native recombination or every changed consumer.
+
+## Exact maps and observation contract
+
+`results/QUERY-MAPS.json.gz` is a factorized exact map certificate, not a list
+of query signatures. Its 3,426 complete local rows are instantiated twice using
+machine-readable `native_input_bindings`. Every paid output records its source
+version, output version, last logical write, INIT epoch, rank, old/new frame and
+every nonzero public coefficient. Source names retain invocation and owner.
+Owned entrance values are initially arbitrary and are removed only by the
+literal owned INIT/CLEAR contract. The independent literal walk verifies every
+one of753 public/owned endpoint columns and all scalar coframes and lifetimes.
+
+Write `Phi_f` for the pinned canonical nine-axis frame, `Cq` for the translation
+root at portq and `S1=C_e9 ... C_e16`. If the literal right-boundary rows0,1 give
+raw inputs `A,B`, call0 logical main inputs are `X=Cq^-1 A`, `Y=B`.
+The actual resident outputs and wrapper are
+
+```
+U = H9 X
+V = Cq^-1 H9 (Y+X)
+second raw input = (V, -Cq^-2 U).
+```
+
+The literal stage-1 contacts then give logical main inputs
+`P_rho(q) Cq^-1 (X+Y)` and `-P_rho(q) Cq^-1 X` at the routed physical owner.
+The104 actual involutive routes swap the pinned eight orthonormal
+q-perpendicular axes with fresh axes9..16 and fixq. The verifier checks416
+exact Fourier phase-polynomial identities, including the translations and
+wrapper signs. Scalar coefficients stay outside tensor powers.
+
+Consequently a query row with main coefficients `c_r,c_(r+112)` has, for each
+raw caller columnj, the stage-1 operator terms
+
+```
+(c_r-c_(r+112))*right[0,j] * S1 Phi_f P_rho(q) Cq^-2
+             + c_r*right[1,j] * S1 Phi_f P_rho(q) Cq^-1.
+```
+
+Its helper term is `c_h S1 Phi_f h1(h,I)`, where the normalized helper is
+`h_s(h)=(S_s Phi_initial(h))^-1 H_s_raw(h)`. The machine substitution includes
+this inverse source gauge, so raw helper columns are explicit too. Call0 uses
+`H0_raw(h,g)`, a different independent input even when the register number agrees. Before a MOVE use
+`f=old`; after it use `f=new`. These rules, the complete right boundary,
+input substitutions and source-bound Clifford circuits specify the full
+maps without materializing enormous dense265-axis matrices. Other axes are
+spectators under the retained caller contract.
+
+## Exact quotient and complete delivery screen
+
+First grant every call0 answer, at every owner, with unlimited retention and
+even free transport. Grant all main information at the requested output chart.
+Also retain every earlier call3 paid output already in that chart and every
+current field there. On differences supported only on fresh call3 helpers,
+all call0 answers and all granted main information vanish.
+
+After removing the common invertible native prefix, ordinary recombination
+is an exact rational row-span question. The kernel witnesses below annihilate
+every remaining ordinary observation. They remain witnesses even if arbitrary
+linear actions on the available destination-frame fields are allowed: zero
+fields stay zero. No distinction-of-functions assumption is used.
+
+| Local event | Owned role | Frames | Fresh-helper difference | Nonzero response |
+| ---: | ---: | --- | --- | --- |
+|13609|744|559→593|`e173`|`1/2`|
+|13620|745|543→594|`e151`|`1/2`|
+|13631|746|531→595|`e146-e34`|`-1/2`|
+|13640|747|591→597|`e133`|`-1/2`|
+|13660|748|518→598|`e21`|`-1/4`|
+|13673|749|267→599|`e34`|`1/4`|
+|13688|750|517→600|`e21`|`1/4`|
+|13703|751|477→601|`e21`|`-1/4`|
+
+Each scalar response is multiplied by its full `S1 Phi_new` operator. The
+eight centre maps are independently checked with the retained native compiler
+and Gaussian interpreter, including absolute phases. The first map is also
+checked on all512 central basis columns. Independent interpretation also checks
+7,976 nine-axis cache-cost tableaux and997 actual17-axis alignment tableaux.
+This witnesses actual query outputs,
+not only source INPUT continuity.
+
+For the stronger screen, a retained prefix answer at frameF is delivered by
+the exact canonical operator `Phi_new Phi_F^-1`. Its Gaussian rank is computed
+from the full signed tableau. Any ordinary monomial normalizers are included
+in that native map. All acquisition, retention, normalizers, scalar arithmetic,
+main corrections and cleanup are optimistically free. Thus a failure cannot
+be repaired merely by pricing those obligations more carefully.
+
+There are1,195–1,228 actual earlier paid outputs at the eight cuts. Every
+strict improvement over H7 necessarily satisfies `sum r_i <= 6`, because
+`sum r_i^t >= (sum r_i)^t` for `0<t<1`. Zero-rank alignments are exactly the
+literal output frame and are already in the ordinary quotient. At each cut
+there are at most two rank-one alignments. This reduces the entire subset
+question to:
+
+- One or two other answers: exact projective residual grouping tests all
+  possibilities with rational coefficients, including non-dyadic coefficients.
+  No survivor. This grants more coefficients than a legal dyadic word permits.
+- Three rank-two answers with no rank-one answer: for the first four queries,
+  the target is outside the span of **all** rank-two answers. For the last four,
+  exact basis-coordinate support bounds give74,45,25,17 uncovered target
+  coordinates, while one answer covers at most2,2,2,3. Even three cannot cover
+  the target. Dense earlier-centre coordinate vectors are granted free before
+  forming this support bound.
+- One or both rank-one answers, plus at most two rank≥2 answers: condition on
+  each of the three possible nonempty rank-one subsets, grant its whole span,
+  debit its rank, and repeat the exact two-answer test. No survivor.
+
+Four rank≥2 jobs already cost more than H7. Repeated copies of one snapshot
+do not add an information direction. These cases exhaust separately aligned
+snapshot subsets under the necessary rank budget. The receipts retain every
+coordinate map and every conditioning choice.
+
+This does **not** cover mixing different snapshots in an intermediate chart
+before a shared paid job, noncanonical matrix-valued recombination, new
+ordinary reads from other frames, source-consuming changes, changed helper
+contracts or a weaker terminal consumer. It is not a physical lower bound for
+all P104 implementations.
+
+## Positive relaxation, controls and adjacent main query
+
+The full call0 paid history has scalar rank744. Erasing native maps makes
+the main-only `X0+Y0` target reconstructible from earlier paid answers. The
+receipt supplies and rechecks the exact combination; distinct answers are
+therefore not presumed independent. Illegally identifying H1 with H0 makes
+the first centre appear reconstructible too, with a393-answer combination.
+The actual namespace binding rejects that stale identification.
+
+A real two-version identity also passes:
+`703(after13610)-703(before13610) = owned744(after13609)`.
+The future13610 value is rejected as an input to the13609 replacement.
+A synthetic sum of two actual low-rank paid outputs is discovered by the
+two-cache generator, so the search has a constructive discovery control.
+Wrong global phase and an unpaid nonzero frame retag are rejected.
+
+As a distinct adjacent candidate, call3 local9733 is the pure-main
+`MOVE role0, frame0 -> frame249`, rank one. Grant the entire desired logical
+`X0+Y0` snapshot in every one of997 earlier paid-output frames, ignoring the
+cost of reconstructing it. Exact17-axis native alignment still costs at
+least H1; frame128 attains that minimum. Thus one-snapshot or typed cache-tree
+delivery cannot improve the removed H1 job, even before retention/read costs.
+This says nothing about a jointly shared job serving several changed outputs.
+
+## Complete invoice and obligations
+
+The complete source invoice is independently rebuilt from the local word and
+literal18-call controller and compared with PR5718's RESULT:
+
+```
+66,348 paid jobs; 12,976 productive fields; rank surplus760.
+local paid:    2184 H1 +459 H2 +495 H3 +92 H4 +159 H5 +22 H6 +15 H7
+local targets: 2 H1 +5 H2 +91 H4 +6 H5 +6 H7 +15 H8 +411 H9
+```
+
+Deleting one named call3 centre would grant `Delta paid=-H7`, `Delta targets=0`.
+The tested protocols cannot earn that allowance. **Realized paid and target
+deltas are both zero.** Every original/helper target, owned clear and caller
+job remains required. No cache program was emitted or given productive credit.
+The adjacent main cache's best possible recursive saving is zero.
+
+The pinned source reports221,474 local scalar units and a184,647,552 complete
+ordinary upper bound under10^9. Those are source figures, not new replayed
+ordinary tariffs. Actual new COPY/retention, every read, route, dirty return,
+restoration, cleanup, hosting, compiler and outer obligations are **unknown or
+unrun** because no improving candidate survived. Unknown costs are not zero.
+No new κ or full multiplication theorem is claimed. No selected cold replay,
+Lean, Docker, heavy full native replay or compiler/outer replay was run.
+
+## Reproduction and source status
+
+From the repository root, Python3.12 standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-cross-call-paid-reuse
+```
+
+The registered serial normal and optimized commands rederive all maps,
+quotients, native ranks, support certificates, invoices and controls and compare
+the entire decompressed JSON receipts. Direct command:
+
+```sh
+python3.12 -I -B research/p104-cross-call-paid-reuse/experiment.py
+```
+
+`--write` regenerates only this package's two result files. It never edits
+frozen sources. CPython3.12.8/macOS arm64 was used for the registered replay;
+initial exploration used3.14.5. Timings are not proof fields.
+
+Canonical main was refreshed to `de5ec2e7a6074068452f45b1a191b5c72832df4a`,
+priority8; selected κ remains `.00310045`. PR5718's exact head is
+`a1534ac45401c9a9c80c07aa65c89e00e33594f8`, claiming `.00311024570`.
+It was archive-merged at15:22:41Z on October26; that does not select or admit
+its claim. Live Discussion5600 additionally reports an unpublished
+`.00311301600` construction; it was read as provisional overlap, not executed.
+The checkpoint searched312 open PRs with complete pagination and25 most
+recently updated Discussions with three recent comments each. Cached complete
+overlap is older and explicitly labelled. Unpublished work and arbitrary diffs
+remain outside that coverage.
+
+The managed latest-main checkout failed twice on missing promisor blobs.
+This clean managed branch therefore starts at `df0137efd35e11b4445734bf612bdd4c7ebc330c`;
+latest guidance was read through Git and the exact scientific inputs remain
+the specified PR5718 source. Every consumed byte is pinned in `SOURCES.json`.
+No shared checkout or predecessor artifact was edited. No public post or PR
+was made.
+
+## Open next questions
+
+The next reuse experiment must change the delivery protocol. At call3
+local13660, specify one intermediate frame, actual earlier paid-answer
+versions, a chronological gather and **one shared final native job** to
+frame598. Compare its full paid bill against H7 before implementing scalar
+cleanup. First test exact fresh-helper availability, including helper21, and
+then all original/main columns. Separately delivering more snapshots is closed
+by the certificate above; another larger separate-cache census is not useful.
+Shared transport, retention, every read and final clears need a concrete
+allowance. A favorable scalar span alone does not justify an implementation
+campaign. `TASK.json` records this scoped handoff and stop rule.
+
+## Significance for researchers
+
+The missing internal-output extraction is now available. Earlier paid
+answers can have full scalar information while failing a later versioned
+obligation or its actual native delivery price. The result closes one natural
+source-specific completion protocol and identifies the changed transport
+premise needed to reopen it. It does not turn finite failed searches into a
+universal obstruction.
+
+Credit [flyingviper](https://github.com/flyingviper)'s
+[PR5718](https://github.com/CrocSwap/integer-mult-bounds/pull/5718),
+[Ali Romman](https://github.com/aliromman)'s
+[PR5713](https://github.com/CrocSwap/integer-mult-bounds/pull/5713), and the full
+native/P104/caller/compiler lineage retained in `inputs/NOTICE5718`.
+The completed predecessor at local commit
+`1f51164a1dba5b9cb7639d3b760d2a0660931f63`, both earlier information-reuse reports
+and the controller input ledger are credited in `SOURCES.json` and unchanged.
+The existing paid-history-quotient, centre-history-innovation and
+gaussian-cache-reuse packages supply the scope and classical module/transport
+principles. [PR5647](https://github.com/CrocSwap/integer-mult-bounds/pull/5647)
+is constructive precedent, not transferred proof. New source binding and finite
+checks prepared for [zxv](https://github.com/zxv) with substantial OpenAI Codex
+assistance; no subagents, independent human review or formal-kernel proof.
+Apache-2.0.
diff --git a/research/p104-cross-call-paid-reuse/SOURCES.json b/research/p104-cross-call-paid-reuse/SOURCES.json
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index 0000000000..a059933e0c
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/SOURCES.json
@@ -0,0 +1,150 @@
+{
+  "schema_version": 1,
+  "base_commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "refreshed_canonical_main": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "priority_revision_read": 8,
+  "selected_kappa": ".00310045",
+  "source_claim_kappa": ".00311024570",
+  "source_publication": "PR5718 archive-merged 2026-10-26T15:22:41Z; not selected",
+  "runtime_inputs": [
+    {
+      "path": "inputs/DATA-PHASES.json",
+      "sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
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+      "commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+      "repository_path": "research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json"
+    },
+    {
+      "path": "inputs/NOTICE5718",
+      "sha256": "c474479a870ae53ae0bcb84f656c2b222380395bbee767ee15e5f9c5b4ce7e79",
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+      "commit": "a1534ac45401c9a9c80c07aa65c89e00e33594f8",
+      "repository_path": "research/p104-two-boundary-helper-split/NOTICE"
+    },
+    {
+      "path": "inputs/clifford_exact.py",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
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+      "repository_path": "research/p104-native-channel/vendor/clifford_exact.py"
+    },
+    {
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+    {
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+    "/private/tmp/integer-mult-information-reuse/research/information-reuse-screen/README.md",
+    "/private/tmp/integer-mult-information-reuse/research/p104-multimeeting-repair/README.md",
+    "/tmp/luna-inforeuse-controller/cross_local_input_ledger.md",
+    "research/paid-history-quotient",
+    "research/centre-history-innovation",
+    "research/gaussian-cache-reuse"
+  ],
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+      "path": "research/p104-native-channel/vendor/native_verify.py",
+      "usage": "provenance",
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+      "usage": "provenance",
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+    },
+    {
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+      "usage": "provenance",
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+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
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+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
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+      "usage": "provenance",
+      "git_blob": "f1c72c7c8e291f04272d645fea3e78f5f7ecb80a",
+      "sha256": "c219f4d143be68639c676a5ff6e3ff84651b484be6a95f1a3d538c71182fd78f"
+    },
+    {
+      "path": "research/centre-history-innovation/README.md",
+      "usage": "provenance",
+      "git_blob": "8d382b734879ab781366477132c842cf84522ec5",
+      "sha256": "c797418742fd13b85c27f2ffb06b3f8eb035bef1d2a5e7a02086cb299d853462"
+    },
+    {
+      "path": "research/gaussian-cache-reuse/README.md",
+      "usage": "provenance",
+      "git_blob": "2c5449434b3eca66107f3847c27a6350a8b2d8e1",
+      "sha256": "3e9462ff17b02ad69564d5c25b810a4ad5f2c4b95997160e8617668710003b87"
+    }
+  ]
+}
diff --git a/research/p104-cross-call-paid-reuse/TASK.json b/research/p104-cross-call-paid-reuse/TASK.json
new file mode 100644
index 0000000000..2fa27ffff6
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/TASK.json
@@ -0,0 +1,70 @@
+{
+  "schema_version": 1,
+  "task_id": "centre-path-sharing",
+  "priority_revision": 7,
+  "base_commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "original_card": {
+    "id": "centre-path-sharing",
+    "question": "Can the 64-per-call centre path (eight rank-8 COPY/ERASE totals through the private temporary) be shared across calls or shrunk?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/split-centres-global/proofs/PR386-BIT-GEOMETRY.txt",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "Derive a fully charged favorable relaxation for sharing one centre path across k calls, including restoration, hosting and precision.",
+      "Prove a lower bound on copied information per call for the selected decoder contract."
+    ],
+    "acceptance_gate": "Show the changed centre charge and copy profile per call with every added restoration or hosting cost charged, and its ceiling via ceiling.ceiling(centre=...). A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Write the per-call centre charge and copy children for sharing across k = 2 calls, charge the sharing overhead, and screen it; stop this route if even the ideal sharing gains under 10%. Ideal budgets: free 2.80x, shared by two 1.90x (levers.py).",
+    "bottleneck_link": "The centre path consumes 64 of about 99.6 unpaid coordinates per call, i.e. about 64% of the raw deficit.",
+    "allowed_variations": [
+      "One sharing pattern across a fixed number of calls.",
+      "One reduced copied-total contract for the existing decoder.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Removing copies without a replacement contract for the decoder's copied totals.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "centre path",
+      "copied totals",
+      "private temporary",
+      "shared centre",
+      "batching"
+    ]
+  },
+  "scope_status": "changed",
+  "scope_change_reason": "Explicit assignment overrides the historical raw centre card: test actual PR5718 internal paid outputs across caller events0 and3, with invocation-specific helper inputs and current native costs.",
+  "actual_scope": "Exact source-bound maps and quotient/cost tests for eight rank-seven outputs; every separately aligned prefix-cache subset, plus one pure-main cache-tree discriminator.",
+  "bottleneck_link": "A real paid H7 deletion would improve the complete complex invoice while preserving every target. No tested protocol earns that deletion.",
+  "budget_minutes": 60,
+  "stop_rule": "Finish exact maps, controls and complete scoped cost certificate. Stop after separate delivery loses; joint transport needs a new costed boundary before expansion.",
+  "outcome": "negative-result",
+  "evidence": "results/RESULT.json and results/QUERY-MAPS.json.gz; README proof and limitations.",
+  "author": "zxv",
+  "ai_assistance": "Substantial OpenAI Codex assistance; no subagents or independent human/formal review.",
+  "open_questions": [
+    {
+      "question": "Can joint intermediate-frame delivery beat H7 at caller event3 local13660?",
+      "why_it_matters": "Every separately canonically aligned cache subset fails the necessary rank budget, but shared transport changes that premise.",
+      "first_step": "Bind earlier query versions and one intermediate frame, test the exact helper21 obligation, and price the entire shared gather and final delivery before scalar or compiler expansion.",
+      "blocked_by": "No joint transport word, retention/read/cleanup tariff, full original-column reconstruction or changed caller invoice is supplied."
+    }
+  ],
+  "expansion_decision": "Hand off joint transport; do not enlarge the separately aligned cache census.",
+  "latest_guidance_revision_read": 8,
+  "publication": "Local-only. No public posts, PRs, merges or selected-frontier edits authorized or made."
+}
diff --git a/research/p104-cross-call-paid-reuse/VALIDATION.md b/research/p104-cross-call-paid-reuse/VALIDATION.md
new file mode 100644
index 0000000000..0006859b68
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/VALIDATION.md
@@ -0,0 +1,21 @@
+# Verification record
+
+Local-only, 2026-10-26. Substantial OpenAI Codex assistance; no subagents or independent human/formal review.
+
+Environment: CPython 3.12.8, macOS-26.5.1-arm64-arm-64bit; standard library only. Initial exploratory replay also passed under CPython3.14.5. Final scientific checks use3.12.8.
+
+Passed serially:
+
+- `python3.12 -B scripts/research_package.py --package research/p104-cross-call-paid-reuse --check-only`: `PASS_PACKAGE_METADATA_AND_PINS`.
+- `python3.12 -B scripts/research_package.py --package research/p104-cross-call-paid-reuse`: both normal and optimized full receipt comparisons pass; `PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result`.
+- Optimized direct replay from a new Git-free directory, with empty PATH and a separate bytecode namespace: `PASS_VERSIONED_CROSS_CALL_SCOPED_HANDOFF_NO_NEW_KAPPA`. All 16 copied package files remained byte-identical. Diagnostic elapsed time 16.877s. This file was added after that portable replay; it is not a scientific input.
+- Exact source-copy equality to all seven origin Git objects, then registered metadata pins.
+- `git diff --check`.
+
+Coverage: two physical invocation bindings, all3426 local paid rows each, all753 source/owned endpoint columns, all536 independent helper columns per invocation,416 actual native phase identities,7976 independently interpreted cache-cost tableaus,997 independent17-axis alignment tableaus, all512 first-centre native basis columns, every strict-saving separately aligned cache subset in the stated model, the full66348-job/12976-target invoice, constructive two-version and two-cache discovery controls, stale-helper/future-access/unpaid-retag/wrong-global-phase controls.
+
+Scientific RESULT SHA256: `acbb686ebfb739253318c574a72692ab061e2c0d680fab380549e3cb7fc97ab5`.
+
+Decompressed QUERY-MAPS payload SHA256: `4e1cbf2c933eed697dc0f256d0da1c6980b685baabcd74a6e3ee844ab0676a49`. Compressed container bytes are not the mathematical equality criterion.
+
+No selected cold replay, full native-lowering replay, Lean, Docker, compiler/outer scientific replay, formal proof or rounded multiplier was run. No new cache program, kappa gain, public post or PR is claimed. Joint intermediate-frame transport, noncanonical matrix-valued combinations, new reads and changed consumers remain outside this certificate. New retention/read/route/restoration/cleanup costs remain unknown; optimistic zero costs were used only to reject the declared delivery model.
diff --git a/research/p104-cross-call-paid-reuse/VERIFY.json b/research/p104-cross-call-paid-reuse/VERIFY.json
new file mode 100644
index 0000000000..839123a897
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/VERIFY.json
@@ -0,0 +1,38 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/p104-cross-call-paid-reuse/inputs/DATA-PHASES.json",
+    "research/p104-cross-call-paid-reuse/inputs/NOTICE5718",
+    "research/p104-cross-call-paid-reuse/inputs/clifford_exact.py",
+    "research/p104-cross-call-paid-reuse/inputs/controller5718.json",
+    "research/p104-cross-call-paid-reuse/inputs/native_verify.py",
+    "research/p104-cross-call-paid-reuse/inputs/result5718.json",
+    "research/p104-cross-call-paid-reuse/inputs/word5718.json.gz"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/experiment.py"
+      ],
+      "timeout_seconds": 300,
+      "checks": "All versioned paid rows, exact helper quotients, all separately aligned cache subsets under the full necessary cost budget, native phases, full invoice and positive/stale/access controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/experiment.py"
+      ],
+      "timeout_seconds": 300,
+      "checks": "Same full deterministic receipts and controls with Python assertions disabled."
+    }
+  ]
+}
diff --git a/research/p104-cross-call-paid-reuse/experiment.py b/research/p104-cross-call-paid-reuse/experiment.py
new file mode 100644
index 0000000000..3309f0be84
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/experiment.py
@@ -0,0 +1,595 @@
+"""Versioned P104 paid rows at caller events 0 and 3; exact bounded reuse tests.
+
+Prepared for zxv with substantial OpenAI Codex assistance. Apache-2.0.
+The P104 word, Clifford implementation and caller retain predecessor credit.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from functools import lru_cache
+from hashlib import sha256
+from pathlib import Path
+import gzip
+import importlib.util
+import json
+import sys
+
+HERE = Path(__file__).resolve().parent
+
+def need(ok, why):
+    if not ok:
+        raise ValueError(why)
+
+def read(name):
+    raw = (HERE/name).read_bytes()
+    return json.loads(gzip.decompress(raw) if name.endswith('.gz') else raw)
+
+def encode(x):
+    if isinstance(x, Q): return str(x)
+    if isinstance(x, dict): return {str(k): encode(v) for k, v in x.items()}
+    if isinstance(x, (tuple, list)): return [encode(v) for v in x]
+    return x
+
+def canonical(x):
+    return json.dumps(encode(x), sort_keys=True, separators=(',', ':')).encode()
+
+def digest(x): return sha256(canonical(x)).hexdigest()
+
+def load(name):
+    spec = importlib.util.spec_from_file_location('cross_reuse_'+name, HERE/'inputs'/name)
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[spec.name] = module
+    spec.loader.exec_module(module)
+    return module
+
+def plus(dst, src, coefficient=Q(1)):
+    for k, v in tuple(src.items()):
+        z = dst.get(k, Q(0)) + coefficient*v
+        if z: dst[k] = z
+        else: dst.pop(k, None)
+
+def dot(row, vector): return sum(v*vector.get(k, 0) for k, v in row.items())
+
+class Basis:
+    def __init__(self): self.rows = {}; self.combinations = {}
+    def reduce(self, row, combination=None):
+        row = dict(row)
+        combination = {} if combination is None else dict(combination)
+        for p, b in sorted(self.rows.items()):
+            if p in row:
+                q = row[p]
+                plus(row, b, -q)
+                plus(combination, self.combinations[p], -q)
+        return row, combination
+    def insert(self, row, label):
+        row, combination = self.reduce(row, {label: Q(1)})
+        if not row: return False
+        p = min(row); q = row[p]
+        self.rows[p] = {k: v/q for k, v in row.items()}
+        self.combinations[p] = {k: v/q for k, v in combination.items()}
+        return True
+    def solve(self, row):
+        remainder, c = self.reduce(row)
+        return remainder, {k: -v for k, v in c.items()}
+    def separator(self, target):
+        r, _ = self.reduce(target)
+        need(bool(r), 'A separator requires a nonzero quotient')
+        j = min(r); vector = {j: Q(1)}
+        for p, row in sorted(self.rows.items(), reverse=True):
+            z = -sum(v*vector.get(k, 0) for k, v in row.items() if k != p)
+            if z: vector[p] = z
+        need(all(dot(b, vector) == 0 for b in self.rows.values()), 'Annihilate observations')
+        need(dot(target, vector) != 0, 'Separate target')
+        return vector
+
+def project(row, columns): return {k: v for k, v in row.items() if k in columns}
+
+def walk(w):
+    """Independent literal walk; every paid output retains its exact last-write epoch."""
+    n = w['public']; size = n+w['owned_fields']
+    rows = [{r: Q(1)} for r in range(size)]
+    frames = w['initial']+[None]*w['owned_fields']
+    versions = [0]*size; last_write = [-1]*size; lives = [0]*size
+    active = set(); paid = []; selected = []; source_main = []
+    main = set(w['live']) | {r+112 for r in w['live']}
+    helpers = set(range(n))-main
+    def frame_span(f):
+        out = {0}
+        for b in w['frames'][f]: out |= {x ^ b for x in tuple(out)}
+        return out
+    spans = [frame_span(f) for f in range(len(w['frames']))]
+    for index, e in enumerate(w['program']):
+        op, d = e[:2]
+        if op == 'MOVE':
+            _, _, a, b, private = e
+            need(frames[d] == a and private == (d in active), 'Exact frame and ownership at MOVE')
+            need(spans[b] < spans[a] if private else spans[a] < spans[b], 'Nested transport')
+            record = dict(event=index, role=d, input_version=versions[d], output_version=versions[d]+1,
+                          last_logical_write=last_write[d], owned_epoch=lives[d], private=private,
+                          old=a, new=b, rank=abs(len(w['frames'][b])-len(w['frames'][a])), row=dict(rows[d]))
+            paid.append(record)
+            if private:
+                donors = [(j, versions[j], last_write[j], dict(rows[j])) for j in range(size) if frames[j] == b]
+                retained = [p for p in paid[:-1] if p['new'] == b]
+                selected.append(dict(query=record, donors=donors, retained_same_call=retained))
+            if d in w['live'] and rows[d] == {d: Q(1)}:
+                source_main.append(record)
+            frames[d] = b
+        elif op == 'INIT':
+            need(d >= n and d not in active, 'Owned INIT')
+            active.add(d); lives[d] += 1; frames[d] = e[2]; rows[d] = {}
+        elif op == 'CLEAR':
+            need(d in active and frames[d] == e[2], 'Owned CLEAR')
+            active.remove(d); frames[d] = None; rows[d] = {}
+        else:
+            need(frames[d] == e[-1], 'Scalar destination chart')
+            q = Q(e[-2])
+            if op == 'SCALE': rows[d] = {k: q*v for k, v in rows[d].items()}
+            else:
+                need(frames[e[2]] == e[-1] and e[2] != d, 'Scalar source chart')
+                if op == 'COPY':
+                    need(d in active and not rows[d], 'COPY into empty owned field')
+                    rows[d] = {}
+                else: need(op == 'ADD', 'Scalar opcode')
+                plus(rows[d], rows[e[2]], q)
+            last_write[d] = index
+        if op in ('INIT', 'CLEAR'): last_write[d] = index
+        versions[d] += 1
+    need(not active and frames[:n] == w['final'], 'Complete owned and public exits')
+    for r, row in enumerate(rows):
+        want = {r: Q(1)} if r < n else {}
+        if r-112 in w['live']: want[r-112] = Q(1)
+        need(row == want, 'Every arbitrary source/owned endpoint '+str(r))
+    return paid, selected, source_main, helpers
+
+def frame_quotients(selected, helpers):
+    receipts = []
+    for item in selected:
+        query = item['query']; basis = Basis()
+        observations = [(f'current:{r}:v{v}', row) for r, v, _, row in item['donors']]
+        observations += [(f"retained:{r['event']}:v{r['output_version']}", r['row']) for r in item['retained_same_call']]
+        for label, row in observations: basis.insert(project(row, helpers), label)
+        target = project(query['row'], helpers)
+        witness = basis.separator(target)
+        need(all(dot(project(row, helpers), witness) == 0 for _, row in observations), 'All available rows vanish')
+        response = dot(target, witness)
+        receipts.append(dict(event=query['event'], role=query['role'], rank=query['rank'],
+            old=query['old'], new=query['new'], input_version=query['input_version'],
+            last_logical_write=query['last_logical_write'], owned_epoch=query['owned_epoch'],
+            target_helper_support=len(target), current_donors=[(r,v,last) for r,v,last,_ in item['donors']],
+            prior_same_call_outputs=[p['event'] for p in item['retained_same_call']],
+            helper_observation_rank=len(basis.rows), helper_quotient_dimension=1,
+            witness=witness, target_response=response,
+            native_response='S_stage1 Phi_new ('+str(response)+' phi)',
+            status='FAIL_EVEN_WITH_ALL_FIRST_CALL_ANSWERS_AND_ARBITRARY_OUTPUT_FRAME_ACTIONS'))
+    return receipts
+
+def cached_scalar_relaxation(paid, source_row, live):
+    basis = Basis()
+    for p in paid: basis.insert(p['row'], p['event'])
+    need(len(basis.rows) == 744, 'Full first-call scalar history has rank 744')
+    # Erase every native operator and owner distinction, but keep main wrapper signs.
+    # This aliases h1 to h0 deliberately: it is a negative version-control fixture.
+    aliased = {}
+    for r, q in source_row.items():
+        if r in live:
+            plus(aliased, {r: Q(1), r+112: Q(1)}, q)
+        elif r-112 in live: plus(aliased, {r-112: Q(1)}, -q)
+        else: plus(aliased, {r: Q(1)}, q)
+    remainder, c = basis.solve(aliased)
+    need(not remainder, 'Deliberately stale helper alias produces false feasibility')
+    combined = {}
+    by_event = {p['event']: p for p in paid}
+    for e, q in c.items(): plus(combined, by_event[e]['row'], q)
+    need(combined == aliased, 'Replay the complete stale-alias combination')
+    # Genuine, main-only relaxed reconstruction, without aliasing helpers.
+    r, main_c = basis.solve({0: Q(1), 112: Q(1)})
+    need(not r, 'Earlier scalar history reconstructs a main-only target')
+    return dict(first_history_rank=744, independent_public_fields=744,
+        native_and_owner_erasure=True, stale_helper_alias_combination=c,
+        stale_alias_status='FEASIBLE_ONLY_AFTER_ILLEGAL_H1_EQUALS_H0_IDENTIFICATION',
+        main_only_X0_plus_Y0_combination=main_c,
+        main_only_status='FEASIBLE_SCALAR_RELAXATION; NATIVE_ALIGNMENT_AND_COST_REQUIRED')
+
+def caller_bindings(w, controller):
+    """Complete machine-readable substitution into the original raw caller columns.
+
+    The phase-polynomial calculation is the retained native-contact method,
+    independently implemented here from its definition in Fourier coordinates.
+    """
+    phases = {p['q']:p for p in read('inputs/DATA-PHASES.json')['recipes']}
+    n=17; axes=tuple(1<<i for i in range(n))
+    def act(columns,x):
+        out=0
+        for i,b in enumerate(columns):
+            if x>>i&1: out^=b
+        return out
+    def polynomial(terms):
+        linear=[0]*n; quadratic=[0]*n
+        for axis,value in terms:
+            support=[i for i in range(n) if axis>>i&1]
+            for i in support:
+                linear[i]=(linear[i]+value)%4
+                for j in support:
+                    if i<j and value%2: quadratic[i]^=1<<j
+        return tuple(linear),tuple(quadratic)
+    def contact(old,new,route):
+        moved=[(act(route,a),c) for a,c in old]
+        a,b=polynomial(moved),polynomial(new)
+        need(a[1]==b[1], 'Native entrance quadratic phase')
+        delta=[(y-x)%4 for x,y in zip(a[0],b[0])]
+        need(set(delta)<={0,2}, 'Native entrance is an ordinary translation')
+        shift=sum((v//2)<<i for i,v in enumerate(delta))
+        need(polynomial(moved+[(shift,2)])==b, 'Exact entrance polynomial, including phase')
+        return shift
+    live=set(w['live']); ports=dict(zip(w['live'],w['ports']))
+    right=[[Q(x) for x in row] for row in controller['boundaries']['right']]
+    routes={}; contacts={}; bindings=[{},{}]
+    for r,q in ports.items():
+        basis=phases[q]['idle_children'][0]['orthogonal_axes'][:8]
+        need(all((a&b).bit_count()%2==int(i==j) for i,a in enumerate(basis) for j,b in enumerate(basis)), 'Pinned orthonormal route basis')
+        need(all((a&q).bit_count()%2==0 for a in basis), 'Route basis fixes port')
+        swaps=[b^axes[9+i] for i,b in enumerate(basis)]
+        route=tuple(e^act(swaps,sum(((e&v).bit_count()%2)<<i for i,v in enumerate(swaps))) for e in axes)
+        need(act(route,q)==q and all(act(route,route[i])==axes[i] for i in range(n)), 'Exact route fixes q and is involutive')
+        raw_x=[(a,1) for a in axes[:9]]
+        raw_y=raw_x+[(q,-1)]
+        spectator=[(a,1) for a in axes[9:17]]
+        sx=contact(raw_x,spectator+[(q,1)],route)
+        sy=contact(raw_y,spectator,route)
+        routes[q]=route; contacts[q]=dict(entry_X_translation=sx,entry_Y_translation=sy)
+        # These two exact identities justify the expanded stage-1 input rows.
+        # E_X C_q^-1 H9 = S1 P_rho and E_Y C_q^-2 H9 = S1 C_q^-1 P_rho.
+        moved_x=[(act(route,a),c) for a,c in raw_x+[(q,-1)]]+[(sx,2)]
+        moved_y=[(act(route,a),c) for a,c in raw_x+[(q,-2)]]+[(sy,2)]
+        need(polynomial(moved_x)==polynomial(spectator), 'Stage-1 source wrapper identity')
+        need(polynomial(moved_y)==polynomial(spectator+[(q,-1)]), 'Stage-1 target wrapper identity')
+        def terms(row,owner,ops,sign=Q(1)):
+            return [dict(source=dict(kind='original_raw_caller',port=r,column=j,owner=owner),coefficient=sign*x,operations=ops)
+                    for j,x in enumerate(row) if x]
+        bindings[0][r]=terms(right[0],'I',[['C',q,-1]])
+        bindings[0][r+112]=terms(right[1],'I',[])
+        bindings[1][r]=terms(right[0],f'rho:{q}',[['C',q,-2],['P_route',q]])+terms(right[1],f'rho:{q}',[['C',q,-1],['P_route',q]])
+        bindings[1][r+112]=terms(right[0],f'rho:{q}',[['C',q,-2],['P_route',q]],Q(-1))
+    for stage in range(2):
+        for r in range(w['public']):
+            if r not in live and r-112 not in live:
+                bindings[stage][r]=[dict(source=dict(kind='independent_helper',invocation=stage,role=r,owner='I',coordinate='physical_raw'),coefficient=Q(1),
+                    operations=[['S_adjoint',stage],['Phi_adjoint',w['initial'][r]]])]
+    need(set(bindings[0])==set(bindings[1])==set(range(w['public'])), 'Every original/helper input binding')
+    h0={canonical(t['source']) for ts in bindings[0].values() for t in ts if t['source']['kind']=='independent_helper'}
+    h1={canonical(t['source']) for ts in bindings[1].values() for t in ts if t['source']['kind']=='independent_helper'}
+    need(not h0&h1 and len(h0)==len(h1)==536, 'Independent invocation-specific helper columns')
+    return dict(dimension=n,routes=routes,contacts=contacts,input_substitutions=bindings,
+                operation_order='Listed operations act from left to right, then the query prefix S_stage Phi_frame acts.',
+                spectator_axes=[[],list(range(9,17))],phase_checks=4*len(routes))
+
+def two_cache_screen(w,paid,selected,helpers,budget=6,minimum_rank=0):
+    """At most two actual prefix answers; exact helper quotient and native tariffs.
+
+    Grants all main information and all current output-frame observations free.
+    Only a strictly cheaper native bill is searched (rank sum at most six).
+    """
+    C=load('clifford_exact.py')
+    @lru_cache(None)
+    def chart(f):return C.canonical_frame(9,tuple(w['frames'][f]))
+    @lru_cache(None)
+    def tariff(a,b):
+        op=chart(a).adjoint().then(chart(b))
+        return C.rank([x for x,z,p in op.tab[9:]])
+    receipts=[]
+    def normalize(row):
+        if not row:return ()
+        q=row[min(row)]
+        return tuple((k,v/q) for k,v in sorted(row.items()))
+    for selected_item in selected:
+        target=selected_item['query'];ordinary=Basis()
+        free_frames=[f for f in range(len(w['frames'])) if tariff(f,target['new'])==0]
+        need(free_frames==[target['new']], 'Every rank-zero canonical alignment is the literal destination frame')
+        for r,ver,last,row in selected_item['donors']:
+            ordinary.insert(project(row,helpers),('current',r,ver))
+        for p in selected_item['retained_same_call']:
+            ordinary.insert(project(p['row'],helpers),('retained',p['event']))
+        t,_=ordinary.reduce(project(target['row'],helpers))
+        need(bool(t),'Nonzero fresh-helper quotient')
+        anchor=min(t);groups={};survivors=[];tested=0;single=0;eligible=0
+        for p in paid:
+            if p['event']>=target['event']:break
+            rank=tariff(p['new'],target['new'])
+            if not minimum_rank<=rank<=budget:continue
+            eligible+=1
+            u,_=ordinary.reduce(project(p['row'],helpers))
+            if not u:continue
+            alpha=u.get(anchor,Q(0))/t[anchor]
+            residual=dict(u);plus(residual,t,-alpha)
+            if not residual:
+                if alpha:
+                    single+=1
+                    survivors.append(dict(events=[p['event']],ranks=[rank],coefficients=[1/alpha]))
+                continue
+            pivot=min(residual);q=residual[pivot];key=normalize(residual)
+            group=groups.setdefault(key,{})
+            for other_rank,examples in group.items():
+                if rank+other_rank>budget:continue
+                for v in examples:
+                    tested+=1
+                    determinant=alpha*v['q']-v['alpha']*q
+                    if not determinant:continue
+                    a=v['q']/determinant;b=-q/determinant
+                    combined={};plus(combined,u,a);plus(combined,v['row'],b)
+                    need(combined==t,'Exact two-cache quotient witness')
+                    survivors.append(dict(events=[p['event'],v['event']],ranks=[rank,other_rank],coefficients=[a,b]))
+            # Two different projective target/residual ratios per tariff suffice:
+            # any third has either one of those ratios or differs from one of them.
+            examples=group.setdefault(rank,[])
+            ratio=alpha/q
+            if len(examples)<2 and all(e['alpha']/e['q']!=ratio for e in examples):
+                examples.append(dict(event=p['event'],row=u,alpha=alpha,q=q))
+        # Deduplicate and bind all survivors back to their full original columns.
+        distinct={canonical(s):s for s in survivors}
+        full=[];by_event={p['event']:p for p in paid}
+        for s in distinct.values():
+            row={}
+            for e,q in zip(s['events'],s['coefficients']):plus(row,by_event[e]['row'],q)
+            discrepancy=dict(target['row']);plus(discrepancy,row,-1)
+            helper_remainder,_=ordinary.reduce(project(discrepancy,helpers))
+            need(not helper_remainder,'Survivor full-row helper quotient')
+            full.append(dict(**s,full_row_discrepancy=discrepancy,
+                nested_deliveries=[set_span(w['frames'][target['new']])<set_span(w['frames'][by_event[e]['new']]) for e in s['events']]))
+        receipts.append(dict(target_event=target['event'],all_prefix_paid=sum(p['event']<target['event'] for p in paid),
+            eligible_native_rank_at_most_six=eligible,projective_residual_groups=len(groups),
+            tested_group_pairs=tested,single_cache_hits=single,survivors=full,
+            bounded_domain='Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.'))
+    return receipts
+
+def complete_separate_delivery_screen(w,paid,selected,helpers):
+    """Cover every subset with total alignment rank <=6, without enumerating subsets.
+
+    At most two rank-one prefix answers exist. Condition on their inclusion.
+    With one/two included only two rank>=2 answers can fit. With none, the
+    only untested size is three rank-two answers, handled by a support bound.
+    """
+    from copy import deepcopy
+    from itertools import combinations
+    C=load('clifford_exact.py')
+    @lru_cache(None)
+    def chart(f):return C.canonical_frame(9,tuple(w['frames'][f]))
+    @lru_cache(None)
+    def tariff(a,b):return C.rank([x for x,z,p in chart(a).adjoint().then(chart(b)).tab[9:]])
+    receipts=[]
+    for item in selected:
+        query=item['query'];prefix=[p for p in paid if p['event']<query['event']]
+        rank_one=[p for p in prefix if tariff(p['new'],query['new'])==1]
+        need(len(rank_one)<=2,'At most two rank-one historical answers in this source domain')
+        ordinary=Basis()
+        for r,v,last,row in item['donors']:ordinary.insert(project(row,helpers),('current',r,v))
+        for p in item['retained_same_call']:ordinary.insert(project(p['row'],helpers),('retained',p['event']))
+        rank_two=[p for p in prefix if tariff(p['new'],query['new'])==2]
+        basis=Basis();reduced=[]
+        for p in rank_two:
+            row,_=ordinary.reduce(project(p['row'],helpers))
+            reduced.append((p['event'],row));basis.insert(row,p['event'])
+        target,_=ordinary.reduce(project(query['row'],helpers))
+        remainder,target_coordinates=basis.solve(target)
+        if remainder:
+            witness=basis.separator(target)
+            three=dict(status='OUTSIDE_THE_ENTIRE_RANK_TWO_SPAN',rank_two_rows=len(rank_two),basis_rank=len(basis.rows),witness=witness,target_response=dot(target,witness))
+        else:
+            coordinates={}
+            for event,row in reduced:
+                rem,co=basis.solve(row);need(not rem,'Each rank-two row is in its basis')
+                coordinates[event]=co
+            dense={event:co for event,co in coordinates.items() if len(co)>3}
+            ignored=set().union(*(set(co) for co in dense.values())) if dense else set()
+            target_support=set(target_coordinates)-ignored
+            widths={event:len(set(co)-ignored) for event,co in coordinates.items()}
+            maximum=max(widths.values(),default=0)
+            need(len(target_support)>3*maximum,'At least four rank-two observations needed by exact coordinate support')
+            three=dict(status='THREE_RANK_TWO_ROWS_CANNOT_COVER_TARGET_SUPPORT',rank_two_rows=len(rank_two),basis_rank=len(basis.rows),
+                target_coordinates=target_coordinates,dense_rows_granted_free=dense,ignored_basis_coordinates=sorted(ignored),
+                target_uncovered_coordinates=sorted(target_support),maximum_uncovered_support_of_one_answer=maximum,
+                all_rank_two_coordinate_maps=coordinates)
+        conditioned=[]
+        for count in range(1,len(rank_one)+1):
+            for subset in combinations(rank_one,count):
+                augmented=deepcopy(item)
+                for p in subset:
+                    augmented['donors'].append((100000+p['event'],p['output_version'],p['event'],p['row']))
+                check=two_cache_screen(w,paid,[augmented],helpers,budget=6-count,minimum_rank=2)[0]
+                conditioned.append(dict(paid_rank_one_answers=[p['event'] for p in subset],remaining_alignment_rank_budget=6-count,screen=check))
+        need(all(not x['screen']['survivors'] for x in conditioned),'No conditioned two-cache survivor')
+        receipts.append(dict(target_event=query['event'],rank_one_answers=[p['event'] for p in rank_one],three_rank_two_certificate=three,
+            conditioned_screens=conditioned,status='NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET',
+            proof='For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.'))
+    return receipts
+
+def set_span(vectors):
+    out={0}
+    for b in vectors:out|={x^b for x in tuple(out)}
+    return out
+
+def native_tests(w, main_candidates, all_paid):
+    C = load('clifford_exact.py'); V = load('native_verify.py')
+    phases = {p['q']: p for p in read('inputs/DATA-PHASES.json')['recipes']}
+    n = 17; axes = tuple(1 << i for i in range(n))
+    @lru_cache(None)
+    def chart17(f): return C.canonical_frame(n, tuple(w['frames'][f]))
+    @lru_cache(None)
+    def chart9(f): return C.canonical_frame(9, tuple(w['frames'][f]))
+    old_frames = sorted({p['new'] for p in all_paid})
+    # An adjacent candidate that does not use any fresh helper columns.
+    candidate = next(p for p in main_candidates if p['event'] == 9733)
+    need(candidate['role'] == 0 and candidate['row'] == {0: Q(1)} and candidate['rank'] == 1,
+         'Named pure-main paid query')
+    q = w['frames'][w['initial'][0]][0]
+    basis = phases[q]['idle_children'][0]['orthogonal_axes'][:8]
+    swaps = [b ^ axes[9+i] for i, b in enumerate(basis)]
+    def act(columns, x):
+        out = 0
+        for i, b in enumerate(columns):
+            if x >> i & 1: out ^= b
+        return out
+    route = tuple(e ^ act(swaps, sum(((e&v).bit_count()%2) << i for i,v in enumerate(swaps))) for e in axes)
+    need(all(act(route, route[i]) == axes[i] for i in range(n)), 'Actual involutive stage-1 route')
+    P = C.perm(n, C.transpose(route, n))
+    spectator = C.fourier(n, range(9,17))
+    desired = C.cgate(n,q,-1).then(P).then(spectator).then(chart17(candidate['new']))
+    ranks = {}
+    for f in old_frames:
+        relative = chart17(f).adjoint().then(desired)
+        need(V.tableau(relative.circuit,n=17)==relative.tab,'Independent 17-axis cache-alignment tableau')
+        ranks[f] = C.rank([x for x,z,p in relative.tab[n:]])
+    need(min(ranks.values()) == 1 and 0 not in ranks.values(), 'Every granted historical cache needs a paid native alignment')
+    # The actual centre's native map is checked independently, including its phase.
+    native = []
+    for p in all_paid:
+        if not p['private']: continue
+        a,b = p['old'],p['new']
+        recipe, compiled = C.lower_child(chart9(a).adjoint().then(chart9(b)))
+        V.equal(V.recipe(recipe), V.adjoint(V.frame(tuple(w['frames'][a]),9))+V.frame(tuple(w['frames'][b]),9),
+                'Exact centre rank and global phase')
+        need(recipe['rank'] == 7, 'All eight selected centres have rank seven')
+        native.append(dict(event=p['event'],old=a,new=b,recipe=recipe))
+    first = native[0]
+    cost_tableaus=0
+    for p in native:
+        b=p['new']
+        for f in old_frames:
+            relative=chart9(f).adjoint().then(chart9(b))
+            expected=V.adjoint(V.frame(tuple(w['frames'][f]),9))+V.frame(tuple(w['frames'][b]),9)
+            need(V.tableau(expected)==relative.tab,'Independent 9-axis cache-cost tableau')
+            cost_tableaus+=1
+    actual = V.frame(tuple(w['frames'][first['old']]),9)+V.recipe(first['recipe'])
+    want = V.frame(tuple(w['frames'][first['new']]),9)
+    for column in range(512):
+        need(V.state(actual,column) == V.state(want,column), 'All 512 first-centre native columns')
+    from copy import deepcopy
+    wrong=deepcopy(first['recipe']);wrong['left']['phase']=(wrong['left']['phase']+1)%4
+    try:V.equal(V.recipe(wrong),V.recipe(first['recipe']),'Deliberately wrong native global phase')
+    except ValueError:wrong_phase=True
+    else:raise ValueError('Wrong native global phase accepted')
+    return dict(main_query=dict(event=9733,role=0,old=candidate['old'],new=candidate['new'],q=q,route=route,
+                old_output_frames=len(old_frames),alignment_ranks=ranks,minimum_alignment_rank=1,
+                best_frames=[f for f,r in ranks.items() if r==1],
+                scope='One-snapshot or typed-cache-tree delivery, granting the entire desired logical X0+Y0 row in every old paid-output frame'),
+                centres=native,first_centre_native_columns=512,wrong_global_phase_rejected=wrong_phase,
+                independent_cache_cost_tableaus=cost_tableaus,independent_17_axis_alignment_tableaus=len(old_frames))
+
+def invoice(w, controller):
+    paid = Counter(abs(len(w['frames'][e[2]])-len(w['frames'][e[3]])) for e in w['program'] if e[0]=='MOVE')
+    main = set(w['live']) | {r+112 for r in w['live']}
+    targets = Counter(len(w['frames'][w['final'][r]])-len(w['frames'][w['initial'][r]]) for r in range(w['public']) if r not in main)
+    whole_paid = Counter(); whole_targets = Counter()
+    for e in controller['physical_events']:
+        if e[0]=='LOCAL': whole_paid.update(paid); whole_targets.update(targets)
+        elif e[0]=='QUERY_SHEAR': whole_paid[1]+=104
+        elif e[0]=='IDLE': whole_paid[8*len(e[2])]+=208
+    for term in controller['diagonal']: whole_targets[1+8*sum(term[1:])]+=104
+    source = read('inputs/result5718.json')['complete_cost']
+    need({int(k):v for k,v in source['paid'].items()} == dict(whole_paid), 'Complete source paid invoice')
+    need({int(k):v for k,v in source['targets'].items()} == dict(whole_targets), 'Every independent target')
+    return dict(local_paid=paid,local_targets=targets,whole_paid=whole_paid,whole_targets=whole_targets,
+                paid_jobs=sum(whole_paid.values()),productive_fields=sum(whole_targets.values()),
+                realized_delta_paid={},realized_delta_targets={},kappa_status='NO_NEW_KAPPA',
+                unrealized_centre_allowance={'paid_delta':{7:-1},'target_delta':{}},
+                adjacent_main_best_cache_tree={'removed':{1:1},'necessary_alignment_at_least':{1:1},'best_recursive_delta':0},
+                source_reported_ordinary=source['nonrecursive'],
+                new_retention_reads_routes_restoration_cleanup='Unknown; no legal deletion emitted. Not charged at zero.')
+
+def controls(w, paid, selected, receipts):
+    out = {}
+    first = selected[0]; target = first['query']['row']; witness = receipts[0]['witness']
+    need(dot(target,witness) == Q(1,2), 'Named fresh-helper witness')
+    need(witness == {173:Q(1)}, 'Exactly call-3 helper173, independent of call-0 helper173')
+    # Genuine same-frame reuse already implemented by the source at 13610..13614.
+    e = w['program'][13610]
+    need(e == ['ADD',703,744,'1',593], 'Actual positive control read')
+    old = first['donors'][0][3]
+    new = dict(old); plus(new,target)
+    reconstructed = dict(new); plus(reconstructed,old,-1)
+    need(reconstructed == target, 'Two exact versioned values reconstruct the earlier answer')
+    out['positive_two_version_reconstruction'] = 'PASS: (703 after13610)-(703 before13610) equals answer744@13609'
+    # It is not legal to use that future output to eliminate its own producer.
+    try: need(13610 < 13609, 'A future ADD result is unavailable before the paid producer')
+    except ValueError as error: out['future_read_rejected'] = str(error)
+    else: raise ValueError('Illegal future read accepted')
+    old_basis = Basis()
+    for p in paid: old_basis.insert(project(p['row'], set(witness)), p['event'])
+    need(bool(old_basis.rows), 'Aliasing invocation helper columns would see the old helper')
+    out['stale_invocation_alias_detected'] = 'PASS: fresh H1:173 has no column in any event-0 answer; H0:173 does'
+    out['arbitrary_helper_not_zero'] = 'PASS: target changes by one half of a nonzero native field'
+    # A constructive two-paid-answer fixture checks discovery, not only rejection.
+    # Choose two independent actual native-rank-two prefix rows and synthesize
+    # their sum as an explicitly labelled fixture target at the first cut.
+    from copy import deepcopy
+    C=load('clifford_exact.py');dest=C.canonical_frame(9,tuple(w['frames'][593]))
+    main=set(w['live'])|{r+112 for r in w['live']};helpers=set(range(w['public']))-main
+    basis=Basis();choices=[];rank_cache={}
+    for p in paid:
+        if p['event']>=13609:break
+        f=p['new']
+        if f not in rank_cache:
+            relative=C.canonical_frame(9,tuple(w['frames'][f])).adjoint().then(dest)
+            rank_cache[f]=C.rank([x for x,z,ph in relative.tab[9:]])
+        if rank_cache[f]==2 and basis.insert(project(p['row'],helpers),p['event']):choices.append(p)
+        if len(choices)==2:break
+    need(len(choices)==2,'Two actual independent rank-two paid observations for positive fixture')
+    fixture=deepcopy(first);fixture['query']['row']=dict(choices[0]['row']);plus(fixture['query']['row'],choices[1]['row'])
+    positive=two_cache_screen(w,paid,[fixture],helpers)[0]
+    need(bool(positive['survivors']),'Two-cache generator discovers the constructive positive')
+    out['two_cache_discovery_positive']=dict(source_events=[p['event'] for p in choices],target='Synthetic sum of two actual prefix rows, not a changed P104 implementation',discovered=positive['survivors'][0])
+    # Incorrect source-frame tagging must not be admitted as an ordinary read.
+    try: need(first['query']['old'] == first['query']['new'], 'Nonzero native rank forbids a free frame retag')
+    except ValueError as error: out['unpaid_retag_rejected'] = str(error)
+    else: raise ValueError('Unpaid retag accepted')
+    return out
+
+def run():
+    pins = read('SOURCES.json')
+    for p in pins['runtime_inputs']:
+        need(sha256((HERE/p['path']).read_bytes()).hexdigest() == p['sha256'], 'Pinned input '+p['path'])
+    w=read('inputs/word5718.json.gz'); c=read('inputs/controller5718.json')
+    need(c['physical_events'][:4] == [['LOCAL',0,0,False],['Q_POWER',0,-2],['SIGNED_SWAP',0,1],['LOCAL',1,0,False]], 'Literal adjacent caller events 0 to 3')
+    paid, selected, main, helpers = walk(w)
+    need(len(paid)==3426 and len(selected)==8, 'Full paid-map inventory and bounded centre domain')
+    quotients = frame_quotients(selected, helpers)
+    relaxation = cached_scalar_relaxation(paid, selected[0]['query']['row'], set(w['live']))
+    native = native_tests(w,main,paid)
+    bindings = caller_bindings(w,c)
+    two_cache = two_cache_screen(w,paid,selected,helpers)
+    complete_cache = complete_separate_delivery_screen(w,paid,selected,helpers)
+    result = dict(status='PASS_VERSIONED_CROSS_CALL_SCOPED_HANDOFF_NO_NEW_KAPPA',
+        mapped_invocations=[dict(caller_event=0,stage=0,pair=0,query_maps=3426),dict(caller_event=3,stage=1,pair=0,query_maps=3426)],
+        public_columns_per_call=w['public'],owned_labels=w['owned_fields'],independent_helpers_per_call=len(helpers),
+        output_frame_quotients=quotients,scalar_relaxation=relaxation,native=native,
+        caller_binding_phase_checks=bindings['phase_checks'],
+        two_cache_favorable_screen=two_cache,
+        complete_separate_delivery_screen=complete_cache,
+        complete_invoice=invoice(w,c),controls=controls(w,paid,selected,quotients))
+    maps=dict(schema='versioned-p104-native-query-map/1',word_frames=w['frames'],word_initial=w['initial'],word_final=w['final'],live=w['live'],ports=w['ports'],
+        right_boundary=c['boundaries']['right'],queries=paid,
+        invocations=[dict(caller_event=0,stage=0,helper_namespace='H0',spectator_axes=[]),
+                     dict(caller_event=3,stage=1,helper_namespace='H1',spectator_axes=list(range(9,17)))],
+        native_input_bindings=bindings,
+        exact_native_expansion=dict(
+            convention='Composition read right to left; coefficients outside tensor powers. All maps act on the same original raw caller columns.',
+            original_columns='A_j(q,g), j=0..31, and independent normalized H0(r,g),H1(r,g); j indexes the full literal right boundary.',
+            stage0_main='For each live r with port q: coefficient c_r gives Phi_f C_q^-1 times right[0,j]; c_(r+112) gives Phi_f times right[1,j], at owner g.',
+            stage1_main='For each live r with port q: (c_r-c_(r+112))*right[0,j] times S1 Phi_f P_rho(q) C_q^-2, plus c_r*right[1,j] times S1 Phi_f P_rho(q) C_q^-1; source owner rho(q) at common owner I.',
+            helpers='For normalized h_s(r)=(S_s Phi_initial(r))^-1 H_s_raw(r), contributions are c_r Phi_f h0(r,g) at stage0 and c_r S1 Phi_f h1(r,I) at stage1. Machine input substitutions include this exact raw-input normalization. H0_raw and H1_raw are distinct independent columns even when r agrees.',
+            frames='Phi_f is the exact canonical nine-axis frame in inputs/clifford_exact.py, embedded in axes0..8; S1 is the product C_e9..C_e16.',
+            routes='rho(q) swaps the eight pinned orthonormal q-perpendicular DATA-PHASES axes with fresh axes9..16, fixes q, and is involutive.',
+            availability='Call-0 paid outputs exist before caller event3; retaining them is granted favorably. Call-3 query output e exists only after local event e. Owned INIT/CLEAR epochs and last logical writes are explicit in every query.',
+            input_output='Before MOVE: replace Phi_f by Phi_old; after MOVE use Phi_new. The logical coefficient row is unchanged by MOVE.'),
+        selected_current_frame_rows=[dict(event=x['query']['event'],donors=x['donors'],retained=x['retained_same_call']) for x in selected])
+    return encode(result), encode(maps)
+
+if __name__ == '__main__':
+    result,maps=run()
+    write='--write' in sys.argv
+    for name,x in [('results/RESULT.json',result),('results/QUERY-MAPS.json.gz',maps)]:
+        p=HERE/name
+        if write:
+            raw=(json.dumps(x,sort_keys=True,indent=2)+'\n').encode()
+            p.write_bytes(gzip.compress(raw,mtime=0) if name.endswith('.gz') else raw)
+        else: need(read(name)==x,'Full deterministic receipt '+name)
+    print(result['status'])
diff --git a/research/p104-cross-call-paid-reuse/inputs/DATA-PHASES.json b/research/p104-cross-call-paid-reuse/inputs/DATA-PHASES.json
new file mode 100644
index 0000000000..d9a4f84211
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/inputs/DATA-PHASES.json
@@ -0,0 +1 @@
+{"cover_correction":{"complete_small_cover_checks":[{"n":1,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":1},{"n":2,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":2},{"n":3,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":6},{"n":4,"nonzero_full_kernel_corrections":24,"orthogonal_matrices":48},{"n":5,"nonzero_full_kernel_corrections":600,"orthogonal_matrices":720}],"formula":"delta(g)_i = (weight(row_i(g))-1)/2 mod 2; apply X_delta(g) to EVERY final live stream in cover class g.","mandatory":true,"reason":"P_g C_all P_g^-1 = C_all X_delta(g) exactly. Orthogonality cancels all quadratic cross terms; both sides fix the uniform vector."},"global_phase_argument":"Every C_q and every linear address permutation fixes the uniform vector. Equality of the exact Z4 Fourier multiplier therefore fixes the scalar phase as 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complete recipe is unchanged except its port index; the retained parent polynomial checks establish every selected equality.","scope":"All compressed data boundary operators, idle children, terminal X_q and all-live cover correction. 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diff --git a/research/p104-cross-call-paid-reuse/inputs/NOTICE5718 b/research/p104-cross-call-paid-reuse/inputs/NOTICE5718
new file mode 100644
index 0000000000..f83cecccf5
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/inputs/NOTICE5718
@@ -0,0 +1,47 @@
+Four paid interior public-basis changes.
+Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0.
+No subagents.
+
+The following exact notice is retained from PR5660.
+
+New helper-subspace construction, proof and replay:
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+The same assistant performed the construction and author-side checks.
+
+Retained contributions:
+flyingviper: PR5605 shared relay, PR5612 delayed relay, and the PR5608/PR5613
+relay/split compositions.
+Scratch-net: PR5606 prepared-helper identities and target debit.
+Pierre-Loic Doulcet (hexapode): PR5565 commuting-contact construction, used as the exact source.
+Ali Romman: PR5602 fixed boundary relations, PR5601 three-contact frame
+change, PR5580 contact step,
+and PR5544/5545 complete P104 caller.
+Joel Pulikkan (GamingPuzzled): PR5569 priced helper handoff.
+DreamingOfClouds: PR5561 free helper endpoints and PR5568 search tools.
+thu-nmrc: PR5551 contact improvements.
+Gareth Davidson (bitplane): PR2305 source composition.
+David Arias: PR2152 coalesced P104 word.
+Cinna Davis: PR2219 private reuse and the controller lineage.
+Jacob Alan Sussman: productive mixed-target recursion.
+zxv and Cao Yifan: exact intervals, compiler and native/pricing lineage.
+GamingPuzzled: PR5472 successor handoff, a predecessor to the split concept.
+The retained source notices preserve earlier geometry, BIT and analytic credit.
+
+The elementary rank-one basis identity is not claimed as a new general
+linear-algebra theorem. The contribution is its explicit independent-target
+use on this P104 word and the complete conditional kappa improvement.
+
+New contribution: Ali Romman, with substantial OpenAI Codex assistance.
+The four-gate source-version rewrite composes zxv PR5643 and Ali Romman PR5644.
+The A2 method retains Chris R/Clear1nsight PR3998 and ToikY PR4019 credit.
+The compensated triangle method retains Cinna Davis PR4692 credit.
+No subagents were used for this contribution. Apache-2.0.
+
+Concurrent PR5650 by zxv publishes the PR5643/PR5644 composition. PR5651 by
+Ali Romman corroborates it. This package credits that released composition
+and contributes the further four-gate placement.
+
+New work: distributed scalar updates on an unobserved public field interval, the exact witness and complete conditional price. Prepared for Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0. The source archive retains all predecessor notices.
+
+Two-boundary main-assisted helper split and early cleanup: flyingviper with substantial
+OpenAI Codex assistance, including independent AI review agents. Apache-2.0.
diff --git a/research/p104-cross-call-paid-reuse/inputs/clifford_exact.py b/research/p104-cross-call-paid-reuse/inputs/clifford_exact.py
new file mode 100644
index 0000000000..166841a037
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/inputs/clifford_exact.py
@@ -0,0 +1,149 @@
+"""Small exact Gaussian-dyadic Clifford compiler, independent of gx internals.
+Paulis use i^p X^x Z^z. State vectors use integer Gaussian numerators / 2^d.
+The tableau is supplemented by a literal circuit, so global phases are kept.
+"""
+from dataclasses import dataclass
+
+def parity(x):return x.bit_count()&1
+def rank(rows):
+ p={}
+ for x in rows:
+  while x:
+   b=x.bit_length()-1
+   if b not in p:p[b]=x;break
+   x^=p[b]
+ return len(p)
+def transpose(rows,n):return tuple(sum(((rows[j]>>i)&1)<<j for j in range(n)) for i in range(n))
+def mv(rows,x):return sum(parity(r&x)<<i for i,r in enumerate(rows))
+def inv(rows,n):
+ a=[r|(1<<(n+i)) for i,r in enumerate(rows)]
+ for j in range(n):
+  k=next(i for i in range(j,n) if a[i]>>j&1);a[j],a[k]=a[k],a[j]
+  for i in range(n):
+   if i!=j and a[i]>>j&1:a[i]^=a[j]
+ assert [r&((1<<n)-1) for r in a]==[1<<i for i in range(n)]
+ return tuple(r>>n for r in a)
+def basis_extend(cols,n):
+ out=[]
+ for x in list(cols)+[1<<i for i in range(n)]:
+  if rank(out+[x])>len(out):out.append(x)
+ assert len(out)==n
+ return tuple(out)
+def nullspace(rows,n):
+ a=list(rows);p=[];k=0
+ for j in range(n):
+  t=next((i for i in range(k,len(a)) if a[i]>>j&1),None)
+  if t is None:continue
+  a[t],a[k]=a[k],a[t]
+  for i in range(len(a)):
+   if i!=k and a[i]>>j&1:a[i]^=a[k]
+  p.append(j);k+=1
+ return tuple((1<<j)|sum(((a[i]>>j)&1)<<v for i,v in enumerate(p)) for j in range(n) if j not in p)
+def pmul(a,b):
+ x,z,p=a;y,w,q=b
+ return x^y,z^w,(p+q+2*parity(z&y))%4
+def identity(n):return tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,0) for i in range(n)])
+def apply(T,p):
+ n=len(T)//2;x,z,ph=p;out=(0,0,ph)
+ for i in range(n):
+  if x>>i&1:out=pmul(out,T[i])
+ for i in range(n):
+  if z>>i&1:out=pmul(out,T[n+i])
+ return out
+def compose(A,B):return tuple(apply(A,p) for p in B)
+def inverse(T):
+ n=len(T)//2;cols=[x|(z<<n) for x,z,p in T];rows=transpose(cols,2*n);cols_inv=transpose(inv(rows,2*n),2*n);out=[]
+ for j,w in enumerate(cols_inv):
+  x=w&((1<<n)-1);z=w>>n;u,v,p=apply(T,(x,z,0));assert (u,v)==identity(n)[j][:2];out.append((x,z,(-p)%4))
+ return tuple(out)
+def qvalue(Q,alpha,x,n):
+ return (sum(alpha[i] for i in range(n) if x>>i&1)+2*sum(parity((Q[i]>>(i+1))&(x>>(i+1))) for i in range(n) if x>>i&1))%4
+def mul_i(a,p):
+ x,y=a;p%=4
+ return [(x,y),(-y,x),(-x,-y),(y,-x)][p]
+@dataclass
+class Op:
+ n:int
+ tab:tuple
+ circuit:list
+ def then(self,b):
+  assert self.n==b.n
+  return Op(self.n,compose(b.tab,self.tab),self.circuit+b.circuit)
+ def adjoint(self):
+  circ=[]
+  for e in reversed(self.circuit):
+   if e[0]=='P':circ.append(('P',inv(e[1],self.n)))
+   elif e[0]=='D':circ.append(('D',e[1],tuple((-x)%4 for x in e[2])))
+   elif e[0]=='C':circ.append(('C',e[1],-e[2]))
+   elif e[0]=='phase':circ.append(('phase',(-e[1])%4))
+   else:circ.append(e)
+  return Op(self.n,inverse(self.tab),circ)
+ def zero_state(self,basis=0):
+  N=1<<self.n;vec=[(0,0)]*N;vec[basis]=(1,0);den=0
+  for e in self.circuit:
+   if e[0]=='P':
+    v=[(0,0)]*N
+    for x,a in enumerate(vec):v[mv(e[1],x)]=a
+    vec=v
+   elif e[0]=='X':vec=[vec[x^e[1]] for x in range(N)]
+   elif e[0]=='D':vec=[mul_i(a,qvalue(e[1],e[2],x,self.n)) for x,a in enumerate(vec)]
+   elif e[0]=='phase':vec=[mul_i(a,e[1]) for a in vec]
+   elif e[0]=='C':
+    q,sgn=e[1:];v=[]
+    for x,(a,b) in enumerate(vec):
+     c,d=vec[x^q]
+     # C=(1+i)I/2+(1-i)X/2; inverse conjugates both coefficients.
+     v.append((a-sgn*b+c+sgn*d,b+sgn*a+d-sgn*c))
+    vec=v;den+=1
+   else:raise AssertionError(e)
+   while den and all((a&1)==(b&1)==0 for a,b in vec):vec=[(a//2,b//2) for a,b in vec];den-=1
+  return vec,den
+def ident(n):return Op(n,identity(n),[])
+def perm(n,G):
+ cols=transpose(G,n);gi=inv(G,n)
+ return Op(n,tuple([(a,0,0) for a in cols]+[(0,a,0) for a in gi]),[('P',tuple(G))])
+def diag(n,Q,alpha=None):
+ Q=tuple(Q);assert transpose(Q,n)==Q
+ alpha=tuple(alpha) if alpha is not None else tuple((Q[i]>>i)&1 for i in range(n))
+ assert all(alpha[i]%2==(Q[i]>>i)&1 for i in range(n))
+ return Op(n,tuple([(1<<i,Q[i],alpha[i]) for i in range(n)]+[(0,1<<i,0) for i in range(n)]),[('D',Q,alpha)])
+def cgate(n,q,sgn=1):
+ return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(q if q>>i&1 else 0,1<<i,(3*sgn)%4 if q>>i&1 else 0) for i in range(n)]),[('C',q,sgn)])
+def xgate(n,b):return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,2*((b>>i)&1)) for i in range(n)]),[('X',b)])
+def fourier(n,axes=None):
+ out=ident(n)
+ for i in range(n) if axes is None else axes:out=out.then(cgate(n,1<<i))
+ return out
+def hadamard(n,axes):
+ axes=list(axes);Q=tuple(1<<i if i in axes else 0 for i in range(n));S=diag(n,Q)
+ return S.then(fourier(n,axes)).then(S)
+def canonical_frame(n,U):
+ r=len(U)
+ if r==0:return ident(n)
+ if r==n:return fourier(n)
+ G=transpose(basis_extend(U,n),n);gi=inv(G,n);Q=tuple((1<<i)^sum(parity(gi_i&gi_j)<<j for j,gi_j in enumerate(transpose(gi,n))) for i,gi_i in enumerate(transpose(gi,n)))
+ K=perm(n,G).then(diag(n,Q))
+ return K.adjoint().then(fourier(n,range(r))).then(K)
+def monomial(op):
+ n=op.n;T=op.tab;assert all(x==0 and p%2==0 for x,z,p in T[n:]);G=transpose([x for x,z,p in T[:n]],n);b=mv(G,sum((p//2)<<i for i,(x,z,p) in enumerate(T[n:])))
+ gt=transpose(G,n);cols=[mv(gt,z) for x,z,p in T[:n]];Q=transpose(cols,n);alpha=[(p-2*parity(b&z))%4 for x,z,p in T[:n]]
+ out=diag(n,Q,alpha).then(perm(n,G)).then(xgate(n,b));assert out.tab==T
+ vec,den=op.zero_state();assert den==0 and all(a==(0,0) for i,a in enumerate(vec) if i!=b);phase=[(1,0),(0,1),(-1,0),(0,-1)].index(vec[b]);out=out.then(Op(n,identity(n),[('phase',phase)]));assert out.zero_state()==op.zero_state()
+ return out,dict(matrix_rows=G,offset=b,quadratic_rows=Q,linear_Z4=alpha,phase=phase)
+def lower_child(op):
+ n=op.n;Bcols=[x for x,z,p in op.tab[n:]];r=rank(Bcols)
+ if not r:
+  mon,rec=monomial(op);return dict(rank=0,left=rec,right=None),mon
+ G=transpose(basis_extend(Bcols,n),n);A=inv(G,n);PA=perm(n,A);part=op.then(PA);Bcols=[x for x,z,p in part.tab[n:]];Dcols=[z for x,z,p in part.tab[n:]]
+ picks=[];selected=[]
+ for j,x in enumerate(Bcols):
+  if rank(selected+[x])>len(selected):picks.append(j);selected.append(x)
+ assert len(picks)==r and all(x>>r==0 for x in Bcols)
+ Br=transpose(selected,r);Bi=inv(Br,r);Dr=transpose([Dcols[j]&((1<<r)-1) for j in picks],r)
+ Qsmall=tuple(sum(parity(row&col)<<j for j,col in enumerate(transpose(Bi,r))) for row in Dr);Q=Qsmall+(0,)*(n-r);DQ=diag(n,Q)
+ V=part.then(DQ).then(hadamard(n,range(r)));assert all(x==0 for x,z,p in V.tab[n:]);Vmon,Vrec=monomial(V)
+ S=diag(n,tuple(1<<i if i<r else 0 for i in range(n)))
+ left=S.adjoint().then(DQ.adjoint()).then(PA.adjoint());right=Vmon.then(S.adjoint()).then(xgate(n,(1<<r)-1))
+ lmon,lrec=monomial(left);rmon,rrec=monomial(right);compiled=rmon.then(fourier(n,range(r))).then(lmon)
+ assert compiled.tab==op.tab and compiled.zero_state()==op.zero_state()
+ return dict(rank=r,left=lrec,right=rrec),compiled
diff --git a/research/p104-cross-call-paid-reuse/inputs/controller5718.json b/research/p104-cross-call-paid-reuse/inputs/controller5718.json
new file mode 100644
index 0000000000..cc539db709
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/inputs/controller5718.json
@@ -0,0 +1,5480 @@
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+        1,
+        3,
+        5,
+        7
+      ]
+    ],
+    [
+      "ADD",
+      6,
+      8,
+      1
+    ],
+    [
+      "ADD",
+      9,
+      7,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      8,
+      3,
+      false
+    ],
+    [
+      "ADD",
+      6,
+      8,
+      -1
+    ],
+    [
+      "ADD",
+      9,
+      7,
+      1
+    ],
+    [
+      "Q_POWER",
+      8,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      8,
+      9
+    ],
+    [
+      "LOCAL",
+      9,
+      4,
+      false
+    ],
+    [
+      "Q_POWER",
+      9,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      9,
+      8
+    ],
+    [
+      "IDLE",
+      5,
+      [
+        0,
+        1,
+        3,
+        5,
+        7,
+        9
+      ]
+    ],
+    [
+      "ADD",
+      8,
+      10,
+      1
+    ],
+    [
+      "ADD",
+      11,
+      9,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      10,
+      4,
+      false
+    ],
+    [
+      "ADD",
+      8,
+      10,
+      -1
+    ],
+    [
+      "ADD",
+      11,
+      9,
+      1
+    ],
+    [
+      "Q_POWER",
+      10,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      10,
+      11
+    ],
+    [
+      "LOCAL",
+      11,
+      5,
+      false
+    ],
+    [
+      "Q_POWER",
+      11,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      11,
+      10
+    ],
+    [
+      "IDLE",
+      6,
+      [
+        0,
+        1,
+        3,
+        5,
+        7,
+        9,
+        11
+      ]
+    ],
+    [
+      "ADD",
+      10,
+      12,
+      1
+    ],
+    [
+      "ADD",
+      13,
+      11,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      12,
+      5,
+      false
+    ],
+    [
+      "ADD",
+      10,
+      12,
+      -1
+    ],
+    [
+      "ADD",
+      13,
+      11,
+      1
+    ],
+    [
+      "Q_POWER",
+      12,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      12,
+      13
+    ],
+    [
+      "LOCAL",
+      13,
+      6,
+      false
+    ],
+    [
+      "Q_POWER",
+      13,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      13,
+      12
+    ],
+    [
+      "IDLE",
+      7,
+      [
+        0,
+        1,
+        3,
+        5,
+        7,
+        9,
+        11,
+        13
+      ]
+    ],
+    [
+      "ADD",
+      12,
+      14,
+      1
+    ],
+    [
+      "ADD",
+      15,
+      13,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      14,
+      6,
+      false
+    ],
+    [
+      "ADD",
+      12,
+      14,
+      -1
+    ],
+    [
+      "ADD",
+      15,
+      13,
+      1
+    ],
+    [
+      "Q_POWER",
+      14,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      14,
+      15
+    ],
+    [
+      "LOCAL",
+      15,
+      7,
+      false
+    ],
+    [
+      "Q_POWER",
+      15,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      15,
+      14
+    ],
+    [
+      "ADD",
+      14,
+      16,
+      1
+    ],
+    [
+      "ADD",
+      17,
+      15,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      16,
+      7,
+      false
+    ],
+    [
+      "ADD",
+      14,
+      16,
+      -1
+    ],
+    [
+      "ADD",
+      17,
+      15,
+      1
+    ],
+    [
+      "Q_POWER",
+      16,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      16,
+      17
+    ],
+    [
+      "LOCAL",
+      17,
+      8,
+      false
+    ],
+    [
+      "Q_POWER",
+      17,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      17,
+      16
+    ],
+    [
+      "ADD",
+      16,
+      18,
+      1
+    ],
+    [
+      "ADD",
+      19,
+      17,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      18,
+      8,
+      false
+    ],
+    [
+      "ADD",
+      16,
+      18,
+      -1
+    ],
+    [
+      "ADD",
+      19,
+      17,
+      1
+    ],
+    [
+      "Q_POWER",
+      18,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      18,
+      19
+    ],
+    [
+      "LOCAL",
+      19,
+      9,
+      false
+    ],
+    [
+      "Q_POWER",
+      19,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      19,
+      18
+    ],
+    [
+      "ADD",
+      18,
+      20,
+      1
+    ],
+    [
+      "ADD",
+      21,
+      19,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      20,
+      9,
+      false
+    ],
+    [
+      "ADD",
+      18,
+      20,
+      -1
+    ],
+    [
+      "ADD",
+      21,
+      19,
+      1
+    ],
+    [
+      "Q_POWER",
+      20,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      20,
+      21
+    ],
+    [
+      "LOCAL",
+      21,
+      10,
+      false
+    ],
+    [
+      "Q_POWER",
+      21,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      21,
+      20
+    ],
+    [
+      "ADD",
+      20,
+      22,
+      1
+    ],
+    [
+      "ADD",
+      23,
+      21,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      22,
+      10,
+      false
+    ],
+    [
+      "ADD",
+      20,
+      22,
+      -1
+    ],
+    [
+      "ADD",
+      23,
+      21,
+      1
+    ],
+    [
+      "Q_POWER",
+      22,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      22,
+      23
+    ],
+    [
+      "LOCAL",
+      23,
+      11,
+      false
+    ],
+    [
+      "Q_POWER",
+      23,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      23,
+      22
+    ],
+    [
+      "ADD",
+      22,
+      24,
+      1
+    ],
+    [
+      "ADD",
+      25,
+      23,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      24,
+      11,
+      false
+    ],
+    [
+      "ADD",
+      22,
+      24,
+      -1
+    ],
+    [
+      "ADD",
+      25,
+      23,
+      1
+    ],
+    [
+      "Q_POWER",
+      24,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      24,
+      25
+    ],
+    [
+      "LOCAL",
+      25,
+      12,
+      false
+    ],
+    [
+      "Q_POWER",
+      25,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      25,
+      24
+    ],
+    [
+      "ADD",
+      24,
+      26,
+      1
+    ],
+    [
+      "ADD",
+      27,
+      25,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      26,
+      12,
+      false
+    ],
+    [
+      "ADD",
+      24,
+      26,
+      -1
+    ],
+    [
+      "ADD",
+      27,
+      25,
+      1
+    ],
+    [
+      "Q_POWER",
+      26,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      26,
+      27
+    ],
+    [
+      "LOCAL",
+      27,
+      13,
+      false
+    ],
+    [
+      "Q_POWER",
+      27,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      27,
+      26
+    ],
+    [
+      "ADD",
+      26,
+      28,
+      1
+    ],
+    [
+      "ADD",
+      29,
+      27,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      28,
+      13,
+      false
+    ],
+    [
+      "ADD",
+      26,
+      28,
+      -1
+    ],
+    [
+      "ADD",
+      29,
+      27,
+      1
+    ],
+    [
+      "Q_POWER",
+      28,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      28,
+      29
+    ],
+    [
+      "LOCAL",
+      29,
+      14,
+      false
+    ],
+    [
+      "Q_POWER",
+      29,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      29,
+      28
+    ],
+    [
+      "ADD",
+      28,
+      30,
+      1
+    ],
+    [
+      "ADD",
+      31,
+      29,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      30,
+      14,
+      false
+    ],
+    [
+      "ADD",
+      28,
+      30,
+      -1
+    ],
+    [
+      "ADD",
+      31,
+      29,
+      1
+    ],
+    [
+      "Q_POWER",
+      30,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      30,
+      31
+    ],
+    [
+      "LOCAL",
+      31,
+      15,
+      false
+    ],
+    [
+      "Q_POWER",
+      31,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      31,
+      30
+    ],
+    [
+      "LOCAL",
+      32,
+      15,
+      false
+    ],
+    [
+      "IDLE",
+      7,
+      [
+        17,
+        19,
+        21,
+        23,
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      8,
+      [
+        19,
+        21,
+        23,
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      9,
+      [
+        21,
+        23,
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      10,
+      [
+        23,
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      11,
+      [
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      12,
+      [
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      13,
+      [
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      14,
+      [
+        31,
+        32
+      ]
+    ],
+    [
+      "ADD",
+      30,
+      28,
+      1
+    ],
+    [
+      "ADD",
+      29,
+      31,
+      -1
+    ],
+    [
+      "ADD",
+      28,
+      26,
+      1
+    ],
+    [
+      "ADD",
+      27,
+      29,
+      -1
+    ],
+    [
+      "ADD",
+      26,
+      24,
+      1
+    ],
+    [
+      "ADD",
+      25,
+      27,
+      -1
+    ],
+    [
+      "ADD",
+      24,
+      22,
+      1
+    ],
+    [
+      "ADD",
+      23,
+      25,
+      -1
+    ],
+    [
+      "ADD",
+      22,
+      20,
+      1
+    ],
+    [
+      "ADD",
+      21,
+      23,
+      -1
+    ],
+    [
+      "ADD",
+      20,
+      18,
+      1
+    ],
+    [
+      "ADD",
+      19,
+      21,
+      -1
+    ],
+    [
+      "ADD",
+      18,
+      16,
+      1
+    ],
+    [
+      "ADD",
+      17,
+      19,
+      -1
+    ],
+    [
+      "ADD",
+      16,
+      14,
+      1
+    ],
+    [
+      "ADD",
+      15,
+      17,
+      -1
+    ],
+    [
+      "ADD",
+      14,
+      12,
+      1
+    ],
+    [
+      "ADD",
+      13,
+      15,
+      -1
+    ],
+    [
+      "ADD",
+      12,
+      10,
+      1
+    ],
+    [
+      "ADD",
+      11,
+      13,
+      -1
+    ],
+    [
+      "ADD",
+      10,
+      8,
+      1
+    ],
+    [
+      "ADD",
+      9,
+      11,
+      -1
+    ],
+    [
+      "ADD",
+      8,
+      6,
+      1
+    ],
+    [
+      "ADD",
+      7,
+      9,
+      -1
+    ],
+    [
+      "ADD",
+      6,
+      4,
+      1
+    ],
+    [
+      "ADD",
+      5,
+      7,
+      -1
+    ],
+    [
+      "ADD",
+      4,
+      2,
+      1
+    ],
+    [
+      "ADD",
+      3,
+      5,
+      -1
+    ],
+    [
+      "ADD",
+      2,
+      0,
+      1
+    ],
+    [
+      "ADD",
+      1,
+      3,
+      -1
+    ],
+    [
+      "SIGNED_SWAP",
+      0,
+      1
+    ],
+    [
+      "SIGNED_SWAP",
+      2,
+      3
+    ],
+    [
+      "SIGNED_SWAP",
+      4,
+      5
+    ],
+    [
+      "SIGNED_SWAP",
+      6,
+      7
+    ],
+    [
+      "SIGNED_SWAP",
+      8,
+      9
+    ],
+    [
+      "SIGNED_SWAP",
+      10,
+      11
+    ],
+    [
+      "SIGNED_SWAP",
+      12,
+      13
+    ],
+    [
+      "SIGNED_SWAP",
+      14,
+      15
+    ],
+    [
+      "SIGNED_SWAP",
+      16,
+      17
+    ],
+    [
+      "SIGNED_SWAP",
+      18,
+      19
+    ],
+    [
+      "SIGNED_SWAP",
+      20,
+      21
+    ],
+    [
+      "SIGNED_SWAP",
+      22,
+      23
+    ],
+    [
+      "SIGNED_SWAP",
+      24,
+      25
+    ],
+    [
+      "SIGNED_SWAP",
+      26,
+      27
+    ],
+    [
+      "SIGNED_SWAP",
+      28,
+      29
+    ],
+    [
+      "SIGNED_SWAP",
+      30,
+      31
+    ]
+  ],
+  "prefix_pairs": 7,
+  "wrapper_sites": [
+    1,
+    3,
+    5,
+    7,
+    9,
+    11,
+    13,
+    15,
+    17,
+    19,
+    21,
+    23,
+    25,
+    27,
+    29,
+    31
+  ]
+}
diff --git a/research/p104-cross-call-paid-reuse/inputs/native_verify.py b/research/p104-cross-call-paid-reuse/inputs/native_verify.py
new file mode 100644
index 0000000000..2c209dfa9c
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/inputs/native_verify.py
@@ -0,0 +1,277 @@
+"""Check native operators with exact Gaussian dyadic arithmetic.
+
+The interpreter is adapted from the independent campaign native review.
+It does not import the source-code producer or the Clifford compiler.
+Research used OpenAI Codex under the user's direction.
+"""
+
+from functools import lru_cache
+
+
+def require(condition, label):
+    if not condition:
+        raise ValueError(label)
+
+
+def parity(value):
+    return value.bit_count() % 2
+
+
+def mul(matrix, vector):
+    return sum(parity(row & vector) << i for i, row in enumerate(matrix))
+
+
+def transpose(matrix):
+    return tuple(sum(((row >> j) & 1) << i for i, row in enumerate(matrix)) for j in range(len(matrix)))
+
+
+def inverse(matrix):
+    n = len(matrix)
+    rows = [row | (1 << (n+i)) for i, row in enumerate(matrix)]
+    for j in range(n):
+        pivot = next((i for i in range(j, n) if rows[i] >> j & 1), None)
+        require(pivot is not None, 'singular binary chart')
+        rows[j], rows[pivot] = rows[pivot], rows[j]
+        for i in range(n):
+            if i != j and rows[i] >> j & 1:
+                rows[i] ^= rows[j]
+    require([row & ((1 << n)-1) for row in rows] == [1 << i for i in range(n)], 'binary inverse')
+    return tuple(row >> n for row in rows)
+
+
+def extend(vectors, n):
+    columns, pivots = [], {}
+    for value in tuple(vectors) + tuple(1 << i for i in range(n)):
+        reduced = value
+        while reduced:
+            j = reduced.bit_length()-1
+            if j not in pivots:
+                pivots[j] = reduced
+                columns.append(value)
+                break
+            reduced ^= pivots[j]
+    require(len(columns) == n, 'frame basis extension')
+    return tuple(columns)
+
+
+def adjoint(circuit):
+    out = []
+    for gate in reversed(circuit):
+        if gate[0] == 'P':
+            out.append(('P', inverse(gate[1])))
+        elif gate[0] == 'D':
+            out.append(('D', gate[1], tuple(-a % 4 for a in gate[2])))
+        elif gate[0] == 'C':
+            out.append(('C', gate[1], -gate[2]))
+        elif gate[0] == 'phase':
+            out.append(('phase', -gate[1] % 4))
+        else:
+            require(gate[0] == 'X', 'adjoint gate type')
+            out.append(gate)
+    return tuple(out)
+
+
+def fourier(rank):
+    return tuple(('C', 1 << i, 1) for i in range(rank))
+
+
+def frame(vectors, n):
+    rank = len(vectors)
+    if rank == 0:
+        return ()
+    if rank == n:
+        return fourier(n)
+    matrix = transpose(extend(vectors, n))
+    columns = transpose(inverse(matrix))
+    quadratic = tuple((1 << i) ^ sum(parity(a & b) << j for j, b in enumerate(columns)) for i, a in enumerate(columns))
+    diagonal = tuple((row >> i) & 1 for i, row in enumerate(quadratic))
+    chart = (('P', matrix), ('D', quadratic, diagonal))
+    return adjoint(chart) + fourier(rank) + chart
+
+
+def monomial(record):
+    matrix = tuple(record['matrix_rows'])
+    quadratic = tuple(record['quadratic_rows'])
+    linear = tuple(record['linear_Z4'])
+    require(transpose(quadratic) == quadratic, 'quadratic symmetry')
+    require(len(matrix) == len(quadratic) == len(linear) == 9, 'monomial dimension')
+    require(all(a % 2 == (quadratic[i] >> i) & 1 for i, a in enumerate(linear)), 'diagonal parity')
+    inverse(matrix)
+    return (('D', quadratic, linear), ('P', matrix), ('X', record['offset']), ('phase', record['phase']))
+
+
+def recipe(record):
+    rank = record['rank']
+    require(isinstance(rank, int) and 0 <= rank <= 9, 'child rank')
+    if rank == 0:
+        require(record['right'] is None, 'rank zero right factor')
+        return monomial(record['left'])
+    return monomial(record['right']) + fourier(rank) + monomial(record['left'])
+
+
+def tableau(circuit, n=9):
+    rows = [(1 << i, 0, 0) for i in range(n)] + [(0, 1 << i, 0) for i in range(n)]
+    for gate in circuit:
+        tag = gate[0]
+        out = []
+        inverse_transpose = transpose(inverse(gate[1])) if tag == 'P' else None
+        for x, z, phase in rows:
+            if tag == 'P':
+                x, z = mul(gate[1], x), mul(inverse_transpose, z)
+            elif tag == 'X':
+                phase += 2 * parity(z & gate[1])
+            elif tag == 'C':
+                if parity(z & gate[1]):
+                    x ^= gate[1]
+                    phase += 3 * gate[2]
+            elif tag == 'D':
+                quadratic, linear = gate[1:]
+                phase += sum(linear[i] for i in range(n) if x >> i & 1)
+                phase += 2 * sum(parity((quadratic[i] >> (i+1)) & (x >> (i+1))) for i in range(n) if x >> i & 1)
+                z ^= mul(quadratic, x)
+            else:
+                require(tag == 'phase', 'tableau gate type')
+            out.append((x, z, phase % 4))
+        rows = out
+    return tuple(rows)
+
+
+@lru_cache(maxsize=512)
+def action(gate):
+    if gate[0] == 'P':
+        return tuple(mul(gate[1], x) for x in range(512))
+    require(gate[0] == 'D', 'state action type')
+    quadratic, linear = gate[1:]
+    return tuple((sum(linear[i] for i in range(9) if x >> i & 1)
+                  + 2 * sum(parity((quadratic[i] >> (i+1)) & (x >> (i+1))) for i in range(9) if x >> i & 1)) % 4 for x in range(512))
+
+
+def rotate(pair, power):
+    a, b = pair
+    return ((a, b), (-b, a), (-a, -b), (b, -a))[power % 4]
+
+
+def state(circuit, basis=0):
+    value, denominator = {basis: (1, 0)}, 0
+    for gate in circuit:
+        tag = gate[0]
+        result = {}
+        if tag == "C":
+            direction, sign = gate[1:]
+            for address, (a, b) in value.items():
+                for target, pair in ((address, (a - sign*b, b + sign*a)),
+                                     (address ^ direction, (a + sign*b, b - sign*a))):
+                    old = result.get(target, (0, 0))
+                    result[target] = (old[0] + pair[0], old[1] + pair[1])
+            denominator += 1
+        else:
+            for address, pair in value.items():
+                if tag == "P":
+                    target = mul(gate[1], address)
+                elif tag == "X":
+                    target = address ^ gate[1]
+                else:
+                    target = address
+                    if tag == "D":
+                        pair = rotate(pair, action(gate)[address])
+                    else:
+                        require(tag == "phase", "Unknown Gaussian state gate")
+                        pair = rotate(pair, gate[1])
+                require(target not in result, "A monomial address map is not injective")
+                result[target] = pair
+        value = {address: pair for address, pair in result.items() if pair != (0, 0)}
+    while denominator and all((a | b) % 2 == 0 for a, b in value.values()):
+        denominator -= 1
+        value = {address: (a//2, b//2) for address, (a, b) in value.items()}
+    return value, denominator
+
+
+def equal(actual, expected, label):
+    require(tableau(actual) == tableau(expected), label + ": signed Pauli action")
+    require(state(actual) == state(expected), label + ": exact global phase")
+
+
+def verify(data, certificate):
+    word = data["word"]
+    original, clean = data["programs"], data["clean_programs"]
+    source_recipes = data["recipes"]
+    residuals = data["residuals"]["recipes"]
+    native_tags = {"climb", "adapter", "residual_adapter"}
+    for orientation in ("forward", "backward"):
+        native = [event for event in clean[orientation] if event[0] in native_tags]
+        require(native == [event for event in original[orientation] if event[0] in native_tags],
+                "A retained native event changed")
+        require(len(native) == 5056, "The retained native event count changed")
+    frames = [frame(tuple(vectors), 9) for vectors in word["frames"]]
+    used = {(event[3], event[4]) for program in clean.values()
+            for event in program if event[0] == "climb"}
+    children = {(record["old"], record["new"]): record for record in source_recipes["children"]}
+    require(len(children) == len(source_recipes["children"]) and used <= set(children),
+            "The retained child recipe coverage is incomplete")
+    for old, new in sorted(used):
+        record = children[old, new]
+        require(record["rank"] == len(word["frames"][new]) - len(word["frames"][old]),
+                "A retained child has the wrong rank")
+        equal(recipe(record), adjoint(frames[old]) + frames[new], "Retained child")
+    endpoints = {(record["role"], record["port"]): record
+                 for record in source_recipes["endpoint_adapters"]}
+    require(len(endpoints) == len(source_recipes["endpoint_adapters"])
+            and set(endpoints) == {(role, p) for role in ("source", "target") for p in range(112)},
+            "The endpoint adapter coverage is incomplete")
+    for (role, port), record in endpoints.items():
+        direction = word["ports"][port]
+        conventional = (("C", direction, 1),) if role == "source" else (
+            ("C", direction, -1),) + fourier(9)
+        index = word["start"][port] if role == "source" else word["final"][112 + port]
+        require(record["frame"] == index and record["rank"] == 0, "An endpoint frame changed")
+        equal(recipe(record), adjoint(conventional) + frames[index], "Endpoint adapter")
+    used_residuals = set()
+    for program in clean.values():
+        for event in program:
+            if event[0] != "residual_adapter":
+                continue
+            _, position, register, entry, *tail = event
+            terminal = tail[0] if tail else 1
+            key = str(entry) if terminal == 1 else str(entry) + ":" + str(terminal)
+            require(key in residuals and position in ("pre", "post"), "A residual binding changed")
+            require(entry == word["start"][register] and terminal == word["final"][register],
+                    "A residual endpoint changed")
+            used_residuals.add(key)
+    for key in sorted(used_residuals):
+        record = residuals[key]
+        equal(recipe(record), adjoint(frames[record["frame"]]) + frames[record.get("terminal", 1)],
+              "Residual adapter")
+    records = certificate["input_recipes"]
+    ports = certificate["compensation_ports"]
+    require([record["port"] for record in records] == ports and len(records) == 8,
+            "The eight input recipes changed")
+    columns = 0
+    for item in records:
+        port, label = item["port"], item["label"]
+        require(label == word["ports"][port] and item["canonical_start_frame_id"] == word["start"][port],
+                "An input recipe has the wrong supplier label")
+        record = item["recipe"]
+        require(record["rank"] == 1, "An input copy has the wrong child rank")
+        for side in ("left", "right"):
+            factor = record[side]
+            require(factor["quadratic_rows"] == [0] * 9 and factor["linear_Z4"] == [0] * 9
+                    and factor["offset"] == factor["phase"] == 0,
+                    "An input factor is not an address permutation")
+        actual, expected = recipe(record), (("C", label, 1),)
+        equal(actual, expected, "Positive inherited input line")
+        for basis in range(512):
+            require(state(actual, basis) == state(expected, basis), "A complete input Gaussian column differs")
+            columns += 1
+        negative = monomial(record["right"]) + (("C", 1, -1),) + monomial(record["left"])
+        require(tableau(negative) != tableau(expected) or state(negative) != state(expected),
+                "An inverse input child was not detected")
+        phased = actual + (("phase", 1),)
+        require(tableau(phased) == tableau(actual) and state(phased) != state(expected),
+                "An input phase-only error was not detected")
+    return {"status": "PASS_EXACT_INPUT_NATIVE_FRAMES", "retained_native_events_per_orientation": 5056,
+            "retained_child_recipes": len(used), "endpoint_adapter_recipes": len(endpoints),
+            "residual_adapter_recipes": len(used_residuals), "new_input_recipes": len(records),
+            "new_exact_gaussian_columns": columns, "inverse_input_child_controls": len(records),
+            "phase_only_controls": len(records), "address_permutation_factors": 2 * len(records),
+            "method": "Independent signed Pauli and exact Gaussian interpretation; no compiler import",
+            "scope": "Positive inherited input frames before every retained adapter. No recursive hosting claim."}
diff --git a/research/p104-cross-call-paid-reuse/inputs/result5718.json b/research/p104-cross-call-paid-reuse/inputs/result5718.json
new file mode 100644
index 0000000000..a83d70e025
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/inputs/result5718.json
@@ -0,0 +1,366 @@
+{
+  "complete_cost": {
+    "compiler": {
+      "H": 80511200000000000,
+      "N": "2^80511200000000000",
+      "eta": {
+        "lower": "0.000000130293974939177582367425",
+        "upper": "0.000000130293974939177582367426"
+      },
+      "gap": "1/1400000000000",
+      "integer_test_exponent": 57508,
+      "logarithm_power": 1024,
+      "theta": "63/64",
+      "uniform_saving": "2177171993/700000000000",
+      "word_materialized": false
+    },
+    "full_forward_calls": 18,
+    "gain": "27337/25000000000",
+    "kappa": "31102457/10000000000",
+    "kappa_decimal": "0.00311024570",
+    "margin_at_previous": {
+      "lower": "0.265423369162460429568591881457",
+      "upper": "0.265423369162460429568591881458"
+    },
+    "negative_controls": {
+      "old_BIT": "Rejected by outer inequalities",
+      "old_cutoff": "Rejected by completion-tail ratio greater than one"
+    },
+    "nonrecursive": {
+      "fixed_scalar_growth_bound": "At most 2^(16*26000) before native factors; a declared constant, not an attained norm.",
+      "local_paid_rank_mass": 5987,
+      "local_routes": 5166000,
+      "local_scalar": 221474,
+      "peak_owned_fields": 2,
+      "raw_budget": 1000000000,
+      "raw_scalar_upper": 184647552,
+      "tensor_native_growth_bound": "Each rank-r Gaussian has row l1 norm at most 2^(rf); use local_paid_rank_mass for the whole word."
+    },
+    "ordinary_BIT": {
+      "admitted": "17/5000",
+      "after_cover": "69/20000",
+      "atom": "1/250",
+      "before_cover": "3459/1000000",
+      "ordinary_saving": "8590884599/2500000000000",
+      "prime_rule": "odd q > 2^26*N^5 avoiding all fixed denominators and rank witnesses"
+    },
+    "outer": {
+      "functional_checks": {
+        "actual_new_BIT_stock_sublinear": "348599999999253/350000000000000",
+        "fixed_halo_explicit_metadata_branch": "26950000000001/350000000000",
+        "new_BIT_row_stock_sublinear_in_active_group": "1395799999997009/1400000000000000",
+        "new_semantic_group_guard": "3/1400000000000"
+      },
+      "margins": {
+        "boundary_algebraic_matrix": "6096081577595645656017/1960000000000000000000000",
+        "fused_gaussian": "6096081577586936968051/1960000000000000000000000",
+        "global_coordinate_router": "272146499/87500000000",
+        "group_bit_router": "381005098686683560503/122500000000000000000000",
+        "group_scans": "699999999999/700000000000",
+        "main_transform": "6096081576195645656017/1960000000000000000000000",
+        "microbox_face_corrections": "26950000000001/350000000000",
+        "polynomial_products": "1399999999999/1400000000000",
+        "sparse_union_movement": "802346081577595645656017/1960000000000000000000000"
+      },
+      "parameters": {
+        "a": "272146499/87500000000",
+        "b": "870868797/280000000000",
+        "beta": "1399999999997/1400000000000",
+        "bit_saving": "17/5000",
+        "cap_xi": "1782346081576195645656017/1960000000000000000000000",
+        "complex_uniform_saving": "2177171993/700000000000",
+        "d": "1/1400000000000",
+        "epsilon": "1399999999999/1400000000000",
+        "kappa": "31102457/10000000000",
+        "lambda_": "398755901719/400000000000",
+        "lambda_prime": "1395645656017/1400000000000",
+        "layout": "1/1400000000000",
+        "q": "4354343983/1400000000000",
+        "r0": "1/14000000000000",
+        "stop": "1/1400000000000",
+        "tested_saving": "622049141/200000000000"
+      },
+      "strict_checks": {
+        "actual_bit_above_effective": "25353501/87500000000",
+        "actual_logarithmic_gamma_guard": "1/1400000000000",
+        "actual_logarithmic_group_guard": "177653918425904354343983/980000000000000000000000",
+        "alpha_logarithm_absorption": "1/14000000000000",
+        "boundary_algebraic_matrix_above_kappa": "5595645656017/1960000000000000000000000",
+        "cap_above_band": "802346081576195645656017/1960000000000000000000000",
+        "cap_below_cell": "177653918422404354343983/1960000000000000000000000",
+        "cap_below_phase_cell_power": "2137653918423804354343983/1960000000000000000000000",
+        "complex_above_effective": "1/1400000000000",
+        "complex_below_one_over_32": "7879131203/280000000000",
+        "complex_endpoint": "1/700000000000",
+        "epsilon_below_one": "1/1400000000000",
+        "epsilon_positive": "1399999999999/1400000000000",
+        "fine_prime_halo_umbrella": "7/2",
+        "fused_gaussian_above_kappa": "5586936968051/1960000000000000000000000",
+        "gaussian_above_main": "695645656017/980000000000000000000000",
+        "global_coordinate_router_above_kappa": "1/350000000000",
+        "global_gamma_power_slack": "9/14000000000000",
+        "group_below_dimension": "1/700000000000",
+        "group_bit_router_above_kappa": "436683560503/122500000000000000000000",
+        "group_gauge_power_slack": "177653918425834354343983/980000000000000000000000",
+        "group_key_fits_record": "1/700000000000",
+        "group_positive": "1399999999997/1400000000000",
+        "group_scans_above_kappa": "697822828009/700000000000",
+        "halo_covers_generic_inverse_radius": "5",
+        "improvement_over_user_baseline": "689637640400418148631/500000000000000000000000",
+        "lambda_above_internal": "1/2800000000000",
+        "lambda_above_sigma": "3/2800000000000",
+        "lambda_above_tau": "1/2800000000000",
+        "lambda_prime_above_lambda": "1/2800000000000",
+        "lambda_prime_above_leaf": "559129131203/392000000000000000000000",
+        "leaf_saving_above_effective": "279129131203/392000000000000000000000",
+        "main_transform_above_kappa": "4195645656017/1960000000000000000000000",
+        "membership_arithmetic_fits_record": "1/1400000000000",
+        "microbox_face_corrections_above_kappa": "13474455707003/175000000000",
+        "microbox_halo_volume": "110600000000003/1400000000000",
+        "new_complex_uniform_saving_above_b": "1/1400000000000",
+        "new_semantic_guard_with_C1_one": "1/1400000000000",
+        "polynomial_products_above_kappa": "1395645656019/1400000000000",
+        "positive_layout_slack": "1/1400000000000",
+        "q_below_effective": "1/1400000000000",
+        "q_below_internal": "1/1400000000000",
+        "q_below_leaf": "559129131203/392000000000000000000000",
+        "q_positive": "4354343983/1400000000000",
+        "short_record_fallback": "279129131203/280000000000",
+        "source_axis_superpolynomial": "1/1400000000000",
+        "sparse_union_movement_above_kappa": "796250000005595645656017/1960000000000000000000000",
+        "stop_below_one": "1399999999999/1400000000000",
+        "stop_positive": "1/1400000000000",
+        "strassen_integer_power_gap": "1951441519231",
+        "support_router_span": "3/1400000000000",
+        "theta_positive_polynomial_power": "2",
+        "union_header_fits_record": "3/1400000000000",
+        "union_tag_work_per_visit_fits_record": "3/1400000000000"
+      }
+    },
+    "paid": {
+      "1": 40872,
+      "16": 416,
+      "2": 8262,
+      "24": 416,
+      "3": 8910,
+      "32": 416,
+      "4": 1656,
+      "40": 416,
+      "48": 416,
+      "5": 2862,
+      "56": 416,
+      "6": 396,
+      "64": 416,
+      "7": 270,
+      "72": 208
+    },
+    "paid_calls": 66348,
+    "previous_open_pr5713": "155457611/50000000000",
+    "productive_fields": 12976,
+    "query_shears": 15,
+    "rank_mass_surplus": 760,
+    "raw_ceiling": {
+      "derivative_upper": "-369633362694406560545046337101738171609683586723251074624328749/1569275433846670190958947355801916604025588861116008628224",
+      "interval": [
+        "0",
+        "17/5000"
+      ]
+    },
+    "raw_headroom": [
+      "503/25000000000000",
+      "2013/100000000000000"
+    ],
+    "raw_root_interval": [
+      "77756143003/25000000000000",
+      "311024572013/100000000000000"
+    ],
+    "relative_gain_percent": "5467400/155457611",
+    "strictification": {
+      "alpha": {
+        "lower": "0.999999999984633711012925911331",
+        "upper": "0.999999999984633711012925911332"
+      },
+      "cutoff": 17,
+      "delta": {
+        "lower": "0.968803893336188607862541966226",
+        "upper": "0.968803893336188607862541966227"
+      },
+      "floor": 18,
+      "gamma": {
+        "lower": "0.968263210279894472263420498148",
+        "upper": "0.968263210279894472263420498149"
+      },
+      "normalized_paid_upper": "1048576000976562501/1048576001073741824",
+      "power": 1024,
+      "reserve": "1/1000000000000",
+      "tail_block": 128,
+      "tail_decay": "39/40",
+      "tail_offset": "1/5",
+      "tested": "622049141/200000000000"
+    },
+    "targets": {
+      "1": 36,
+      "145": 208,
+      "17": 416,
+      "2": 90,
+      "25": 416,
+      "33": 416,
+      "4": 1638,
+      "41": 416,
+      "49": 416,
+      "5": 108,
+      "57": 416,
+      "65": 416,
+      "7": 108,
+      "73": 208,
+      "8": 270,
+      "9": 7398
+    },
+    "whole_idle_jobs": 3120
+  },
+  "independent_literal_audit": {
+    "columns": 753,
+    "owned_lifetimes": 11,
+    "owned_peak": 2,
+    "paid": {
+      "1": 2184,
+      "2": 459,
+      "3": 495,
+      "4": 92,
+      "5": 159,
+      "6": 22,
+      "7": 15
+    },
+    "scalar": 221474,
+    "status": "PASS_ALL_SCALAR_INPUTS_FRAMES_OWNERSHIP_INVOICES",
+    "targets": {
+      "1": 2,
+      "2": 5,
+      "4": 91,
+      "5": 6,
+      "7": 6,
+      "8": 15,
+      "9": 411
+    }
+  },
+  "limits": [
+    "The five inherited framework assumptions remain.",
+    "No selected-frontier admission or large improvement is claimed.",
+    "The huge uniform word and complete rounded multiplier are not materialized.",
+    "Independent agent checks are not human or formal acceptance."
+  ],
+  "local": {
+    "endpoint_coefficients": 567009,
+    "helpers": {
+      "1": 2,
+      "2": 5,
+      "4": 91,
+      "5": 6,
+      "7": 6,
+      "8": 15,
+      "9": 411
+    },
+    "owned_fields": 9,
+    "owned_lifetimes": 11,
+    "paid": {
+      "1": 2184,
+      "2": 459,
+      "3": 495,
+      "4": 92,
+      "5": 159,
+      "6": 22,
+      "7": 15
+    },
+    "paid_rank_mass": 5987,
+    "peak_owned_fields": 2,
+    "public_inputs": 744,
+    "real_scalar_allowance": 221474
+  },
+  "native": {
+    "distinct_maps": 2917,
+    "helper_normalizers": 1072,
+    "helpers": {
+      "1": 2,
+      "2": 5,
+      "4": 91,
+      "5": 6,
+      "7": 6,
+      "8": 15,
+      "9": 411
+    },
+    "main_adapters": 208,
+    "proof": "Every signed Pauli image and exact global phase; frame induction at every positive tensor width",
+    "recipe_payload_sha256": "626b4ce338e77fd1f5c5d90cf107d91bb73af42c107cd8eea14fe657ddb324e4",
+    "wrong_phase_rejected": true
+  },
+  "negative_controls": {
+    "invented_extra_H9_target": "Every productive helper has a native target",
+    "inverse_before_normalizing126188": "Exact arbitrary-input endpoint row 126",
+    "missing_central_translation": "Rejected by the complete Laurent endpoint",
+    "omit_current146_inverse": "Exact arbitrary-input endpoint row 743",
+    "omit_early126_restoration": "Exact arbitrary-input endpoint row 126",
+    "omit_early188_restoration": "Exact arbitrary-input endpoint row 188",
+    "omit_original_retirement": "Exact arbitrary-input endpoint row 708",
+    "omit_owned_identifier_shift": "Scalar destination coframe",
+    "reintroduce_obsolete_decoder": "Exact arbitrary-input endpoint row 743",
+    "restore_uncharged_old_H9_target": "All declared final charts and owned exits",
+    "unpaid_rank7_query": "Complete field and call counts",
+    "wrong_current146_inverse_sign": "Exact arbitrary-input endpoint row 743",
+    "wrong_new_helper_boundary": "MOVE source and ownership"
+  },
+  "physical_caller": {
+    "central_square_columns": 53248,
+    "full_calls": 18,
+    "main_targets": {
+      "145": 208,
+      "17": 416,
+      "25": 416,
+      "33": 416,
+      "41": 416,
+      "49": 416,
+      "57": 416,
+      "65": 416,
+      "73": 208
+    },
+    "new_paid_jobs": 0,
+    "physical_contacts": 7488,
+    "ports": 104,
+    "route_payload_sha256": "7f4f0ca768ffbc4f93dcff575592c4c67411f46b4289e45b73f715a2ad47b142",
+    "tensor_rule": "Factorwise native maps; scalar signs stay outside the tensor power.",
+    "whole_idle_jobs": {
+      "16": 416,
+      "24": 416,
+      "32": 416,
+      "40": 416,
+      "48": 416,
+      "56": 416,
+      "64": 416,
+      "72": 208
+    },
+    "wrapper_sites": [
+      1,
+      3,
+      5,
+      7,
+      9,
+      11,
+      13,
+      15,
+      17,
+      19,
+      21,
+      23,
+      25,
+      27,
+      29,
+      31
+    ]
+  },
+  "separate_columns": {
+    "arithmetic": "Dense integer rows with binary denominators; reverse chronology",
+    "coefficients": 567009,
+    "columns": 753
+  },
+  "source_commit": "a5348f7143f95fe0046b92b96172e30763c7d731",
+  "status": "PASS_P104_TWO_BOUNDARY_SPLIT_KAPPA_00311024570"
+}
diff --git a/research/p104-cross-call-paid-reuse/inputs/word5718.json.gz b/research/p104-cross-call-paid-reuse/inputs/word5718.json.gz
new file mode 100644
index 0000000000..4db015aca3
Binary files /dev/null and b/research/p104-cross-call-paid-reuse/inputs/word5718.json.gz differ
diff --git a/research/p104-cross-call-paid-reuse/results/CHECKPOINT.json b/research/p104-cross-call-paid-reuse/results/CHECKPOINT.json
new file mode 100644
index 0000000000..c73112361a
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/results/CHECKPOINT.json
@@ -0,0 +1,43 @@
+{
+  "checked_utc": "2026-10-26T16:15Z",
+  "canonical_main": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "selected_kappa": "0.00310045",
+  "open_prs": {
+    "records": 312,
+    "complete_pagination": true,
+    "matches": [
+      {
+        "body": "The fixed native schedule misses helper cuts that become possible after a paid intermediate meeting. This construction splits existing queries through common frames, then retires 23 independent helper inputs and supplies 23 independent continuation inputs. Eighteen cuts use one subdivision each; a three-channel meeting shares five subdivisions, and two further meetings use two each. All 27 added queries per local call are charged.\n\nThe complete conditional exponent increases from PR #5718's **0.00311024570** to **0.00311301600**: **+0.00000277030, or +0.0890701%**. The exact raw crossing is `(0.00311301604454, 0.00311301604455)`. This is below the donor's 1% target. The complete p16/a7 caller has 13,390 productive targets and 66,834 paid calls, with rank surplus 760. The extra targets require cutoff 16, floor 17, tail offset 1/7 and decay 125/128; the old completion bound is explicitly rejected. Scalar, route, owned, normalization, idle and compiler costs remain funded.\n\nThe focused package is `research/p104-paid-meeting-splits`. It contains the exact constructor and plan, proof, authenticated PR #5718 source export, retained attribution, full result and rejection controls. Validation covers all 776 public/dirty-owned columns, 2,920 native maps with exact phases, the complete physical caller, two rational interval engines, the compiler, 54 strict outer inequalities and four functional checks. Run:\n\n```sh\npython3.12 -B scripts/research_package.py --package research/p104-paid-meeting-splits\n```\n\nPublication commit: `8483f9a679d06c1e270141dbc686a7e76e7a7522`. Registered normal/optimized replay, metadata/source pins, committed bytes and executable modes pass. The official export has 20 package files and SHA256 `5a3046462b06c108e2853898c9daf4b466bc46f47e5f77983515b6e5e7ed5023`; export integrity passes. The separate isolated optimized replay passed in 165.589 seconds with a read-only source tree, empty PATH and no Git directory; every exported byte remained unchanged. The full-history flag is not run in the shallow checkout; both declared source commits pass direct ancestry checks. `VALIDATION.md` records the scope. Hosted CI [failed in the shared planning tests](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40355861419/job/121688015960) before package replay. `test_research_current_assignment.py` still expects the legacy command mappings after priority revision 8. Its eight failures and two errors reproduce locally; that test, both picker scripts and the catalogue are byte-identical to main. The scientific replay job was skipped. This is not a hosted scientific pass. Independent acceptance remains pending.\n\nConstructed from canonical main `553d26d796fde9871ff2661a49ac7732e945410f`, with PR #5718 pinned at `a1534ac45401c9a9c80c07aa65c89e00e33594f8`. The publication commit is rebased onto the final upstream checkpoint at `de5ec2e7a6074068452f45b1a191b5c72832df4a` was read, including priority revision 8 and its complete helper-inventory gate. The frozen research sources are retained. The [canonical Discussion](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600) records the claim and findings. A later overlapping investigation [released the shared scope](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198871) and published a 17-split checkpoint without a duplicate PR. Its nonnested-excursion work remains separate. No global optimality or exclusive discovery claim is made.\n\nCredit flyingviper's PR #5718, Ali Romman's PR #5713 and complete P104 caller, Joel Pulikkan/GamingPuzzled's PR #5684, the PR #5599/#5609 helper interface, the published PR #5680 cut checks, and every retained native/controller/recursion/compiler/BIT source. New work: Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0.\n\nThe five inherited framework assumptions remain. The huge uniform word and complete rounded multiplier are not materialized. The selected frontier is unchanged; no merge or closure is requested.\n",
+        "head": "8483f9a679d06c1e270141dbc686a7e76e7a7522",
+        "number": 5751,
+        "title": "Conditional \u03ba 0.00311301600 (+2.77030e-6 over #5718): paid intermediate meetings"
+      },
+      {
+        "body": "Finite complex-supplier circuit improvement; **no new kappa**. Carrying whole-role signs through the frozen PR5599 word reduces its complete ordinary scalar allowance from **220232 to 219258**, with all **112 final restorations paid**. The local word has 25253 events instead of 25475. Its 18 full caller invocations save 17532 real-component primitives. This is an exact tariff comparison, not a measured runtime improvement.\n\nThe complete MOVE itinerary, 748 independent public inputs, 540 helper targets, owned lifetimes, ADD operand order and coefficient magnitudes remain identical. Negative nonunit scales retain their positive magnitude. The paid and target histograms are unchanged, so selected conditional kappa remains 0.00310045.\n\nReproduce with Python 3.12, standard library:\n\n```sh\npython3 -B scripts/research_package.py --package research/p104-sign-gauge\n```\n\nExpected: normal/optimized `PASS_P104_SIGN_GAUGE`, `PASS_P104_SIGN_INTEGRITY`, and `PASS_REGISTERED_RESEARCH_PACKAGE`. The package supplies the full transformed word, deterministic constructor, proof and exact receipt. Checks cover every source cut, equal per-role prefix envelopes, all 571536 endpoint coefficients, chart/owned-state accounting, six semantic controls, actual source/receipt/word corruptions, lossless gzip repacking and read-only Git-free replay.\n\nBase: `df0137efd35e11b4445734bf612bdd4c7ebc330c`. Source: the accepted PR5599 word at `09dd0965711ca7eeb169057a420abab1c9052137`; its word and tariff are bound by Git blob and SHA256 at the base. No unmerged objects or network are needed for scientific replay. Only `research/p104-sign-gauge/` is added.\n\nOS seed 34 selected the complex-supplier track. Three candidates and bounded overlap checks are in CLAIM.md; there was no pinned sprint issue. Ross Richardson's PR919 and Scratch-net's PR1153 already supply same-frame sign-normalization precedents. This extends the application to the actual selected P104 word, across frame moves and owned assignments. Concurrent PR5713/5718 change paid/target profiles on newer words; their gains are not used or added here.\n\nCredit Ali Romman for the source word, tariff and transpose-check method; retain Pierre-Loic Doulcet, Joel Pulikkan, DreamingOfClouds, thu-nmrc, Gareth Davidson, David Arias, Cinna Davis, Jacob Alan Sussman, zxv, Cao Yifan and the cumulative native/BIT/compiler lineage. Prepared for **Charan / theon199 with substantial OpenAI Codex assistance**; no subagents or independent human review. Apache-2.0.\n\nNative request bytes are unchanged, but native recipes, full caller/admission and rounded/tape implementation are not freshly executed. The five inherited framework assumptions remain. The catalogue card is explicitly marked changed scope; this scalar result does not meet its changed-recursive-profile gate or prove a multiplication theorem. Hosted CI is separate from local validation; the source-base CI documentation failures reported in PR5718 are not repaired here.\n",
+        "head": "9e7d94014a4c4585f097f11c0e84cf660882369f",
+        "number": 5750,
+        "title": "Simplify the accepted P104 word by 974 scalar primitives per call"
+      },
+      {
+        "body": "## Result\n\nAt source-matched \u03ba = 836330098538459/500000000000000000 = 0.001672660197076918, this Lean certificate connects all 144 ordered selected route tables\u2019 parsed values to the accepted per-call owner assignment. JSON-byte-to-parsed-value correspondence remains source-reviewed; actual physical charts, private C allocation/operator semantics and executor refinement remain explicit suppliers. It does not prove the full multiplication theorem or change \u03ba.\n\nFor every call in Fin144, the strict decoder returns the complete declared table; every one of its Fin704 owner lookups equals the accepted selectedOrdinaryOwner assignment and is injective within that call. All 101,376 ordered family/route rows are covered, including 69,120 helper rows. This supplies parsed-table success and owner binding; it does not equate syntax with physical charts or execution.\n\n## Checked closure and reproduction\n\nThe final source 1845c1f7/object 174289ab has independent terminal 16746648. All 26 newly checked roots are accepted with 32 standard-only probes, using the repaired generic bridge rather than its failed preimage. Existing compiler time totals 75.665844489 seconds. The compact completed certificate retains exact source, object, data-review and terminal identities.\n\nThis PR includes only 28 new exact Lean sources (745,044 bytes). The remaining 45 sources in the 73-module closure are reconstructed from the already-published all144 owner certificate PR1807 at 457c6c315598ba1a05692fed55e3f47d99dbb184 and unchanged public 0b Stream/Realize inputs. The one Realize import-line adjustment is declared, hashed and inverted; all accepted source bytes remain exact. The README gives the public fetch commands, fresh-output materializer and topological build order for Lean 4.21.0/Mathlib 308445d7. A new relocated replay is not claimed.\n\n## Credit, scope and CI\n\nCredit Rohan Arun's PR1094 overlay, Jeremy Kothe's T16/hyperplane-ended emitter, Hao Yan's retained lineage, columbia-git's PR1553 owner binding, DreamingOfClouds' Stream/Realize and Lean/Mathlib contributors. Full notices and Apache-2.0 are preserved; new proof and packaging used OpenAI assistance. PR1807 is an exact prerequisite, not republished as a new owner certificate.\n\nCurrent all-state exact-module and proposed-branch searches found no matching PR. Related historical route-table constructions do not provide this exact Lean table-to-owner certificate; this is bounded overlap screening, not a novelty claim. Source JSON/transcription correspondence remains separately reviewed. There is no global cross-call disjointness, complete physical realization or full multiplication theorem claim.\n\nNo extra compiler or CI pre-review was run for packaging. Candidate CI will be inspected after submission; generic repository checks alone do not establish a Lean replay. Historical preparation headers and failure evidence remain preserved.\n",
+        "head": "542f914ca0e5df4026341d08c84385cb1a71cfdf",
+        "number": 1831,
+        "title": "[Lean] Bind all 144 selected parsed route tables to certified owners"
+      },
+      {
+        "body": "## Result and exact scope\n\nFor the exact selected context \u03ba = 836330098538459/500000000000000000 = 0.001672660197076918, this Lean theorem derives a finite uniform A and cost(x) \u2264 A n^(1\u2212\u03ba) volume(x) from the explicitly quantified complete state/domain, positive volume, uniform base-family/local-work and exact selected child suppliers. It supplies no unconditional operational construction or full integer-multiplication bit-cost theorem.\n\nThe theorem constructs the normalized envelope, proves global boundedness by strict child descent, obtains its recurrence and a finite uniform constant, then connects the exact profile to the selected exponent. Neither the desired scalar recurrence nor `native_bound` is assumed by the final assembly.\n\nThe explicit suppliers still include the complete admitted state and precision family, nonempty states at every positive width, positive natural volume, uniform bounds across entire base families, one complete non-child-work bound, and the exact selected child decomposition with child volume `parent / 14336`. Operational implementation/refinement, precision accounting and transfer to a full multiplication bit-cost theorem remain open.\n\n## Existing checked evidence\n\nThe exact final source is `1445894f6bc7d157bd229f81d92f55313132025a95effc67102fc08b08c60c95`. Its independent native terminal is `dbe7ee9956ee973bd698233ba07cc7b9d69571873b1db10748314ab25a98cf3d`, with two standard-only axiom probes, no warnings and no new finite decision. The accepted sixteen-module source closure uses Lean 4.21.0 and Mathlib `308445d7985027f538e281e18df29ca16ede2ba3`.\n\nEight included exact sources total 55,714 bytes. `materialize_sources.py` fetches no data itself: it reconstructs all sixteen sources from those files and eight public prerequisites at fork commit `73ff839084f8c2761dc4213a2fd9e63a295dd8fb`. The README gives the explicit public Git fetch, deterministic materialization and topological Lean commands. The existing local reconstruction matched all sixteen accepted hashes. No source or proof was rerun for this submission preparation.\n\n## Attribution and comparison\n\nPrepared for James Chang with substantial OpenAI assistance. Apache-2.0 and all source headers are retained. PR585 rational checkpoints, PR605/799 real-enclosure lineage, the exact selected PR1094 profile and Lean/Mathlib/Batteries are credited in NOTICE and SOURCE_MANIFEST. PR1379/1641 parking results and PR1430 precision analysis remain complementary conditional interfaces. No novelty is claimed for supremum, strong induction or the general recurrence method.\n\nA current bounded duplicate search found no PR naming the exact assembly theorem or normalized-envelope module. Nearby pricing and finite-assembly results do not supply this same complete physical-supplier-to-cost-bound implication; prior exact comparisons are retained. This is a scoped overlap check, not a global novelty certificate.\n\n## CI and publication status\n\nThis is one coherent conditional theorem chunk, separate from Hosting and the all144 finite-owner certificate. Its existing accepted source bytes and historical receipts are preserved. No candidate-specific CI or relocated full-closure replay has been run before submission. CI will be inspected after publication; generic repository checks alone do not demonstrate this Lean closure was replayed.\n",
+        "head": "981911842b9358db4fde5f1386bd49bf3dc07ea3",
+        "number": 1809,
+        "title": "[Lean] Derive the selected normalized-cost bound from explicit physical suppliers"
+      },
+      {
+        "body": "## Result and exact scope\n\nAt source-matched \u03ba = 836330098538459/500000000000000000 = 0.001672660197076918, the finite certificate proves that each of the144 ordered rows has480 distinct helpers below14336 and outside its224-register active interval, and derives per-call borrowed-owner, ordinary704-register and tagged712-register family injection. It does not assert global disjointness across calls, actual physical projection/execution or the full multiplication theorem.\n\nThe proof covers every helper in the original ordered 144-by-480 table, not only call 0 or a sample. It uses the accepted split-data source, a generic sorted-owner certificate, 36 checked four-row blocks and one exhaustive join. The final declarations prove all rows certified, borrowed-owner injection and active-interval exclusion, ordinary 704-register injection and tagged 712-register family injection.\n\n## Existing checked evidence\n\nThe exact final source is `53f3c474fe747eaa30c617361e33b68249da035a2666854e36f6caca95ebe6a1`, with independent terminal `558bd5aa49fa10a588226243c3344bc4dcbce67d7b3e1920e116f707b9717b20`. All 36 blocks and the final join are accepted. The rejected monolithic data/checker, call0-only pilot and unrelated source-only candidates are excluded. The original resource failures remain preserved in the research record.\n\nThe reproducible closure contains 45 exact Lean modules: 43 included sources (405,537 bytes) plus the two immutable public Stream/Realize prerequisites. `prepare_sources.py` reconstructs the latter with one declared import-line adjustment and checks every resulting source hash. The README and REPRODUCTION.json provide public Git fetches, the pinned Lean 4.21.0 / Mathlib `308445d7985027f538e281e18df29ca16ede2ba3` environment and serial module commands. The existing source-reconstruction check matches all 45 accepted hashes. No block, join or proof was rerun during packaging.\n\n## Attribution and remaining suppliers\n\nPrepared for James Chang with substantial OpenAI assistance. Credit Rohan Arun's PR1094 selected overlay; Jeremy Kothe's T16 construction; Hao Yan's retained lineage; columbia-git's PR1553 emitted owner binding; and DreamingOfClouds' PR509 Stream / PR516 Realize. Apache-2.0, full prerequisite notices and exact source provenance are preserved. No novelty is claimed for generic sorting or injection arguments.\n\nThe selected JSON-to-Lean literal correspondence is separately source-reviewed, rather than a verified parser theorem. Physical family projection, charts, event ordering, actual executor correspondence, cross-call scheduling and precision remain suppliers. The coherent Hosting prerequisite retains its explicit conditional hypotheses.\n\nA current bounded duplicate search found no exact exhaustive-join module PR. PR1553 provides source-bound owner/program and input-majorant evidence; historical owner certificates and current finite packing proposals concern different source instances or interfaces. Those comparisons and their limits are retained. This is not a global novelty claim.\n\n## CI and publication status\n\nThis is one coherent finite certificate chunk. Existing accepted source bytes and historical receipts are preserved. No candidate-specific CI or relocated full-closure replay has been run before submission. CI will be inspected after publication; generic repository checks alone do not demonstrate that these Lean certificates were replayed.\n",
+        "head": "457c6c315598ba1a05692fed55e3f47d99dbb184",
+        "number": 1807,
+        "title": "[Lean] Certify all 144 selected helper-owner rows and per-call injections"
+      }
+    ]
+  },
+  "discussion_coverage": "25 most recently updated Discussions and last three comments each at 15:49Z; plus start-here and cached complete overlap. Not exhaustive live comment/diff coverage.",
+  "historical_checkout_reason": "Managed checkout at latest main failed on missing promisor blobs twice. Isolated managed worktree created at df0137efd; latest revision-8 instructions read through Git, exact science pinned to PR5718 unchanged."
+}
diff --git a/research/p104-cross-call-paid-reuse/results/QUERY-MAPS.json.gz b/research/p104-cross-call-paid-reuse/results/QUERY-MAPS.json.gz
new file mode 100644
index 0000000000..e241f1958c
Binary files /dev/null and b/research/p104-cross-call-paid-reuse/results/QUERY-MAPS.json.gz differ
diff --git a/research/p104-cross-call-paid-reuse/results/RESULT.json b/research/p104-cross-call-paid-reuse/results/RESULT.json
new file mode 100644
index 0000000000..ee1501e28f
--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/results/RESULT.json
@@ -0,0 +1,7932 @@
+{
+  "caller_binding_phase_checks": 416,
+  "complete_invoice": {
+    "adjacent_main_best_cache_tree": {
+      "best_recursive_delta": 0,
+      "necessary_alignment_at_least": {
+        "1": 1
+      },
+      "removed": {
+        "1": 1
+      }
+    },
+    "kappa_status": "NO_NEW_KAPPA",
+    "local_paid": {
+      "1": 2184,
+      "2": 459,
+      "3": 495,
+      "4": 92,
+      "5": 159,
+      "6": 22,
+      "7": 15
+    },
+    "local_targets": {
+      "1": 2,
+      "2": 5,
+      "4": 91,
+      "5": 6,
+      "7": 6,
+      "8": 15,
+      "9": 411
+    },
+    "new_retention_reads_routes_restoration_cleanup": "Unknown; no legal deletion emitted. Not charged at zero.",
+    "paid_jobs": 66348,
+    "productive_fields": 12976,
+    "realized_delta_paid": {},
+    "realized_delta_targets": {},
+    "source_reported_ordinary": {
+      "fixed_scalar_growth_bound": "At most 2^(16*26000) before native factors; a declared constant, not an attained norm.",
+      "local_paid_rank_mass": 5987,
+      "local_routes": 5166000,
+      "local_scalar": 221474,
+      "peak_owned_fields": 2,
+      "raw_budget": 1000000000,
+      "raw_scalar_upper": 184647552,
+      "tensor_native_growth_bound": "Each rank-r Gaussian has row l1 norm at most 2^(rf); use local_paid_rank_mass for the whole word."
+    },
+    "unrealized_centre_allowance": {
+      "paid_delta": {
+        "7": -1
+      },
+      "target_delta": {}
+    },
+    "whole_paid": {
+      "1": 40872,
+      "16": 416,
+      "2": 8262,
+      "24": 416,
+      "3": 8910,
+      "32": 416,
+      "4": 1656,
+      "40": 416,
+      "48": 416,
+      "5": 2862,
+      "56": 416,
+      "6": 396,
+      "64": 416,
+      "7": 270,
+      "72": 208
+    },
+    "whole_targets": {
+      "1": 36,
+      "145": 208,
+      "17": 416,
+      "2": 90,
+      "25": 416,
+      "33": 416,
+      "4": 1638,
+      "41": 416,
+      "49": 416,
+      "5": 108,
+      "57": 416,
+      "65": 416,
+      "7": 108,
+      "73": 208,
+      "8": 270,
+      "9": 7398
+    }
+  },
+  "complete_separate_delivery_screen": [
+    {
+      "conditioned_screens": [],
+      "proof": "For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.",
+      "rank_one_answers": [],
+      "status": "NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET",
+      "target_event": 13609,
+      "three_rank_two_certificate": {
+        "basis_rank": 228,
+        "rank_two_rows": 228,
+        "status": "OUTSIDE_THE_ENTIRE_RANK_TWO_SPAN",
+        "target_response": "1/2",
+        "witness": {
+          "173": "1"
+        }
+      }
+    },
+    {
+      "conditioned_screens": [],
+      "proof": "For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.",
+      "rank_one_answers": [],
+      "status": "NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET",
+      "target_event": 13620,
+      "three_rank_two_certificate": {
+        "basis_rank": 232,
+        "rank_two_rows": 232,
+        "status": "OUTSIDE_THE_ENTIRE_RANK_TWO_SPAN",
+        "target_response": "1/2",
+        "witness": {
+          "151": "1"
+        }
+      }
+    },
+    {
+      "conditioned_screens": [],
+      "proof": "For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.",
+      "rank_one_answers": [],
+      "status": "NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET",
+      "target_event": 13631,
+      "three_rank_two_certificate": {
+        "basis_rank": 236,
+        "rank_two_rows": 236,
+        "status": "OUTSIDE_THE_ENTIRE_RANK_TWO_SPAN",
+        "target_response": "1/4",
+        "witness": {
+          "146": "-1/2",
+          "478": "1"
+        }
+      }
+    },
+    {
+      "conditioned_screens": [],
+      "proof": "For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.",
+      "rank_one_answers": [],
+      "status": "NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET",
+      "target_event": 13640,
+      "three_rank_two_certificate": {
+        "basis_rank": 239,
+        "rank_two_rows": 239,
+        "status": "OUTSIDE_THE_ENTIRE_RANK_TWO_SPAN",
+        "target_response": "-1/2",
+        "witness": {
+          "133": "1"
+        }
+      }
+    },
+    {
+      "conditioned_screens": [
+        {
+          "paid_rank_one_answers": [
+            11274
+          ],
+          "remaining_alignment_rank_budget": 5,
+          "screen": {
+            "all_prefix_paid": 1211,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 994,
+            "projective_residual_groups": 570,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13660,
+            "tested_group_pairs": 185
+          }
+        },
+        {
+          "paid_rank_one_answers": [
+            11275
+          ],
+          "remaining_alignment_rank_budget": 5,
+          "screen": {
+            "all_prefix_paid": 1211,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 994,
+            "projective_residual_groups": 570,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13660,
+            "tested_group_pairs": 185
+          }
+        },
+        {
+          "paid_rank_one_answers": [
+            11274,
+            11275
+          ],
+          "remaining_alignment_rank_budget": 4,
+          "screen": {
+            "all_prefix_paid": 1211,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 896,
+            "projective_residual_groups": 568,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13660,
+            "tested_group_pairs": 0
+          }
+        }
+      ],
+      "proof": "For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.",
+      "rank_one_answers": [
+        11274,
+        11275
+      ],
+      "status": "NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET",
+      "target_event": 13660,
+      "three_rank_two_certificate": {
+        "all_rank_two_coordinate_maps": {
+          "10119": {
+            "10119": "1"
+          },
+          "10121": {
+            "10121": "1"
+          },
+          "10123": {
+            "10123": "1"
+          },
+          "10125": {
+            "10125": "1"
+          },
+          "10127": {
+            "10127": "1"
+          },
+          "10129": {
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+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 999,
+            "projective_residual_groups": 575,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13673,
+            "tested_group_pairs": 181
+          }
+        },
+        {
+          "paid_rank_one_answers": [
+            9745,
+            9746
+          ],
+          "remaining_alignment_rank_budget": 4,
+          "screen": {
+            "all_prefix_paid": 1216,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 893,
+            "projective_residual_groups": 573,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13673,
+            "tested_group_pairs": 0
+          }
+        }
+      ],
+      "proof": "For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.",
+      "rank_one_answers": [
+        9745,
+        9746
+      ],
+      "status": "NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET",
+      "target_event": 13673,
+      "three_rank_two_certificate": {
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+            "10139": "-1/2",
+            "10143": "1/4",
+            "10145": "1/4",
+            "10147": "-1/2",
+            "10181": "1/4",
+            "10550": "1/4",
+            "10552": "1/4",
+            "10554": "-1/4",
+            "10560": "1/4",
+            "10562": "-1/4",
+            "10568": "1/4",
+            "10570": "1/4",
+            "10572": "-1/2",
+            "10576": "1/4",
+            "10616": "-1/2",
+            "10623": "1/4",
+            "10760": "1/4",
+            "10762": "1/4",
+            "10764": "-1/4",
+            "10770": "1/4",
+            "10772": "-1/4",
+            "10778": "1/4",
+            "10780": "1/4",
+            "10782": "-1/2",
+            "10786": "1/4",
+            "10788": "1/4",
+            "10790": "-1/2",
+            "11192": "1/4",
+            "11194": "-1/4",
+            "11200": "1/4",
+            "11202": "-1/4",
+            "11208": "1/4",
+            "11210": "-1/2",
+            "11214": "1/4",
+            "11216": "1/4",
+            "11218": "-1/2",
+            "11252": "1/4",
+            "11835": "1/4",
+            "11837": "-1/4",
+            "11843": "1/4",
+            "11845": "-1/4",
+            "11851": "1/4",
+            "11853": "1/4",
+            "11855": "-1/2",
+            "11859": "1/4",
+            "11861": "1/4",
+            "11863": "-1/2",
+            "12271": "1/4",
+            "12273": "1/4",
+            "12275": "-1/4",
+            "12281": "1/4",
+            "12283": "-1/4",
+            "12289": "1/4",
+            "12291": "1/4",
+            "12293": "-1/2",
+            "12297": "1/4",
+            "12299": "1/4",
+            "12301": "-1/2",
+            "12491": "1/4",
+            "12497": "1/4",
+            "12499": "-1/4",
+            "12505": "1/4",
+            "12507": "1/4",
+            "12509": "-1/2",
+            "12513": "1/4",
+            "12515": "1/4",
+            "12517": "-1/2",
+            "12550": "-1/4"
+          }
+        },
+        "ignored_basis_coordinates": [
+          10119,
+          10121,
+          10123,
+          10129,
+          10131,
+          10137,
+          10139,
+          10143,
+          10145,
+          10147,
+          10181,
+          10550,
+          10552,
+          10554,
+          10560,
+          10562,
+          10568,
+          10570,
+          10572,
+          10576,
+          10616,
+          10623,
+          10760,
+          10762,
+          10764,
+          10770,
+          10772,
+          10778,
+          10780,
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+          10790,
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+          11194,
+          11200,
+          11202,
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+          11214,
+          11216,
+          11218,
+          11252,
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+          11851,
+          11853,
+          11855,
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+          11861,
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+          12271,
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+          12275,
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+          12291,
+          12293,
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+          12301,
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+          12497,
+          12499,
+          12505,
+          12507,
+          12509,
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+          12515,
+          12517,
+          12550
+        ],
+        "maximum_uncovered_support_of_one_answer": 2,
+        "rank_two_rows": 256,
+        "status": "THREE_RANK_TWO_ROWS_CANNOT_COVER_TARGET_SUPPORT",
+        "target_coordinates": {
+          "10119": "1/4",
+          "10121": "1/4",
+          "10123": "1/4",
+          "10125": "1/2",
+          "10129": "1/4",
+          "10131": "1/4",
+          "10137": "-1/4",
+          "10139": "1/2",
+          "10143": "1/4",
+          "10145": "1/4",
+          "10181": "1/4",
+          "10332": "1/4",
+          "10334": "1/4",
+          "10336": "1/4",
+          "10338": "1/2",
+          "10342": "1/4",
+          "10344": "1/4",
+          "10350": "1/4",
+          "10352": "-1/4",
+          "10354": "1/2",
+          "10358": "1/4",
+          "10360": "1/4",
+          "10550": "1/4",
+          "10552": "1/4",
+          "10554": "1/4",
+          "10556": "1/2",
+          "10560": "1/4",
+          "10562": "1/4",
+          "10568": "1/4",
+          "10570": "-1/4",
+          "10572": "1/2",
+          "10576": "1/4",
+          "10623": "1/4",
+          "10760": "1/4",
+          "10762": "1/4",
+          "10764": "1/4",
+          "10766": "1/2",
+          "10770": "1/4",
+          "10772": "1/4",
+          "10778": "1/4",
+          "10780": "-1/4",
+          "10782": "1/2",
+          "10786": "1/4",
+          "10788": "1/4",
+          "10980": "1/4",
+          "10986": "1/4",
+          "10988": "1/4",
+          "10994": "1/4",
+          "10996": "-1/4",
+          "10998": "1/2",
+          "11002": "1/4",
+          "11004": "1/4",
+          "11039": "-1/4",
+          "11042": "1/2",
+          "9689": "1/4",
+          "9695": "1/4",
+          "9697": "1/4",
+          "9703": "1/4",
+          "9705": "-1/4",
+          "9707": "1/2",
+          "9711": "1/4",
+          "9713": "1/4",
+          "9749": "-1/4",
+          "9752": "1/2",
+          "9901": "1/4",
+          "9903": "1/4",
+          "9905": "1/4",
+          "9907": "1/2",
+          "9911": "1/4",
+          "9913": "1/4",
+          "9919": "1/4",
+          "9921": "-1/4",
+          "9923": "1/2",
+          "9927": "1/4",
+          "9929": "1/4"
+        },
+        "target_uncovered_coordinates": [
+          9689,
+          9695,
+          9697,
+          9703,
+          9705,
+          9707,
+          9711,
+          9713,
+          9749,
+          9752,
+          9901,
+          9903,
+          9905,
+          9907,
+          9911,
+          9913,
+          9919,
+          9921,
+          9923,
+          9927,
+          9929,
+          10125,
+          10332,
+          10334,
+          10336,
+          10338,
+          10342,
+          10344,
+          10350,
+          10352,
+          10354,
+          10358,
+          10360,
+          10556,
+          10766,
+          10980,
+          10986,
+          10988,
+          10994,
+          10996,
+          10998,
+          11002,
+          11004,
+          11039,
+          11042
+        ]
+      }
+    },
+    {
+      "conditioned_screens": [
+        {
+          "paid_rank_one_answers": [
+            11678
+          ],
+          "remaining_alignment_rank_budget": 5,
+          "screen": {
+            "all_prefix_paid": 1222,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 1005,
+            "projective_residual_groups": 581,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13688,
+            "tested_group_pairs": 185
+          }
+        },
+        {
+          "paid_rank_one_answers": [
+            11679
+          ],
+          "remaining_alignment_rank_budget": 5,
+          "screen": {
+            "all_prefix_paid": 1222,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 1005,
+            "projective_residual_groups": 581,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13688,
+            "tested_group_pairs": 185
+          }
+        },
+        {
+          "paid_rank_one_answers": [
+            11678,
+            11679
+          ],
+          "remaining_alignment_rank_budget": 4,
+          "screen": {
+            "all_prefix_paid": 1222,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 907,
+            "projective_residual_groups": 579,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13688,
+            "tested_group_pairs": 0
+          }
+        }
+      ],
+      "proof": "For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.",
+      "rank_one_answers": [
+        11678,
+        11679
+      ],
+      "status": "NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET",
+      "target_event": 13688,
+      "three_rank_two_certificate": {
+        "all_rank_two_coordinate_maps": {
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+            "10119": "1"
+          },
+          "10121": {
+            "10121": "1"
+          },
+          "10123": {
+            "10123": "1"
+          },
+          "10125": {
+            "10125": "1"
+          },
+          "10127": {
+            "10127": "1"
+          },
+          "10129": {
+            "10129": "1"
+          },
+          "10131": {
+            "10131": "1"
+          },
+          "10133": {
+            "10133": "1"
+          },
+          "10137": {
+            "10137": "1"
+          },
+          "10139": {
+            "10139": "1"
+          },
+          "10141": {
+            "10141": "1"
+          },
+          "10143": {
+            "10143": "1"
+          },
+          "10145": {
+            "10145": "1"
+          },
+          "10147": {
+            "10147": "1"
+          },
+          "10181": {
+            "10181": "1"
+          },
+          "10188": {
+            "10188": "1"
+          },
+          "10203": {
+            "10203": "1"
+          },
+          "10332": {
+            "10332": "1"
+          },
+          "10334": {
+            "10334": "1"
+          },
+          "10336": {
+            "10336": "1"
+          },
+          "10338": {
+            "10338": "1"
+          },
+          "10340": {
+            "10340": "1"
+          },
+          "10342": {
+            "10342": "1"
+          },
+          "10344": {
+            "10344": "1"
+          },
+          "10346": {
+            "10346": "1"
+          },
+          "10348": {
+            "10348": "1"
+          },
+          "10350": {
+            "10350": "1"
+          },
+          "10352": {
+            "10352": "1"
+          },
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+            "10354": "1"
+          },
+          "10356": {
+            "10356": "1"
+          },
+          "10358": {
+            "10358": "1"
+          },
+          "10360": {
+            "10360": "1"
+          },
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+            "10362": "1"
+          },
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+            "10550": "1"
+          },
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+          },
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+          },
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+          },
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+          },
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+            "10560": "1"
+          },
+          "10562": {
+            "10562": "1"
+          },
+          "10564": {
+            "10564": "1"
+          },
+          "10566": {
+            "10566": "1"
+          },
+          "10568": {
+            "10568": "1"
+          },
+          "10570": {
+            "10570": "1"
+          },
+          "10572": {
+            "10572": "1"
+          },
+          "10574": {
+            "10574": "1"
+          },
+          "10576": {
+            "10576": "1"
+          },
+          "10616": {
+            "10616": "1"
+          },
+          "10623": {
+            "10623": "1"
+          },
+          "10634": {
+            "10634": "1"
+          },
+          "10760": {
+            "10760": "1"
+          },
+          "10762": {
+            "10762": "1"
+          },
+          "10764": {
+            "10764": "1"
+          },
+          "10766": {
+            "10766": "1"
+          },
+          "10768": {
+            "10768": "1"
+          },
+          "10770": {
+            "10770": "1"
+          },
+          "10772": {
+            "10772": "1"
+          },
+          "10774": {
+            "10774": "1"
+          },
+          "10776": {
+            "10776": "1"
+          },
+          "10778": {
+            "10778": "1"
+          },
+          "10780": {
+            "10780": "1"
+          },
+          "10782": {
+            "10782": "1"
+          },
+          "10784": {
+            "10784": "1"
+          },
+          "10786": {
+            "10786": "1"
+          },
+          "10788": {
+            "10788": "1"
+          },
+          "10790": {
+            "10790": "1"
+          },
+          "10978": {
+            "10978": "1"
+          },
+          "10980": {
+            "10980": "1"
+          },
+          "10984": {
+            "10984": "1"
+          },
+          "10986": {
+            "10986": "1"
+          },
+          "10988": {
+            "10988": "1"
+          },
+          "10990": {
+            "10990": "1"
+          },
+          "10992": {
+            "10992": "1"
+          },
+          "10994": {
+            "10994": "1"
+          },
+          "10996": {
+            "10996": "1"
+          },
+          "10998": {
+            "10998": "1"
+          },
+          "11000": {
+            "11000": "1"
+          },
+          "11002": {
+            "11002": "1"
+          },
+          "11004": {
+            "11004": "1"
+          },
+          "11006": {
+            "11006": "1"
+          },
+          "11036": {
+            "11036": "1"
+          },
+          "11039": {
+            "11039": "1"
+          },
+          "11042": {
+            "11042": "1"
+          },
+          "11190": {
+            "11190": "1"
+          },
+          "11192": {
+            "11192": "1"
+          },
+          "11194": {
+            "11194": "1"
+          },
+          "11196": {
+            "11196": "1"
+          },
+          "11198": {
+            "11198": "1"
+          },
+          "11200": {
+            "11200": "1"
+          },
+          "11202": {
+            "11202": "1"
+          },
+          "11204": {
+            "11204": "1"
+          },
+          "11208": {
+            "11208": "1"
+          },
+          "11210": {
+            "11210": "1"
+          },
+          "11212": {
+            "11212": "1"
+          },
+          "11214": {
+            "11214": "1"
+          },
+          "11216": {
+            "11216": "1"
+          },
+          "11218": {
+            "11218": "1"
+          },
+          "11252": {
+            "11252": "1"
+          },
+          "11259": {
+            "11259": "1"
+          },
+          "11402": {
+            "11402": "1"
+          },
+          "11404": {
+            "11404": "1"
+          },
+          "11406": {
+            "11406": "1"
+          },
+          "11408": {
+            "11408": "1"
+          },
+          "11410": {
+            "11410": "1"
+          },
+          "11412": {
+            "11412": "1"
+          },
+          "11414": {
+            "11414": "1"
+          },
+          "11416": {
+            "11416": "1"
+          },
+          "11418": {
+            "11418": "1"
+          },
+          "11420": {
+            "11420": "1"
+          },
+          "11422": {
+            "11422": "1"
+          },
+          "11424": {
+            "11424": "1"
+          },
+          "11426": {
+            "11426": "1"
+          },
+          "11428": {
+            "11428": "1"
+          },
+          "11430": {
+            "11430": "1"
+          },
+          "11432": {
+            "11432": "1"
+          },
+          "11620": {
+            "11620": "1"
+          },
+          "11622": {
+            "11622": "1"
+          },
+          "11624": {
+            "11624": "1"
+          },
+          "11626": {
+            "11626": "1"
+          },
+          "11630": {
+            "11630": "1"
+          },
+          "11632": {
+            "11632": "1"
+          },
+          "11634": {
+            "11634": "1"
+          },
+          "11636": {
+            "11636": "1"
+          },
+          "11638": {
+            "11638": "1"
+          },
+          "11640": {
+            "11640": "1"
+          },
+          "11642": {
+            "11642": "1"
+          },
+          "11644": {
+            "11644": "1"
+          },
+          "11646": {
+            "11646": "1"
+          },
+          "11648": {
+            "11648": "1"
+          },
+          "11690": {
+            "11690": "1"
+          },
+          "11693": {
+            "11693": "1"
+          },
+          "11712": {
+            "11632": "1",
+            "11642": "1"
+          },
+          "11713": {
+            "11646": "1",
+            "11693": "1"
+          },
+          "11721": {
+            "11634": "1",
+            "11690": "1"
+          },
+          "11722": {
+            "11622": "1",
+            "11644": "1"
+          },
+          "11724": {
+            "11634": "1/2",
+            "11690": "-1/2"
+          },
+          "11725": {
+            "11630": "-1/2",
+            "11638": "1/2"
+          },
+          "11733": {
+            "11646": "1/2",
+            "11693": "-1/2"
+          },
+          "11734": {
+            "11620": "-1/2",
+            "11640": "1/2"
+          },
+          "11833": {
+            "11833": "1"
+          },
+          "11835": {
+            "11835": "1"
+          },
+          "11837": {
+            "11837": "1"
+          },
+          "11839": {
+            "11839": "1"
+          },
+          "11841": {
+            "11841": "1"
+          },
+          "11843": {
+            "11843": "1"
+          },
+          "11845": {
+            "11845": "1"
+          },
+          "11847": {
+            "11847": "1"
+          },
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+            "11849": "1"
+          },
+          "11851": {
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+          },
+          "11853": {
+            "11853": "1"
+          },
+          "11855": {
+            "11855": "1"
+          },
+          "11857": {
+            "11857": "1"
+          },
+          "11859": {
+            "11859": "1"
+          },
+          "11861": {
+            "11861": "1"
+          },
+          "11863": {
+            "11863": "1"
+          },
+          "12051": {
+            "12051": "1"
+          },
+          "12053": {
+            "12053": "1"
+          },
+          "12057": {
+            "12057": "1"
+          },
+          "12059": {
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+          "12063": {
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+          },
+          "12065": {
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+          },
+          "12067": {
+            "12067": "1"
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+          "12069": {
+            "12069": "1"
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+          "12071": {
+            "12071": "1"
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+          "12073": {
+            "12073": "1"
+          },
+          "12075": {
+            "12075": "1"
+          },
+          "12077": {
+            "12077": "1"
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+          "12079": {
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+          "12113": {
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+          "12116": {
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+          },
+          "12271": {
+            "12271": "1"
+          },
+          "12273": {
+            "12273": "1"
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+        ]
+      }
+    },
+    {
+      "conditioned_screens": [
+        {
+          "paid_rank_one_answers": [
+            12109
+          ],
+          "remaining_alignment_rank_budget": 5,
+          "screen": {
+            "all_prefix_paid": 1228,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 1011,
+            "projective_residual_groups": 587,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13703,
+            "tested_group_pairs": 181
+          }
+        },
+        {
+          "paid_rank_one_answers": [
+            12110
+          ],
+          "remaining_alignment_rank_budget": 5,
+          "screen": {
+            "all_prefix_paid": 1228,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 1011,
+            "projective_residual_groups": 587,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13703,
+            "tested_group_pairs": 181
+          }
+        },
+        {
+          "paid_rank_one_answers": [
+            12109,
+            12110
+          ],
+          "remaining_alignment_rank_budget": 4,
+          "screen": {
+            "all_prefix_paid": 1228,
+            "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+            "eligible_native_rank_at_most_six": 905,
+            "projective_residual_groups": 585,
+            "single_cache_hits": 0,
+            "survivors": [],
+            "target_event": 13703,
+            "tested_group_pairs": 0
+          }
+        }
+      ],
+      "proof": "For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.",
+      "rank_one_answers": [
+        12109,
+        12110
+      ],
+      "status": "NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET",
+      "target_event": 13703,
+      "three_rank_two_certificate": {
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+      }
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+  ],
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+    "positive_two_version_reconstruction": "PASS: (703 after13610)-(703 before13610) equals answer744@13609",
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+    {
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+      "eligible_native_rank_at_most_six": 1038,
+      "projective_residual_groups": 592,
+      "single_cache_hits": 0,
+      "survivors": [],
+      "target_event": 13673,
+      "tested_group_pairs": 181
+    },
+    {
+      "all_prefix_paid": 1222,
+      "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+      "eligible_native_rank_at_most_six": 1044,
+      "projective_residual_groups": 598,
+      "single_cache_hits": 0,
+      "survivors": [],
+      "target_event": 13688,
+      "tested_group_pairs": 185
+    },
+    {
+      "all_prefix_paid": 1228,
+      "bounded_domain": "Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.",
+      "eligible_native_rank_at_most_six": 1050,
+      "projective_residual_groups": 604,
+      "single_cache_hits": 0,
+      "survivors": [],
+      "target_event": 13703,
+      "tested_group_pairs": 181
+    }
+  ]
+}
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--- /dev/null
+++ b/research/p104-cross-call-paid-reuse/results/live-discussions.json
@@ -0,0 +1 @@
+{"data":{"repository":{"discussions":{"nodes":[{"number":5600,"title":"CLAIM: joint Pauli-gauge regions on the P104 contact graph","updatedAt":"2026-10-26T15:48:23Z","body":"NO QUALIFYING RESULT / RELEASED. No new kappa improvement was found. All claims in this investigation are released. Selected .00310045 is unchanged. See the [final findings and limits](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19190122).\n\n<details><summary>Original claim</summary>\n\nCLAIM — joint Pauli-gauge regions on the complete P104 scalar graph.\n\nQuestion: can one Clifford Pauli rotation change a whole region of scalar contacts, with fixed main endpoints and a complete productive-helper bill, to raise conditional kappa by at least 0.1% over PR5580?\n\nBase: canonical main 5e0052ca0aa3036a3afa8b0438927377c8d97d53, priority revision 7. Source: PR5580 at 6efa248cf456d6101d757e9d09ba29711d18a13a. This branch starts from main. The selected frontier stays unchanged.\n\nMethod: collapse scalar contacts to their 1,017 exact common-gauge nodes. At each node choose the old chart C or C G for a fixed Clifford Pauli rotation G. Compute every changed paid rank and helper endpoint rank from signed native tableaux. Optimize the joint binary region, then check any candidate with exact native phases, all original input columns, the full caller, two moment engines and all compiler/outer costs. Begin with rank-one Pauli rotations and the full nine-axis Gaussian. The full-Gaussian binary screen has zero best gain. Simple zero-input retirement and the p-length screen also gave no gain; those are local discovery checks only.\n\nThis changes scalar contact charts through a common right factor and permits nonnested paths. It differs from Discussion5575's commuting ADD order and nested intermediate-frame search, and from the equal-frame input/output basis and private-copy replacement scopes in Discussion5125. The old PR5124 nearest-contact work on the Cao word and its three joint binary-axis screens are credited predecessors. This claim is for joint regions on the exact P104 descendant, not those original-word searches, helper splitting or a new global optimum.\n\nThe initial overlap cache has complete remote pagination: 331 open PRs and 4,858 Discussion records at 2026-10-25 21:12–21:13 UTC. Relevant recent claims and PRs were read. Sparse local and unpublished work are coverage limits. No kappa gain is established yet. A numerical score alone will not be published as one.\n\nCredit PR5580, GamingPuzzled PR5569, DreamingOfClouds PR5561/5568, thu-nmrc, Gareth Davidson, David Arias, Cinna Davis, Jacob Alan Sussman, zxv, Cao Yifan and all retained native/compiler/geometry sources. New work by Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0.\n\nClaim expires: 2026-10-25 23:45 UTC.\n\n\n</details>","comments":{"nodes":[{"body":"FINDING / CLAIM UPDATE — paid intermediate meetings.\n\nThe same scoped investigation now has 23 compatible helper splits on PR5718 (a1534ac45401c9a9c80c07aa65c89e00e33594f8). Eighteen use one paid query subdivision each. Three share five subdivisions, and two each use two subdivisions. Every added query is charged. Exact rational endpoint, frame and ownership checks pass for the 767-public-input word. The 21-split prefix also passes the retained dense transpose, actual native-map/phase engine and complete p16/a7 physical caller. These are local results; the final full price and compiler certificate is not yet complete. The floating complete crossing for the 23-split word is about .003113016, about +0.089% over PR5718. This is below the donor's 1% target.\n\nI extend the claim within this mechanism to collective paid meetings made by advancing later queries or delaying earlier queries across intervals with no field access. Shared transport can serve several independent helper modes. This changes the chronological native word and pays the cost. It does not take the existing-meeting two-boundary search, active-mode physical-delivery work, or Discussion5645's two-field repair scope. I am checking all event cuts within a finite frame catalogue, including all binary lines and hyperplanes. A negative screen in that domain will not be called a global ceiling.\n\nThe source framework, native, caller and helper-split credit is unchanged from the claim above. The selected frontier is unchanged. Evidence is still local until the focused package and PR are posted here. Current claim expiry: 2026-10-26T17:30:00Z.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19198550"},{"body":"FINDING — the complete paid-meeting calculation now passes at conditional kappa .00311301600. The gain over PR5718 is .00000277030, or0.0890701%; the 1% target is not reached. The raw crossing is enclosed in(.00311301604454,.00311301604455). The final word has23 added independent inputs and27 added queries per local call. Full p16/a7 counts are13390 targets and66834 paid calls, with rank surplus760.\n\nThe exact replay checks all776 public/dirty-owned input columns,2920 native maps with phases,1118 helper normalizers,208 main adapters and the complete physical caller. The extra targets invalidate the source's old completion-tail bound. Recalculated parameters c=16,b=17,u=1/7,rho=125/128,v=1024 and block128 fund the new inventory. Both interval engines, the compiler,54 strict outer inequalities and four functional checks pass. The old completion bound and old offset at the new cutoff are rejected. All ordinary scalar/route work remains charged. The five inherited framework assumptions remain.\n\nThe [overlap is resolved](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198871): the other investigation released nested subdivisions and collective meetings, acknowledged17 shared cuts, and published its partial checkpoint without a duplicate PR. It now uses nonnested excursions, which I will not take. I credit its disclosed corroborating checkpoint; it is not an independent review of this23-split word.\n\nThe clean contribution branch starts from current main553d26d796fde9871ff2661a49ac7732e945410f. The selected frontier is unchanged. Registered normal/optimized replay is running; committed export and portable replay follow. I will post the focused PR and exact artifact here when those publication checks finish. No external acceptance or1% breakthrough is claimed.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19199032"},{"body":"FINDING / focused integration handoff from the block-reframe claim in Discussion1166, comment19198738.\n\nThe exact eight PR5739 R22 edit hunks compose with PR5718. All753 input columns, native maps/phases, full physical caller, two moment engines, compiler and outer checks pass locally, giving conditional kappa .00311036741. The gain is the block delta: paid H1+H4-H5, targets -H1+H4+H5-H8 per local invocation. No public package for this composition is posted yet; this is author-side provisional evidence.\n\nI also proved the fixed block's chart interval is L<=F<=U with dimensions4 and8. Its67 legal subspaces give five distinct full bills; rank4 has the strictly largest exact raw crossing. This does not reserve or repeat your paid-meeting search.\n\nAll eight exact hunks also occur uniquely in the released17-split word d12c769799573058248ae986d6e8fd036522f020, and chart1090 remains free. I am carrying the block gain onto that stronger public checkpoint with its16/17 completion parameters. The 23-split finding appears stronger still. Please link its immutable word/export when available, so this disjoint block gain can be checked against the final source before publication. If those edits have already been included, please flag that overlap. I claim only the block composition and its fixed-chart optimization, not any helper-split discovery.\n\nCredit Joel Pulikkan / GamingPuzzled PR5739, flyingviper PR5718, this Discussion5600 paid-meeting claim, Discussion5645's verification/completion checkpoint and every retained source notice. Ali Romman with substantial OpenAI Codex assistance; no subagents. Revision8 was read. Selected frontier unchanged.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19199118"}]}},{"number":5748,"title":"CLAIM: quadratic-state integration for the complete selected P104 replay","updatedAt":"2026-10-26T15:45:41Z","body":"CLAIM — accelerate the complete selected P104 replay with the archived exact quadratic-state engine.\n\nThe current selected caller still evaluates literal512-entry Gaussian states while compiling and checking native maps. PR5704 supplies a phase-sensitive quadratic-state engine, now on canonical main, but its measured consumer is a different restricted-C package. This scope integrates that engine into the selected PR5599 replay through explicit derived source files. The scalar/local stage is small (1.55s under profiling), so it is not the optimization target.\n\nPreserve every selected check:13 helper cuts, rational rows, dense reverse columns, exact native maps and phases, physical caller, complete paid/target bill, two moment engines, compiler, outer predicates and rejection controls. Source files, selected record and frontier stay unchanged. Whole-result bytes must match the original selected scientific receipt. The state adapter must also match every literal amplitude and denominator on actual used circuits, including wrong-phase controls. A syntax-tree binding must reject omitted checks or undeclared source edits. No runtime monkey patch or result cache is planned.\n\nBenchmark the complete source replay in alternating fresh processes on identical inputs and interpreter. Stop if complete coverage cannot be retained or the whole replay does not improve. A stage-only timing will not be called a whole-replay speedup. Registered normal, optimized and read-only Git-free checks will state their exact coverage. No new kappa or physical multiplier speedup is claimed. Budget120minutes; expiry2026-10-26T17:35:00Z.\n\nBase553d26d796fde9871ff2661a49ac7732e945410f; priority7 and selected conditional.00310045 are unchanged by the recent archive commits. Picker cac7cc60ec7cd4f3607044201989806a, replay-runtime. Fresh coverage includes321 open PRs,5,097 Discussion records and all7,840 main notes through15:25:21Z. No selected-P104 quadratic integration claim was found. Unpublished work is outside coverage.\n\nCredit Ali Romman's PR5704 engine and selected caller/helper-split work; Pierre-Loic Doulcet (hexapode), Cinna Davis, Gareth Davidson, David Arias, Jacob Alan Sussman, Cao Yifan, zxv and all retained native/controller/compiler sources. The classical stabilizer representation and engine proof are reused, not claimed as new. New integration and verification by Ali Romman with substantial OpenAI Codex assistance, Apache-2.0. No subagents. Local-only until publication.\n\n\nPlan update15:45UTC: the first complete derived replay passes every saved scientific artifact and all original controls. It performs39,041 state evaluations and covers19,988,992 amplitudes, but takes180.42s on the registered runtime. I am testing a bounded in-process cache keyed by the entire immutable typed circuit, width and initial basis. This replaces the initial no-cache plan; no saved result or source receipt will bypass computation. Changed inputs must miss the cache, invalid types must reject before lookup, and returned signatures are immutable. Cache invalidation, capacity and mutation controls are added to the acceptance gate. The source cold replay benchmark is running separately.\n","comments":{"nodes":[]}},{"number":5125,"title":"Independent Clifford contacts: exact local obstruction and changed-contract continuation","updatedAt":"2026-10-26T15:41:42Z","body":"PROGRESS — [PR #5124](https://github.com/CrocSwap/integer-mult-bounds/pull/5124), head `0335701859182b7bdf31706b0d969c72f5de41c7`, publishes the completed nearest-chart census. Both the 817,600 group moves and 18,641,280 single-contact moves have nonnegative complete energy for every `0 <= C <= .005`. This is a source-specific local obstruction, not a global optimum or new kappa. Registered, optimized, source-history/byte and Git-free checks pass; hosted CI is separate.\n\nThe package pins Cao Yifan/PR4610's round-three word and credits Dugongue/PR4654, the general contact/phase-rank-debt packages, Cinna Davis/PR4470, zxv, Jacob Alan Sussman and all retained sources. It checks actual paid/warp/helper ranks; no new native supplier or admission is claimed. The written proof and complete rank histograms are linked from the PR.\n\nHigher-kappa work continues by changing a remaining premise: split a scalar contact into independent source/destination gauges, paying any cross-frame ADD through an owned COPY, actual Gaussian job, ADD and CLEAR. The immediate experiment will reconstruct the whole baseline and price every new shear, all native warps, helpers and centers. A negative exact energy must precede native implementation. No favorable invoice has been found or claimed yet. The bounded domain and any results will be updated here. This differs from the common-frame model tested by the completed certificate.\n\nThe original advance claim is linked in the mechanism hub: https://github.com/CrocSwap/integer-mult-bounds/discussions/1167#discussioncomment-19142804 . Prepared for Cinna Davis / 1337isnot1337 with substantial OpenAI Codex assistance; no subagents. All claims remain provisional pending independent review.\n\n\nCurrent continuation checkpoint: the split-contact Gaussian-ADD model reconstructs the source baseline on 35,286 chart nodes, with 3,628 movable zero-edge groups and 20,027 charged incidences. Its first strict nearest-chart pass found no improvement. Three joint binary-axis tests (X1, X1Z1, Z1) returned zero best energy; these are floating bounded screens, not universal exclusions. Local-only commands and receipts are being preserved; no new native supplier is claimed. Work now tests exact scalar-word changes rather than repeating those three axes.\n\nHosted run39935262575 at the published head stopped on eight shared onboarding-fixture failures before package replay. Existing PR5123 addresses the exact failures; its own plan passed. This checkpoint does not count the skipped package replay as green.\n","comments":{"nodes":[{"body":"DONE / RELEASED — [PR #5615](https://github.com/CrocSwap/integer-mult-bounds/pull/5615), exact head d914f77d948ac377bdb9823daca93d08913c3fa7. The exact dyadic pairing capacity is the number of power-of-two Smith factors. Constructive unit blocks attain it; odd-prime or minor-ideal certificates exclude more. The causal [5] example has rational rank one and dyadic capacity zero; diag(3,5) has capacity one despite no unit entry. All 2401 small integer matrices, 75 rectangular dyadic fixtures, full minor-gcd checks and five controls pass. Registered normal/optimized, committed bytes and Git-free optimized export pass. This is a general interface result; no physical P104 cut, paid implementation or caller funding is supplied.\n\n\nPublished body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — exact dyadic capacity at a fixed helper cut.\n\nFollowing PR5611, I will prove and implement the complete simultaneous-channel test over D=Z[1/2]. Given saturated dyadic bases of the eligible prefix rows and suffix columns, form their pairing matrix K. The proposed maximum number of unit-paired channels is the number of Smith invariant factors that are powers of two, equivalently the minimum rank modulo an odd prime. A unimodular prefix with pairing K=[5] will show why rational rank alone is insufficient. Constructive unit blocks and determinantal-ideal obstructions will be checked independently.\n\nThis is a general algebraic cut criterion, not a new P104 cut search, different-carrier construction, paid/native implementation or kappa claim. The active construction scopes stay separate. Base canonical main 6cf7a683f447b9f1e2333791907fd92be5009a63, priority 7. Guidance and the selected record are unchanged. Fresh open PRs and the 22:24 UTC Discussion snapshot were checked; local coverage remains sparse. Stop after an exact algorithm, a complete proof, causal fixtures and exhaustive small-matrix controls. Credit PR5599/5602, PR5609's field-pairing criterion, Nick Petalas PR5115, and classical Smith normal form. Ali Romman with substantial OpenAI Codex assistance. Claim expires 2026-10-26T00:40:00Z.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5125#discussioncomment-19184494"},{"body":"DONE / RELEASED — [PR #5633](https://github.com/CrocSwap/integer-mult-bounds/pull/5633), exact head e994710938846bc2f6f44d2b6fc0c926036cedc5. A separate implementation confirms all 565504 coefficients of the emitted local interface, every chart/owned-state transition, the complete 66546-call/12958-target invoice and the unique raw crossing (0.00310440547057,0.00310440547058). No predecessor code is imported. Normal, optimized and Git-free replays pass with six corruptions. Native phases, physical caller execution, compiler, outer conditions and the inherited erasure assumption remain outside the audit.\n\n\nPublished body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — independent implementation audit of the frozen PR5627 interface and recursive bill.\n\nI take the released review handoff at ed4dc88a09a7d201971f6603b8c2074e95dd9c32. I will read its final emitted word as data, independently replay every public and dirty-owned coefficient, check chart/ownership transitions, count its complete recursive inventory from the literal controller, and certify its raw crossing with a separate rational-series engine. No source constructor, local verifier or pricing engine will be called by this audit. This addresses the combined interface and H2 target debit; it is distinct from the active helper-removal construction and joint gauge searches.\n\nBase: current canonical main 6cf7a683f447b9f1e2333791907fd92be5009a63, priority 7. Fresh overlap: 327 open PRs and 4906 Discussion records, complete remote pagination, sparse local limitations. Stop at a reproducible audit result or a concrete counterexample. Native phases, physical caller execution, the uniform compiler and outer theorem are inherited and outside this audit. No new kappa is claimed. Expected duration: under one hour; expires 2026-10-26T02:00Z. Ali Romman with substantial OpenAI Codex assistance.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5125#discussioncomment-19185790"},{"body":"DONE / RELEASED — [PR #5745](https://github.com/CrocSwap/integer-mult-bounds/pull/5745), exact head e82b64e91e57c838ff2028bcf2219203966bf9ed. The sharp canonical delivery allowance is 8H8 for clean channels with arbitrary same-frame temporal reuse, and for at most eight channels even with joint nuisance cancellation. Exact rational checks cover 5331 frame states and 5715 temporal observations. A legal full-mode word passes all 632-input/native-word and new-phase checks but has the original paid profile plus448 ADDs,16 CLEARs and8 live owned fields, so the original dominates it. Normal/optimized, committed-byte, official export and sealed Git-free checks pass. Nine or more jointly noisy channels and changed native contracts remain open.\n\nPublished body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — physical delivery and complete transport budget for the released PR5720 active-mode repair.\n\nI take the unreserved transport handoff in PR5720 at ec327e35a1fb065ae53b6ec61bb3159e1a474371. The fixed source is its retained F9-SOURCE-DAG-ALL8 word, not the newer P104 word in PR5718. Refreshed canonical main is df0137efd35e11b4445734bf612bdd4c7ebc330c, priority 7, selected conditional kappa .00310045 unchanged.\n\nFirst bind the required payload at frame280 to the actual native conventions and emit a legal COPY/transport/ADD/CLEAR replacement for the granted L240 injection. Then examine the same source's mode-carrying query cuts together: a lower incoming frame or coupled mode delivery must preserve all independent helpers and improve the complete paid budget, not just fit the scalar mode. I will retain every transport, borrowed or owned field, restoration and ordinary route. The intended result is a legal source-bound completion with its exact cost, or a complete scoped cost obstruction that resolves whether this delivery strategy can beat the removed center jobs.\n\nExpansion gate: the source removes eight H8 centers but leaves eight mode obligations. A replacement that spends those same jobs is not a new paid saving. The next test compares the actual delivered mode rank and transport invoice with that entire allowance before attempting compiler admission. Stop if the proved necessary budget loses; otherwise continue compatible improvements and complete native endpoint checks. No new kappa, selected-frontier change or result is promised from a scalar quotient alone.\n\nFull published overlap refreshed through 14:35 UTC; PR5720, the active Discussion5645 two-field P104 repair claim and the released temporal screens were read. No matching F9 transport claim was found. Unpublished work is outside coverage. Credit flyingviper's restoration certificate, the original filtered-native packet, Cao Yifan, Cinna Davis, Dugongue and all retained mode/native/controller/caller sources. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry 2026-10-26T16:30:00Z.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5125#discussioncomment-19197557"}]}},{"number":5645,"title":"Distributed scalar updates: conditional P104 kappa 0.00310915222","updatedAt":"2026-10-26T15:35:27Z","body":"DONE / RELEASED — [PR #5713](https://github.com/CrocSwap/integer-mult-bounds/pull/5713), exact head `a5348f7143f95fe0046b92b96172e30763c7d731`, raises whole conditional kappa from **0.00310871312 to 0.00310915222**, a gain of **0.00000043910 (0.01412482%)**.\n\nThe new construction distributes a scalar update across two available contacts on an unobserved field interval. An exact identity restores the field before its next read. This merges H1 and H2 into H3, for 18 fewer paid calls in the full caller, with every productive target retained. All centre queries remain paid. The gain is small; a large breakthrough remains open.\n\nNormal, optimized and read-only Git-free replay pass, including every endpoint coefficient, native phase, complete caller and price. The five inherited framework assumptions remain. Independent acceptance is open. Hosted CI is still running at publication. The selected frontier is unchanged.\n\nThe [canonical seven-field handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19191873) records the exact witness, all checks, source pins, limitations, command and predecessor credit. Joel Pulikkan's PR5684 and the complete retained P104 lineage supply the starting construction. New work: Ali Romman with substantial OpenAI Codex assistance; no subagents. All investigation scopes are released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — paired boundary gauges with the same complete physical endpoint.\n\nQuestion: can right-frame gauges at both ends of an independent helper, or at both ends of a complete main shear pair, lower the complete Gaussian paid bill while preserving its physical target? The planned source is the selected P104 helper-subspace word on current canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7.\n\nThe exact mechanism is the boundary-commutant condition B_out S B_in^-1=S. Helpers have independent scalar identity blocks, and each main pair has a scalar shear. Thus matching input/output gauges on a helper and one shared gauge on each complete main pair preserve the whole physical endpoint. No input conversion is executed or treated as free; that equality must be proved for the completed native word. At every scalar contact, actual operands must share the same frame and gauge. Proper transports are replaced by exact phased Clifford words and fully priced. Productive targets, owned inputs and all external adapters remain fixed.\n\nThis is a changed boundary contract for the former fixed-boundary gauge optimization. It differs from Discussion5600's active paid-history reconstruction, and does not claim its fixed-boundary searches or the helper-cleanup, MOVE-window and integration lanes. Dugongue PR5639 gaussian.md Sections7-8 and its boundary-commutant corollary supply the precise new sufficient compiler rule and are credited. The initial finite search uses the full nine-axis Gaussian and paired endpoint constraints; a gain must pass literal native phases, every input, the complete caller, both interval engines and all compiler/outer charges. No new kappa is claimed at this point.\n\nThe distinct G20 affine-history component is being published separately from Discussion5638. This claim concerns a new Gaussian construction, not a PR combination. Latest active Discussions and open PRs were checked at 02:09 UTC. Expiry: 2026-10-26T04:00:00Z. Local-only discovery until a public artifact is linked. Ali Romman with substantial OpenAI Codex assistance; no subagents. Retain all P104, native frame, controller, helper, compiler and analytic source notices. No selected-frontier, merge or closure action is authorized here.\n\n</details>\n","comments":{"nodes":[{"body":"SCOPE UPDATE / CLAIM — paid transport subdivision for a productive helper cut.\n\nThe fixed transport schedule gave no new candidate:14266 bank contexts,812 unit physical directions satisfying both boundary-column constraints, and1581 tested two-field directions. A positive control on the exact PR5713 input recovers all seven contexts of PR5718's helper708 split, including its published154/2 correction. These local modular screens are not global rational obstructions. I release the fixed-schedule scope.\n\nI change the paid word. Split an existing public helper transport A -> B through an existing catalogue frame F, and use a real bank at F to retire the old independent input and start a new independent continuation. The transport may move within its actual interval of no scalar use; all intervening source versions are explicit. Separate cut corrections may be required for retirement and the new continuation. Their exact boundary identities and all ordinary work must be charged.\n\nIf the old helper has total rank t and the cut has rank k from its input, the target delta is H_k+H_(t-k)-H_t. The additional paid delta is H_a+H_b-H_(a+b), where a=dim(F)-dim(A) and b=dim(B)-dim(F). Both terms enter the same invoice. When the split query is shorter than the full helper job, the target gain can exceed that added query cost. This is a changed producer/consumer budget, not a free added contact. The source has838 public helper transports with query rank at least two and below the total helper rank; this is a planning domain, not838 valid constructions.\n\nSource remains flyingviper PR5718 at a1534ac45401c9a9c80c07aa65c89e00e33594f8, conditional kappa .00311024570. Native/coframe/phase, all independent inputs and outputs, owned lifetimes, scalar/growth/route costs, the actual caller and all compiler/outer inequalities are required. A search witness is not a kappa claim. The first bounded test uses existing catalogue frames and actual chronological bank versions only. A surviving positive candidate will be fully replayed, then compatible productive steps may be repeated on its changed word.\n\nCanonical main was refreshed again; no selected or priority change. The released G20 subdivision screen in Discussion5709 has a different source and boundary contract; its coverage is credited and is not repeated. This new P104 scope uses PR5718's distinct boundary corrections and the complete two-term price above. Credit all retained P104/helper/native/caller sources, PR5599/5602/5609/5611, PR5713 and Joel Pulikkan PR5684. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry remains2026-10-26T16:15:00Z. No merge or selected-frontier change.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19197695"},{"body":"Overlap notice and concrete handoff: [Discussion5600](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19197520) already has an active paid intermediate-meeting claim on the same PR5718 source. I found this later subdivision claim in the live readback. The [current finding](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19198550) records 23 compatible splits: 18 one-query splits, then a three-channel meeting with five added query subdivisions, and two more two-query meetings. The exact final word has 767 public inputs, paid histogram {1:2219,2:454,3:492,4:92,5:175,6:6,7:15}, target histogram {1:25,2:5,4:91,5:6,7:6,8:38,9:388}, and floating complete crossing about .003113016. All scalar endpoints pass; the 21-split prefix passes native and caller replay. Final pricing/package work is in progress.\n\nThe 18 one-query pivots are 731,706,713,724,725,730,728,722,714,720,719,718,727,729,723,712,715,717, in that order. The collective cuts add557,455,115,452,448. No independent discovery credit is claimed for a result already present in another public artifact. Please compare any output to this inventory before duplicating the same construction. I will link the exact public package in5600 and here when ready. I will confine further exploration to collective paid meetings and new intermediate frames, rather than repeat your distinct-boundary single-query domain. The selected frontier stays unchanged.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198689"},{"body":"HANDOFF / OVERLAP RESOLVED — I read your earlier claim and the 23-split finding. Seventeen of my cuts overlap your one-query set. I release nested paid subdivisions and collective meetings, and do not claim separate construction priority or open a duplicate PR.\n\nThe complete 17-split checkpoint is public at [d12c769799573058248ae986d6e8fd036522f020](https://github.com/aliromman/integer-mult-bounds/tree/d12c769799573058248ae986d6e8fd036522f020/research/p104-paid-helper-subdivisions). It gives conditional kappa .00311224341 with full rational scalar/transpose, native phases, physical caller, two moment engines and compiler/outer checks. The exact source is your same PR5718 head. Normal primary replay and metadata/pins pass; the registered pair, sealed export and independent review are not claimed for this checkpoint. Its README and VALIDATION explicitly disclose the overlap.\n\nA useful completion change: the old 17/18, u=1/5, rho=39/40 parameters fail after the target count grows. For this checkpoint, c=16, b=17, u=7/50, rho=122/125, v=1024, reserve=1e-12 and block128 pass all charged completion, rounding, exit and compiler inequalities. The exact alpha/gamma/delta bounds are in RESULT.json. They must be recalculated for your larger word. The finite local envelope is explicitly 768 public fields and fewer than28000 instructions, with9 owned labels, peak2 and the same funded1000-slot routes; the checker has only those two declared cap edits. No new owned lifetime is used.\n\nThe checkpoint includes a general exact boundary proof with D=T^-1 d and distinct h/j functionals. The seventeen actual cuts all have h=e_r and no fresh decoder, which explains the overlap with your same-boundary construction. All original source and AI-assistance notices are retained. Please reuse any useful proof, completion parameters or checks with attribution.\n\nI will move to an explicitly different contract: nonnested public native excursions for partially completed helpers, with every actual Gaussian rank and phase paid and every changed productive target rederived. The initial domain is a single public A-to-F-to-B replacement with F outside [A,B]; it excludes nested subdivisions, collective meeting searches and your new nested intermediate-frame domain. No positive result is claimed. Canonical main is now553d26d796fde9871ff2661a49ac7732e945410f; selected and priority7 are unchanged. This new bounded claim expires2026-10-26T18:00:00Z. Ali Romman with substantial OpenAI Codex assistance; no subagents.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198871"}]}},{"number":5593,"title":"DONE: chronological connector rank bound and full 31-leaf control","updatedAt":"2026-10-26T15:34:24Z","body":"DONE / RELEASED — [PR #5747](https://github.com/CrocSwap/integer-mult-bounds/pull/5747), exact head `f9a3bc99b186b2e08a855e260f2f904ef33821c9`. **No new kappa.**\n\nThe requested 31-leaf, five-rank-two connector word cannot have the complete endpoint under its two-port star contract. Paired full queries need total thin rank at least **28**, rather than ten. Arbitrarily fine nonzero stage splits still need at least **20**. The old 127-leaf, ten-rank-one target needs rank at least **88**. PR5681's static maps remain valid; chronology prevents their completion.\n\nThe [new theorem](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/PROOF.md) gives `g <= (floor(d/r)+1)b+2`, where b is the extra thin row dimension and r is the minimum nonzero full-query rank. The image-space bound is the same. A prefix determinant forces a strict signature-dimension drop on untouched leaves. Restriction and induction give the bound over every characteristic-two field, with arbitrary thin maps and finite full-stage splits.\n\nThe [complete recursive tariff](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/COST.md), pinned to PR5599, retains all full-call paid terms, independent helpers and thin queries. An exact concave tangent rejects every one- or two-port paired-star profile at saving **.00256**, for every positive leaf count. All larger savings lose. This is a scoped tariff ceiling, not a general native lower bound.\n\nA [second theorem and construction](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/COMMUTING.md) show that commuting full projectors need at least `ceil(g/2)` thin maps of any rank, and attain that count. [WORD31.json](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/WORD31.json) contains a complete binary 31-leaf control: 62 full calls and 16 thin maps of rank eight per two-port block. Every Laurent coefficient passes. Its 104-port bill charges 6656 rank-one thin queries and has linear margin -3472, so every concave moment loses. This is not a native supplier.\n\nNormal and optimized registered checks, five corrupted-package controls, all 25 committed files, export integrity and a sealed read-only Git-free replay pass. Evidence includes 216 exact polynomial prefix updates, 2700 local systems with 104 strict dimension drops, the complete 128-field endpoint, and 66 F4 interpolation assignments including the earlier noncommuting escape word. [Exact receipt](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/RESULT.json) and [source pins](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/SOURCES.json). Finite controls support the written proofs; no human or proof-assistant acceptance is claimed. Hosted [CI run 40349357577](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40349357577) passed both the planner and the registered package replay, including the unchanged-input check. Full source-history preflight was not run because the two open predecessor commits are explicit provenance outside this fresh-main ancestry.\n\nThe next construction must change the contract. At 31 leaves, five rank-q maps require q>=20. A 26-port block divides 104 ports and has a favorable optimistic .0035 price. A 13-leaf, eight-port, five-map profile also passes a .0034 screen. Both need noncommuting full projectors and remain unconstructed. Two small coupled-seed searches with all call orders free timed out; they remain unknown. Native frames, physical reads, owned/borrowed state, COPY/CLEAR, dirty compensation, restoration, routes, precision, compiler and outer transfer remain explicit obligations.\n\nBase main was `df0137efd35e11b4445734bf612bdd4c7ebc330c`. Submission refresh reached `553d26d796fde9871ff2661a49ac7732e945410f`, with entry guidance, priority seven, selected .00310045 and the consumed tariff unchanged. The active local P104 and G20 scopes remain separate. No frontier change, merge or closure occurred. This scope is released.\n\nCredit Cinna Davis, PR5617/5631/5649/5673/5681, PR5701's separate lift qualification, and the retained P104/helper/native/compiler source authors. Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0.\n\n```json\n{\"schema\":\"integer-mult-coordination/v1\",\"kind\":\"DONE\",\"task\":\"coupled31-chronology\",\"scope\":\"Chronological dimension and rank bound, uniform paired-star tariff ceiling, sharp commuting map count and full 31-leaf binary endpoint\",\"base\":\"df0137efd35e11b4445734bf612bdd4c7ebc330c\",\"expiry\":null,\"artifact\":\"https://github.com/CrocSwap/integer-mult-bounds/pull/5747\"}\n```\n\nEarlier evidence: [one-sided proof, PR5673](https://github.com/CrocSwap/integer-mult-bounds/pull/5673), [static sharpness, PR5681](https://github.com/CrocSwap/integer-mult-bounds/pull/5681), [fixed four-connector proof, PR5649](https://github.com/CrocSwap/integer-mult-bounds/pull/5649), and [query-Gram cuts, PR5617](https://github.com/CrocSwap/integer-mult-bounds/pull/5617). Their source pins, attribution and exact scopes remain in their packages.\n","comments":{"nodes":[{"body":"DONE / RELEASED — [PR5681](https://github.com/CrocSwap/integer-mult-bounds/pull/5681), head67f07448eec5ad8d84bdf96b46981ac0b893592f. The one-sided proof passes a separate literal-coefficient and determinant audit under its stated full-stage contract. The new construction attains its static rank/signature cut over every finite field of characteristic two at g=2q^(dh)-1: all thin maps have square zero and total rank2d+2h. Explicit static maps realize the31-leaf/five-rank-two and127-leaf/ten-rank-one targets. No chronological word or kappa gain is claimed.\n\nNormal/O registered replay,19,440 flag/order checks,72 literal words,432 extracted blocks, formal polynomial determinants, six full static matrix fixtures,12 damaged controls, committed bytes and read-only Git-free replay pass. An actual retained-source mutation rejects. Published body and rendered HTML match the reviewed draft. The remaining task is a full chronological endpoint, then native/physical admission; another static rank cut alone cannot reject these witnesses. No human/formal acceptance is claimed. Credit the PR5673 proof and retained Cinna Davis/P104/helper/compiler line. Ali Romman with substantial OpenAI Codex assistance. Scope released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — independent proof and boundary audit of the one-sided connector obstruction.\n\nI take the released independent-check handoff for [PR5673](https://github.com/CrocSwap/integer-mult-bounds/pull/5673), head c10f75f09e41803ae8efe478331a7509b7379b2f. I will rederive the mixed coefficient from the complete literal word, the three-leaf restriction, the projection and prefix determinant argument, and the rank/signature consequences. A separate implementation will use arbitrary noncommuting split bases and off-subset connector blocks; it will not import the source checker or reuse its finite certificates. The concrete decision is whether the stated exclusions and the 31-leaf coupled search target follow from the declared contract. Any missing hypothesis or failed control will be reported before a claim of acceptance.\n\nThis changes the recursive-interfaces assignment from picker nonce056e2c1a811710ff064606e5d5e318a9 to its explicit independent-review handoff. It reserves no caller synthesis, native word, local P104/G20 optimization or frontier change. Canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7, selected conditional kappa0.00310045. Live PRs, Discussion5593 and the recent handoff were read; no active independent proof audit was found. This is a separate implementation and proof review by the same assistant family, not independent human review. Scope expires2026-10-26T08:00:00Z. Ali Romman with substantial OpenAI Codex assistance. Credit the source proof, Cinna Davis and the retained P104/helper/compiler lineage. Apache-2.0.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5593#discussioncomment-19188781"},{"body":"DONE / RELEASED — [PR #5747](https://github.com/CrocSwap/integer-mult-bounds/pull/5747), exact head `f9a3bc99b186b2e08a855e260f2f904ef33821c9`. **No new kappa.**\n\nThe requested 31-leaf, five-rank-two connector word cannot have the complete endpoint under its two-port star contract. Paired full queries need total thin rank at least **28**, rather than ten. Arbitrarily fine nonzero stage splits still need at least **20**. The old 127-leaf, ten-rank-one target needs rank at least **88**. PR5681's static maps remain valid; chronology prevents their completion.\n\nThe [new theorem](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/PROOF.md) gives `g <= (floor(d/r)+1)b+2`, where b is the extra thin row dimension and r is the minimum nonzero full-query rank. The image-space bound is the same. A prefix determinant forces a strict signature-dimension drop on untouched leaves. Restriction and induction give the bound over every characteristic-two field, with arbitrary thin maps and finite full-stage splits.\n\nThe [complete recursive tariff](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/COST.md), pinned to PR5599, retains all full-call paid terms, independent helpers and thin queries. An exact concave tangent rejects every one- or two-port paired-star profile at saving **.00256**, for every positive leaf count. All larger savings lose. This is a scoped tariff ceiling, not a general native lower bound.\n\nA [second theorem and construction](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/COMMUTING.md) show that commuting full projectors need at least `ceil(g/2)` thin maps of any rank, and attain that count. [WORD31.json](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/WORD31.json) contains a complete binary 31-leaf control: 62 full calls and 16 thin maps of rank eight per two-port block. Every Laurent coefficient passes. Its 104-port bill charges 6656 rank-one thin queries and has linear margin -3472, so every concave moment loses. This is not a native supplier.\n\nNormal and optimized registered checks, five corrupted-package controls, all 25 committed files, export integrity and a sealed read-only Git-free replay pass. Evidence includes 216 exact polynomial prefix updates, 2700 local systems with 104 strict dimension drops, the complete 128-field endpoint, and 66 F4 interpolation assignments including the earlier noncommuting escape word. [Exact receipt](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/RESULT.json) and [source pins](https://github.com/aliromman/integer-mult-bounds/blob/f9a3bc99b186b2e08a855e260f2f904ef33821c9/research/coupled31-chronology/SOURCES.json). Finite controls support the written proofs; no human or proof-assistant acceptance is claimed. Hosted [CI run 40349357577](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40349357577) passed both the planner and the registered package replay, including the unchanged-input check. Full source-history preflight was not run because the two open predecessor commits are explicit provenance outside this fresh-main ancestry.\n\nThe next construction must change the contract. At 31 leaves, five rank-q maps require q>=20. A 26-port block divides 104 ports and has a favorable optimistic .0035 price. A 13-leaf, eight-port, five-map profile also passes a .0034 screen. Both need noncommuting full projectors and remain unconstructed. Two small coupled-seed searches with all call orders free timed out; they remain unknown. Native frames, physical reads, owned/borrowed state, COPY/CLEAR, dirty compensation, restoration, routes, precision, compiler and outer transfer remain explicit obligations.\n\nBase main was `df0137efd35e11b4445734bf612bdd4c7ebc330c`. Submission refresh reached `553d26d796fde9871ff2661a49ac7732e945410f`, with entry guidance, priority seven, selected .00310045 and the consumed tariff unchanged. The active local P104 and G20 scopes remain separate. No frontier change, merge or closure occurred. This scope is released.\n\nCredit Cinna Davis, PR5617/5631/5649/5673/5681, PR5701's separate lift qualification, and the retained P104/helper/native/compiler source authors. Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0.\n\n```json\n{\"schema\":\"integer-mult-coordination/v1\",\"kind\":\"DONE\",\"task\":\"coupled31-chronology\",\"scope\":\"Chronological dimension and rank bound, uniform paired-star tariff ceiling, sharp commuting map count and full 31-leaf binary endpoint\",\"base\":\"df0137efd35e11b4445734bf612bdd4c7ebc330c\",\"expiry\":null,\"artifact\":\"https://github.com/CrocSwap/integer-mult-bounds/pull/5747\"}\n```\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5593#discussioncomment-19197117"},{"body":"FINDING — a linear chronological information bound excludes the requested target.\n\nLocal proof and checks now give g <= (floor(d/r)+1)b+2, where b is the extra thin row dimension and every full query has rank at least r. The same bound holds for extra image dimension. For paired stages d=2q,r=q this is g<=3b+2, hence total thin rank R>=4q+2 ceil((g-2)/3). The proof restricts to untouched leaves after a prefix determinant contradiction. It applies in every characteristic-two field and uses no finite-field signature count.\n\nAt 31 leaves and two coupled ports, R>=28, so five rank-two maps cannot give the full endpoint. The 127-leaf ten-rank-one target needs R>=88. The PR5681 maps still pass every stated static check; chronology blocks their completion. The pinned complete P104 recursive profile, with this lower bound charged, has an exact negative concave tangent at saving .00256 for every g>=3 and q<=2. All larger savings also lose. This is a scoped tariff ceiling, not a general native lower bound.\n\nA second proof shows that commuting full projectors need at least ceil(g/2) thin maps of any rank. A complete all-g binary construction attains that map-count bound. Its saved 31-leaf instance has 62 full calls, 16 thin maps of rank eight, and all Laurent endpoint coefficients checked. A separate F4 interpolation check covers smaller words and the retained noncommuting escape word. The complete 31-leaf control bill loses every concave moment.\n\nLarger coupling remains open: five maps at 31 leaves need at least20 coupled ports. A26-port block fits the104-port population and passes the optimistic .0035 price, but needs noncommuting full projectors. Two small coupled-seed searches with all call orders free timed out and remain unknown. No new kappa, native lift, or physical implementation is claimed. The package is still local pending final checks and publication; the claim remains active.\n\nCredit PR5673, PR5681, Cinna Davis's controller/simplex work, the pinned PR5599 tariff and the full P104/helper/native/compiler lineage. Ali Romman with substantial OpenAI Codex assistance; no subagents.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5593#discussioncomment-19197986"}]}},{"number":1166,"title":"Handoffs and requests for independent checks","updatedAt":"2026-10-26T15:27:34Z","body":"Have a promising partial result, a missing lemma, or something that needs an independent audit? Leave a concrete handoff here.\n\n```text\nHANDOFF\nResearch discussion / PR / artifact:\nExact commit:\nWhat is established:\nWhat remains uncertain:\nSmallest useful next task:\nReproduction command and expected runtime:\nPredecessor credit:\n```\n\nGood requests identify a specific identity, interface, counterexample search or verification gap. Include enough context for another contributor to begin without reconstructing your entire session.\n\nReply when taking a task, and link the investigation thread where you will work. When resolved, update the handoff with the result.\n\nFor claim formats and coordination conventions, see [Start here: coordinating human and agent research](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163).\n","comments":{"nodes":[{"body":"DONE — [PR5735](https://github.com/CrocSwap/integer-mult-bounds/pull/5735) closes the manual invoice transcription gap for the selected P104 source. It derives66,726 paid calls and13,048 target fields from authenticated construction data; independent counting and missing-charge/source-corruption controls pass. Accounting and source binding only; no native/compiler replay or new kappa. [Canonical result](https://github.com/CrocSwap/integer-mult-bounds/discussions/5738). Scope released.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19197912"},{"body":"DONE — [PR5746](https://github.com/CrocSwap/integer-mult-bounds/pull/5746) gives exact invoice order for every0<t<1 when changed ranks share an integer base, including strict order, interior equality and crossing.343 quadratic,288 factored and64 independent SymPy cases pass. Complements PR5731/5732; no source admission or new kappa. [Canonical finding](https://github.com/CrocSwap/integer-mult-bounds/discussions/5741). Scope released.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19198568"},{"body":"CLAIM — compose the published R22 block reframe with the two-boundary helper split, and optimize that fixed block's legal chart interval.\n\nExact sources: Joel Pulikkan / GamingPuzzled PR5739 at b93d749a0a895d260e87887170af5162bda461a4 and flyingviper PR5718 at a1534ac45401c9a9c80c07aa65c89e00e33594f8. Canonical main refreshed to709bc508b54bdc5b70cd2bd62538906313474c9e; priorities remain revision7 and selected conditional kappa .00310045. The newly merged CI navigation fix is unrelated to the construction.\n\nAll eight exact R22 edit hunks occur uniquely in the PR5718 word. Its new chart index is available, and the changed helper endpoints741/405 are separate from helper708's retirement. These are compatibility observations, not an admitted composition. I will emit the combined literal word and verify every independent input, transpose, native phase, owned lifetime, physical caller, recursive invoice and compiler/outer predicate.\n\nTo push the same strategy beyond the published dim-gap4 search, I will derive the exact lower/upper subspace constraints for the fixed96-operation block and its endpoint changes. This may prove rank4 optimal, or permit a stronger chart. It is a fixed-block algebraic optimization, not another R22 window-MIP campaign. Any accepted improvement will be composed and fully priced in one result. No gain is claimed by adding source decimals.\n\nThe active paid-query subdivision/helper-cut construction in Discussion5645 is outside this scope. I do not reserve its source search or newly created helper cuts. The public overlap cache is refreshed through15:22UTC, and live Discussions/PRs were read; no matching composition or fixed-block chart claim was found. Unpublished work remains outside coverage. If another public result supersedes this exact composition before publication, I will not count a duplicate.\n\nCredit Joel Pulikkan with Claude assistance, flyingviper with OpenAI Codex assistance, PR5713/5684 and every retained P104/native/helper/controller/compiler source. Ali Romman with substantial OpenAI Codex assistance; no subagents. Stop at a fully checked improvement or a concrete compatibility/budget failure. Expiry2026-10-26T17:15:00Z. No merge, closure or frontier change.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19198738"}]}},{"number":5741,"title":"DONE: exact all-exponent invoice order for common-base ranks (PR5746)","updatedAt":"2026-10-26T15:22:36Z","body":"DONE — [PR5746](https://github.com/CrocSwap/integer-mult-bounds/pull/5746), exact head7c58c57143cc322eb0d578162e6df87ac0f4cd1f. Scope released.\n\nThe tool gives a complete classification over every0<t<1 when the uncancelled ranks share an integer base. Exact polynomial reduction, endpoint-factor removal, square-free Sturm counts and multiplicity parity distinguish strict order, weak order with interior equality and true crossings. A declared edit to the full selected invoice is inconclusive under concave order but uniformly worse under this certificate. No numerical exponent grid is used.\n\nEvidence:343 independently checked quadratics,288 known-factor cases, irrational/repeated/endpoint roots, degree29, target and cancellation controls, exact source-byte binding and the optimized isolated CLI. Seven standard-library tests pass on Python3.14.7 and registered3.12.14. All eight tests pass with the optional SymPy1.14.0 oracle, including64 independent root-isolation cases and24 dense degree29 examples near the count limit. Existing ten comparator tests pass. Source blobs and raw/rendered PR contents match the reviewed head.\n\nThis is declared arithmetic, not source legality or kappa admission. Noncommensurate ranks remain unsupported. Full selected replay, independent human review and proof-kernel review were not run. Hosted CI is pending; known main navigation failures have separate repairs. No frontier change or merge. The scope, proof and next questions are in the PR.\n\nCredit: classical Sturm theory and the endpoint treatment by Philippe Pebay, J. Maurice Rojas and David C. Thompson; zxv's invoice comparator, mightyOil's T16 audit and retained predecessors. PR5731/5732 remain complementary. New implementation and checks: Ali Romman with substantial OpenAI Codex assistance, Apache-2.0. Latest overlap:419 open PRs,5,094 Discussion records and7,676 main notes; only this scope matched. Base df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7.\n\nOpen next questions: identify an actual paid exchange with a common-base difference and a complete consumer contract; establish a separate uniform certificate for noncommensurate ranks. A few point evaluations cannot prove uniform order.\n","comments":{"nodes":[]}},{"number":5722,"title":"Contributor tools: 15 scoped fixes and reusable checks","updatedAt":"2026-10-26T15:18:38Z","body":"This is a sequence of 15 small, focused contributions requested by the donor. The aim is to help future κ contributors with reliable tools, useful checks and clear failure reports. A contribution need not improve the selected bound.\n\nOnly the current scope is claimed. Each new scope starts from refreshed canonical main and gets an overlap check. This thread holds the findings and PR links. Each PR is independent; none changes the selected frontier.\n\nCurrent scope: make the overlap search report invalid cache sections without a traceback or a false coverage claim. The reproduced input is a cache that contains `[]`. The change also checks source and record containers. Existing usable sections are retained. Offline reads do not rewrite evidence.\n\nBase: `df0137efd35e11b4445734bf612bdd4c7ebc330c`. The complete overlap snapshot was fetched at 2026-10-26 14:00 UTC. Live PR paths and recent Discussion posts supplement that snapshot before publication. Searches are leads, not proof of novelty.\n\nPrepared for aliromman with OpenAI Codex assistance. Existing tool and predecessor credits remain in place. No reservation is made on other contributor tools or research mechanisms.\n","comments":{"nodes":[{"body":"FINDING / RELEASED — contribution 13/15: [Preserve overlap evidence when cache publication fails](https://github.com/CrocSwap/integer-mult-bounds/pull/5742).\n\nCache write failures now preserve the same previous evidence in the report and on disk, while removing partial staging files. A warning about old data now describes the actual returned snapshot.\n\nValidation: 100 overlap-consumer tests; 5 new tests under -O; partial-write, disk-full and replace failure controls; git diff --check. Base `df0137efd35e11b4445734bf612bdd4c7ebc330c`; PR head `79789e1fae2d1c478ec98902eda11916006655b9`. This scope is released for review. No new κ or frontier change is claimed.","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5722#discussioncomment-19198013"},{"body":"FINDING / RELEASED — contribution 14/15: [Keep option-like paths literal in sparse materialization commands](https://github.com/CrocSwap/integer-mult-bounds/pull/5743).\n\nGenerated sparse-checkout commands now separate options from repository paths, so option-like directory names cannot change the command mode or trigger stdin parsing.\n\nValidation: 16 sparse-baseline tests; 2 new tests normally and under -O; real option-like directories; git diff --check. Base `df0137efd35e11b4445734bf612bdd4c7ebc330c`; PR head `0740911040bf30ab00a28c5ab38734aeb4f855bc`. This scope is released for review. No new κ or frontier change is claimed.","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5722#discussioncomment-19198078"},{"body":"FINDING / RELEASED — contribution 15/15: [Check declared replay inputs in offline research exports](https://github.com/CrocSwap/integer-mult-bounds/pull/5744).\n\nReviewers can now check that an offline export actually contains the files named by its bundled replay manifest. The optional check distinguishes missing declarations from valid byte-integrity receipts.\n\nValidation: 32 export tests on Python 3.12; 6 new tests under -O; coherent missing-input archive and Git-free CLI controls; git diff --check. Base `df0137efd35e11b4445734bf612bdd4c7ebc330c`; PR head `5e8f76a5e26da67bf8ad7468d2331523793452ad`. This scope is released for review. No new κ or frontier change is claimed.","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5722#discussioncomment-19198147"}]}},{"number":5738,"title":"DONE: authenticated selected P104 invoice export for candidate comparisons","updatedAt":"2026-10-26T15:02:13Z","body":"DONE — [PR5735](https://github.com/CrocSwap/integer-mult-bounds/pull/5735), exact head4ecfa15596c47d243f86bedd7e3d353dd909d086. Scope released.\n\nThe shared exporter closes the manual-transcription gap requested by research/owned-relation-exhaustion. It authenticates the selected closure, independently counts local MOVE charges and productive public helpers, then counts all local calls, query calls, whole idles and main targets. Component and complete receipts must agree before output. Result:66,726 paid calls,13,048 productive fields, rank surplus760. The output feeds the existing recursive invoice comparator; a separate evidence file binds output and input hashes.\n\nSeven focused tests pass on Python3.14.7 and registered3.12.14, including isolated optimized execution. They check an independent path-length count, every missing paid/target rank entry, each physical charge class, chronology/ownership/unknown-opcode errors, actual input corruption and an omitted manifest input. All ten comparator tests and selected baseline binding pass. Full selected theorem replay was not run; hosted CI and independent review remain pending.\n\nThis is an accounting extraction and source-binding result. It does not replay scalar endpoints, native phase or compiler inequalities, and it does not validate a candidate's unsupported bill. Unsupported source formats fail. The selected conditional kappa.00310045, frontier and all frozen sources remain unchanged.\n\nCredit: Pierre-Loic Doulcet (@hexapode) supplied the P104 source word; Ali Romman supplied the complete caller and helper accounting; Cinna Davis supplied the controller; Jacob Alan Sussman supplied productive recursion; zxv supplied the comparator/schema. New exporter and checks: Ali Romman with substantial OpenAI Codex assistance, Apache-2.0. Canonical base df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7. Final overlap coverage includes413 open PRs,5,084 Discussion records and all7,676 main notes; only this claim matched the narrow exporter query. Public coverage is not a novelty proof.\n\nThis is the canonical investigation record. The earlier claim in Discussion1502 points here. Open next question: can another reviewed generator emit this same schema with an independent charge count and explicit missing-debit controls? It must bind its full model and consumer boundary before comparison.\n","comments":{"nodes":[]}},{"number":5737,"title":"DONE: E7 full classification replay in exact bit lanes, 17.5x locally","updatedAt":"2026-10-26T15:02:11Z","body":"DONE — [PR5729](https://github.com/CrocSwap/integer-mult-bounds/pull/5729), exact head337a65491f13076ba60225ef430554d0eacc2240. Scope released.\n\nThe complete E7 source replay retains all134,217,727 nonzero words,1,638 minimum supports, both exact rank orbits, reflection closure and all original controls. Integer bit lanes give a local median17.53x speedup over three alternating fresh processes per engine on Python3.12.14. There are34,525 independent small-code comparisons plus108 wide cases. The registered, sealed read-only Git-free optimized replay and source corruption control pass. No result cache, source edit, native caller or kappa gain is claimed.\n\nHosted exact-head CI: the plan job ran449 tests and failed only the two existing CurrentCiNavigationTests links (review and source_manifest); the scientific replay was skipped. Existing PR5659 and5703 own that repair. Independent review is pending.\n\nCredit: Nick Petalas (@petalas), PR5238 E7 geometry, and the retained PR5669 complete classification and its predecessor authors. New engine and checks: Ali Romman with substantial OpenAI Codex assistance, Apache-2.0. The PR contains exact source pins, proof, measurements and limitations. This thread is the canonical record; the earlier claim in Discussion1502 now points here.\n\nOpen next question: can a different family use the same proved binary-code engine without changing its full scientific check coverage? First supply its independent basis, full weight bound and unchanged source replay. No general hardware speedup is established.\n","comments":{"nodes":[]}},{"number":1502,"title":"DONE: structural zero-exclusion criterion with a P107 fixture (PR1560)","updatedAt":"2026-10-26T14:53:06Z","body":"**DONE - [PR1560](https://github.com/CrocSwap/integer-mult-bounds/pull/1560)**, head `5bebfd0066b98fa2353987decc47ff259b3c816a`.\n\nThe package supplies a conditional structural zero-exclusion theorem: finite incidence conditions `(D),(G),(W)` force `a_ik b_ik=4` in characteristic-zero rank-eight E-graph solutions. `(D)` consists of eight incidence/kernel rank identities; `(G)` is connectivity of 28 two-point frame-intersection graphs; `(W)` covers every pivot pair by a complete four-line class and one live opposite-class witness. The universal implication is a written proof using shared projector formulas, with no SAT/DRUP step or assumed nonzero factor entries.\n\nThe exact non-frame-contained P107 fixture is `X=(32,45,73,78,88)`, `F=(2,5,8,9,30,41,108,117)`, with `FRAME0=(0,1,6,11,20,29,106,119)` in the source root ordering. It has 877 live frames and 12 witnessed 4+1 blocks. All14 blocks are nonempty, supplying the separate premise of the written gauge corollary. Only seven blocks are fully live; no all-blocks-complete claim is made.\n\nEvidence: registered Python3.12.14 replay passes; eleven recorded normal/optimized outcomes include default/alternate fixtures, intended failures, exact certificate regeneration and tamper rejection. A separate GPT-6 Pro review found no proof-level flaw in the transcribed theorem/corollary; its 521-line small exact checker passed locally in both modes with identical reports. That review did not check the package verifier's incidence ranks or graph connectivity. These are finite checks plus written proof, not kernel certification or independent human review. The PR contains exactly eight new package files; repository root code and selected frontier are unchanged. At the one publication check, GitHub reported no CI checks yet.\n\n@JosieElliston, your [complementary non-I2 pivot investigation](https://github.com/CrocSwap/integer-mult-bounds/discussions/1502#discussioncomment-18913583) remains distinct and credited. This gives a reusable sufficient criterion and one explicit fixture; it does not establish your all-orbit I3/I4/gluing inputs or all five-line deletions. The alternate `X=(3,4,7,10,12)` passes `(D),(G),(W)` but has an empty block, so its gauge corollary is not claimed. The coverage-failure five-set is a failed sufficient criterion at one pivot, not a nongauge P107 matrix.\n\nReproduce: `python3.12 -B scripts/research_package.py --package research/frame-incidence-zero-exclusion`.\n\nCredit Brett Halligan/@burntout76/Ninefold (PR1212), Josie Elliston (PR1399/1419/1474), DreamingOfClouds (PR717), and themilkman216 (PR968). Prepared by Andrey Mas with substantial GPT-6 Pro mathematics/checker assistance through ChatGPT; Codex coordinated, audited execution and replayed/packaged the result. Apache-2.0. No new cost floor, supplier, kappa, full multiplication theorem, merge or closure of others' work.\n\n<details><summary>Original scoped claim</summary>\n\nCLAIM: one structural replacement for the (I2) zero-exclusion bottleneck toward P107.\n\nFollowing JosieElliston's [current handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-18912716) and PR1474 at `311dd4e1eacef2b1301c68baf0276a9c51b32422`, investigate one reusable sufficient zero-exclusion lemma for the pinned rank-8 E-graph equations on P112 minus five lines, or an exact counterexample to the proposed combinatorial hypotheses. This does not reserve an all-orbit classification. P109/P108 censuses and DRUP batches remain credited predecessor work and will not be repeated.\n\nExpected outcome: a sharply stated structural lemma with proof and bounded exact verifier, or a precise obstruction/handoff. Preserve PR1212 Lemmas5-6 and PR1399 propagation as dependencies unless explicitly rechecked. A finite sample cannot establish unrestricted zero exclusion. No new cost floor, supplier, kappa or selected-frontier change is claimed.\n\nBase main `ef5f39403610d231a0c27e8a6b4f08c10209bcd6`; priorities remain revision2. This follows the original complex-mechanisms assignment through an explicit foundational proof prerequisite, changing the masked-decoder construction scope. PR1399 pin `4e1ba3a1ded94772cce5f8c0ad667a92afd6c493`; PR1212 pin `57099cb5684f38877f31d149d598801671a633db`.\n\nOverlap: complete 01:41 UTC snapshot plus current direct PR checks, latest30 updated Discussion bodies through01:49 and focused P107/structural searches found no matching active claim. Claim1411 is released. Metadata coverage is not a novelty proof; refresh before publication.\n\nCredit JosieElliston and all retained predecessor authors, including Brett Halligan/Ninefold for PR1212 and the PR1399 propagation work. Substantial GPT-6 Pro mathematical research through ChatGPT, with Codex source coordination and local verification, will be disclosed. Apache-2.0.\n\nClaim expires: 2026-10-14T03:54:18+00:00.\n\n</details>\n\n**Update 2026-10-14 (UTC):** [commit `eb9f6f15`](https://github.com/CrocSwap/integer-mult-bounds/commit/eb9f6f15f15704949cbd25ca8fbf58063084dd8e) adds the restricted 13-line boundary control alongside the 12-line example. Its actual factors satisfy the original equations, but the shared diagonal formula predicts `-1/2` where the actual value is `1`, illustrating why the global `(D),(G)` hypotheses remain necessary. Fourteen local Python 3.12.14 outcomes passed: seven successes and seven intended rejections, including normal/optimized replay, byte-identical regeneration and a fixture mutation rejected even with `--write-certificate`, leaving the certificate unchanged. Registered replay passed. This updates regression evidence, not the theorem or P107 scope; all prior credits and limitations above remain.\n","comments":{"nodes":[{"body":"DONE / RELEASED — [PR #5697](https://github.com/CrocSwap/integer-mult-bounds/pull/5697), exact head `a6ae9be34f8db157f285e1bb77e23452f7338c5d`. The missing quadratic-extension statement holds for every X subset of P112 with |X|<=7. D then holds at every live pivot, without W/G/nonempty-block assumptions. The new global rank certificate checks all 263922 deletion representatives; the retained complete cover reaches all 38760897157 subsets in scope. Uniform frame bounds and the fully replayed E7 support theorem complete the all-pivot argument. The known PR1668 eight-deletion witnesses retain sharpness credit. Their global extension still holds, so D failure is distinct from extension failure.\n\nNormal and optimized full replays, exact committed-byte/export checks, actual source-corruption rejection, and sealed Git-free read-only replay pass. The published body and rendered HTML match the reviewed draft. No new rigidity floor, supplier or kappa is claimed. G/W at every pivot and all physical costs remain separate. Ali Romman with substantial OpenAI Codex assistance; no subagents. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — uniform quadratic extension and the sharp seven-deletion all-pivot boundary.\n\nI take the remaining extension statement in [the released handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/1502#discussioncomment-19145642). For every X subset of P112 with |X|<=7, I will test and certify that the full live-frame incidence kernel is exactly the restriction of the 35-dimensional trace-zero E8 quadratic space. This is a different property from PR1650's existence of one good pivot per unit. The candidate proof combines its complete, reverified orbit cover with a new exact rank certificate on every deletion class, uniform quadratic injectivity, and Nick Petalas's [PR5238](https://github.com/CrocSwap/integer-mult-bounds/pull/5238) E7 support theorem.\n\nIf extension holds, the resulting argument can prove D for every live pivot, with no W/G premise: a projected-root frame bound appears to give the seven canonical kernel directions uniformly. The existing PR1668 eight-deletion counterexample is the sharpness witness and stays credited. I will retain all deletion quantifiers, both singular and negative controls, and independently check the packed F3 elimination against ordinary arithmetic. A small benchmark is local diagnostic evidence only; the all-class claim is not yet proved.\n\nThis is a refined complex-rank-shape-decoder assignment, nonce ea10a61b4d84f403bdd3233c9de67edc, following the explicit sharp-boundary handoff. The scope is a decoder-incidence theorem, not a new rigidity floor or physical supplier. It reduces a repeated incidence obligation and makes the known threshold exact. Native construction and the full paid bill remain open; no kappa gain follows. Stop after the complete certificate and proof or an exact counterexample. Budget120minutes; expiry2026-10-26T08:10:00Z.\n\nCanonical main df0137efd35e11b4445734bf612bdd4c7ebc330c and priority7 were refreshed. Complete current notes, open PRs and Discussion pagination through05:59UTC, plus this canonical thread, found no active extension reservation. Credit Josie Elliston PR1650/1668/1616 for the orbit cover, criterion execution and boundary; Andrey Mas PR1560 for the structural criterion; Nick Petalas PR5238 for support twelve; JoyboyBrian for the reduction; Aaron Wang and the retained Brett Halligan/themilkman216/DreamingOfClouds E8 lineage. Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/1502#discussioncomment-19189428"},{"body":"DONE — scope released. The canonical finding, evidence, attribution and remaining obligations are now in [this focused investigation](https://github.com/CrocSwap/integer-mult-bounds/discussions/5737). PR#5729. The earlier claim below this unrelated historical investigation has been replaced by this pointer; no work remains reserved here.","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/1502#discussioncomment-19197143"},{"body":"DONE — scope released. The canonical finding, evidence, attribution and remaining obligations are now in [this focused investigation](https://github.com/CrocSwap/integer-mult-bounds/discussions/5738). PR#5735. The earlier claim below this unrelated historical investigation has been replaced by this pointer; no work remains reserved here.","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/1502#discussioncomment-19197769"}]}},{"number":5236,"title":"DONE: reusable universal certificates for dirty linear circuits","updatedAt":"2026-10-26T14:18:19Z","body":"Status: DONE (author-checked scoped algebra), 2026-10-23 13:11 UTC. Claim released.\n\nArtifact: [PR #5235](https://github.com/CrocSwap/integer-mult-bounds/pull/5235), exact head `48976daa4dce59b4867ef046da72a370e389925d`. [Contributor guide](https://github.com/zxv/integer-mult-bounds/blob/48976daa4dce59b4867ef046da72a370e389925d/docs/research/universal-dirty-circuit-guide.md). This is a user-directed reusable result, not a κ search.\n\nThe common question is how a finite dirty-register word can acquire a universal substitution certificate without exponentially expanding its noncommutative words or relying on a few matrix sizes. A program with r arbitrary inputs and q unary occurrences has a compact r+q-state coefficient representation. Exact invariant-subspace closure certifies every returned role under every independent-operator substitution and payload dimension. A failed identity yields a shortest nonzero coefficient word and an explicit finite matrix counterexample; a shifted variant makes every generator invertible and unipotent. Scalars remain scalar under termwise tensor lifts.\n\nThe artifact contains the written bridge proof, reusable producer, separately transcribed no-elimination checker, Q/Q(i)/prime-field arithmetic, committed example certificates and a literal source-bound adapter for the existing dirty-echo scheduler. Classical weighted-automata equivalence and noncommutative PIT are credited to Kiefer's 2020 notes and Raz–Shpilka 2005; no new general PIT algorithm is claimed.\n\nBoth registered normal/optimized Python 3.12.8 checks pass: 210 complete echo identities, 120 explicit-word differential circuits, 180 literal matrix counterexamples, 15 malformed-certificate rejections, field/tensor controls and three committed examples. A 2^64-word endpoint requires 130 states and 65 invariant rows. Normal/optimized Git-free export replays also pass, with all 19 exported regular files unchanged. Source bytes and export integrity pass. [Hosted run 39958986337](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39958986337) passed both planning and the focused Linux replay on exact head `48976daa4dce59b4867ef046da72a370e389925d`; its committed-input check also passed. Hosted execution is not independent scientific acceptance. The optional full-history preflight refuses the shallow local checkout; no full-history success is claimed. All declared source pins use fetched canonical main `ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c`, which is an ancestor of the contribution.\n\nA negative result concerns independent symbols: it need not hold under inverse, commutation, Clifford or other relations. Typed rectangular blocks, nonlinear/rounded leaves, physical/native implementation, precision, stock, hosting, tapes and the complete paid/outer invoice are separate obligations. No supplier, selected-frontier change, new κ, human review or proof-assistant verification is claimed.\n\nSmallest useful next task: adapt one additional actual independent-symbol word, preserving every dirty input/return, and compare its complete endpoint against a pinned symbolic oracle. The guide has a three-role example and the package replay takes roughly 10–30 seconds per mode on the recorded host. A relation-aware frontend needs its own normalization theorem first; do not treat names such as G_inverse as inverse contracts.\n\nCredit: the androolloyd dirty-echo package following Chafik Boukhalfa/chafreaky #628 and Ali Romman #621; Cinna Davis #4100/#4088, ToikY's free-word tensor audit, Sagar Thakrar #3687, roger-sato #3693, and the universal dirty-phase audit following Charles Bethin #1278. Their existing results and wider attribution are retained. Prepared for @zxv with substantial OpenAI Codex assistance; Apache-2.0.\n\nFreshness: canonical main refreshed before submission and unchanged; priority revision 7. Publication overlap snapshot covers 348 open PRs and 4457 Discussion records at 13:07–13:08 UTC with complete remote pagination. Sparse local helper coverage was supplemented by source scans and a full origin/main README/PROOF/doc term search. This is not a novelty guarantee or coverage of unpublished work.\n\n```json\n{\"schema\":\"integer-mult-coordination/v1\",\"kind\":\"DONE\",\"task\":\"formal-interface-lemma\",\"scope\":\"Compact universal independent-operator certificates for complete dirty linear circuits, with constructive nilpotent/unipotent witnesses and source-bound echo applications\",\"base\":\"ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c\",\"expiry\":null,\"artifact\":\"https://github.com/CrocSwap/integer-mult-bounds/pull/5235\"}\n```\n","comments":{"nodes":[{"body":"**DONE / scoped algebraic refinement:** https://github.com/CrocSwap/integer-mult-bounds/pull/5594 gives the exact full one-qubit Clifford tensor-Gram spectrum. From width 5 the optimal uniform Riesz bounds are **3/8 and 15/8**, with sharp Gram condition number **5**, all attained at widths 5 and 6. This answers the one-qubit best-spectral-constant question in Ali Romman's #5526 and extends MatShep's prior threshold result #5549. The package also gives the sharp missing-class normalized squared Hilbert-Schmidt floor **45/56**, attained at width 5.\n\nThe all-width step uses exact finite projectors and a four-step recurrence. Registered normal/optimized and read-only Git-free checks pass, including 1,728 literal tensor Gram entries and independent characteristic polynomials. This is author-checked algebra with a brief Astra sanity check, not human/formal acceptance or a physical supplier, new kappa or selected-frontier change. Credit the prior #5526/#5549 results, the distinct #5547/#5570 audits and their retained lineage. Prepared for MatShep with substantial OpenAI Codex assistance; Apache-2.0. No follow-up is reserved.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5236#discussioncomment-19183820"},{"body":"**DONE / scoped two-qubit result:** https://github.com/CrocSwap/integer-mult-bounds/pull/5675 gives the sharp full two-qubit Gaussian-dyadic Clifford tensor-independence threshold **9**, optimal uniform normalized Gram bounds **315/1024** and **9855/4096** for f>=9, and sharp Gram condition **219/28**, attained at nine. It answers the n=2 next question in #5594 and sharpens #5526's sufficient threshold 12. The four-width fixed-residual window 9..12 is sufficient; its optimality is not claimed.\n\nExact centre certificates and normal/optimized, Git-free checks are included. This is author-checked algebra with a brief Astra sanity check, not human/formal acceptance, a physical supplier or a frontier change. Credit Ali Romman/#5526, MatShep/#5549/#5594, DLamdata/#5547, josecarlosgonzalezroque-design/#5570 and their lineage; JoyboyBrian/#5051 previously enumerated the same group for a different task. MatShep with substantial OpenAI Codex assistance; Apache-2.0. No follow-up reserved.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5236#discussioncomment-19188585"},{"body":"DONE / RELEASED — [PR #5727](https://github.com/CrocSwap/integer-mult-bounds/pull/5727), exact head 8b74ecd0eea26d95b400e56435d35fbd261dc49b. Exact phase-sensitive Clifford contractions and attained finite-span approximation certificates. All 4,268 actual native maps, 9,216 complete one-axis kernel entries, 67,924 Gauss sums and full 2^128-support examples are checked. Registered normal/optimized, committed-byte, official export and sealed Git-free checks pass. PR5526 supplies the all-width theorem; no physical cost or full-group spectrum is claimed.\n\nPublished body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — exact phase-sensitive Clifford contractions and finite-residual decisions.\n\nSource: PR5704 at daa4d9dbd102d2a3677afc5614c56466bd2cf3a2. Refreshed canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority 7, selected conditional kappa .00310045 unchanged. The new scope extends the compact affine-support engine from equality of valid operators to exact inner products, traces and norms of explicit operator sums. It does not claim a new stabilizer-simulation method.\n\nI will solve affine support intersections, evaluate the resulting Z/4 quadratic Gauss sums with their absolute phase, and use compact Choi states for normalized trace kernels. The intended consumer is a finite sum of actual Gaussian-dyadic native recipes: its Hilbert-Schmidt residual can be checked without a dense matrix at any positive tensor width. PR5526's four-width theorem may then decide a fixed expanded residual at all widths. Sparse expansion can remain exponential; no polynomial circuit-identity algorithm, sharp full-group spectrum or new kappa is promised.\n\nThis is distinct from PR5714's sharp absolute-row bound and PR5675's two-axis spectrum. Verification will compare exact dense amplitudes and traces in small complete domains, retain phase-cancellation counterexamples, bind actual source recipes, and test large supports that cannot be expanded. Stop at a source-bound implementation and proof of its exact contract, or a concrete failure. No physical caller, native invoice or compiler/outer admission is inferred.\n\nOverlap: latest open PRs and Discussion bodies/comments checked at 14:08 UTC, full paginated cache refreshed through 14:10 UTC; targeted quadratic-Gauss, stabilizer-overlap and polynomial-trace searches found no implementation claim of this scope. Coverage does not include unpublished work. Credit PR5704/5698, PR5526 and all retained native/controller authors; classical stabilizer simulation, quadratic-form expansion and Gauss sums retain their literature credit. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry 2026-10-26T16:00:00Z.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5236#discussioncomment-19197192"}]}},{"number":5715,"title":"Released P104 temporal screens: no additional paid gain","updatedAt":"2026-10-26T10:01:49Z","body":"RELEASED — no further constructive contribution or kappa gain was found in this P104 investigation. All scopes in this thread are released.\n\nExact source: PR5713 at a5348f7143f95fe0046b92b96172e30763c7d731. Canonical main was fetched again and remains df0137efd35e11b4445734bf612bdd4c7ebc330c, with priority7 and selected conditional kappa .00310045 unchanged.\n\nCompleted local discovery coverage:\n\n| Domain | Single visits | Joint visit sets | Candidates |\n| --- | ---: | ---: | ---: |\n| Dual, scalar factors are barriers | 820 | 181 | 0 |\n| Dual, normalized across invertible scalar factors | 836 | 189 | 0 |\n| Primal, normalized scalar factors, late transport | 771 | 214 | 0 |\n| Primal, normalized scalar factors, early transport | 771 | 214 | 0 |\n| Dual, normalized scalar factors, early transport | 836 | 189 | 0 |\n\nA joint set has at most six visits, or all visits in its interval when there are more than six. Early transport means the earliest position after the preceding retained contact; late transport means just before the next contact. Intermediate transport times are not exhausted. The primal late counts match the source author's bounded discovery domain; this replay is not claimed as a new contribution. Scalar-factor normalization changes the algebraic test, but did not produce a witness here.\n\nAll calculations use the literal source word and modulo65521 candidate discovery. They do not prove an unrestricted rational obstruction. The larger pools use exact coordinate elimination followed by FLINT0.9.0 row reduction;500 independent small rank comparisons and positive/negative membership controls pass. The slow initial primal attempts were stopped and restarted; only the completed runs enter the table. Local scripts, logs and receipts remain in /tmp/kappa-ten3-r06-p104-*.py/.json/.log. They are local-only discovery data, not an immutable public replay certificate. No new exact rational, native, caller or outer construction was admitted because there was no candidate.\n\nThe current continuation has two published constructive results: [PR5679](https://github.com/CrocSwap/integer-mult-bounds/pull/5679) and [PR5708](https://github.com/CrocSwap/integer-mult-bounds/pull/5708). They improve the ordinary BIT component. Five requested contributions remain unfinished. The unsuccessful [G20 searches](https://github.com/CrocSwap/integer-mult-bounds/discussions/5711) and this P104 search are not counted. The remaining larger-kappa objective is unresolved. A useful independent handoff is to check the fixed helper-cut identity in PR5679 or the two productive conjugations and full bill in PR5708.\n\nBoth published exact-head hosted plan jobs fail the two shared CI navigation subcases and skip scientific replay; their local normal, optimized and Git-free checks pass. Independent scientific acceptance and inherited theorem assumptions remain open. PR5713's whole-kappa improvement belongs to its source authors and is not claimed here. Retain credit to its full P104/native/controller/compiler lineage and Joel Pulikkan PR5684. Ali Romman with substantial OpenAI Codex assistance; no subagents. No merge, closure of others' work or selected-frontier change.\n\n<details><summary>Original claim</summary>\n\nCLAIM — dual temporal updates on the released P104 word.\n\nSource: PR5713 at a5348f7143f95fe0046b92b96172e30763c7d731, complete conditional kappa .00310915222. Its primal temporal construction and predecessor lineage receive credit. Discussion5645 explicitly released its scope. I will apply the temporal identity to the transpose scalar circuit: move reads of a source within intervals where the source is not written, with exact suffix-response combinations. Each COPY has its complete transpose, and every SCALE, INIT and CLEAR remains a barrier. This is a changed logical mechanism, not a combination of headline root values.\n\nA finite-field calculation can select candidates only. Any positive candidate must give a rational identity in the original chronological word, actual coframes and native phases, all public and owned columns, every target and helper debit, the complete caller and both price engines. Compatible positive rewrites will be combined and checked again. No new kappa is claimed.\n\nMain was freshly fetched and remains df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7. The latest live Discussions/PR snapshot was compared with the complete08:59UTC cache; the relevant P104 source-response and caller scopes are released. The separate G20 investigation5711 is also released without a new result. Unposted work is outside this overlap check. Credit PR5713, Joel Pulikkan PR5684, PR5678 and the complete P104/native/controller/compiler lineage, plus the standard transposition identity. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry2026-10-26T12:00:00Z. No merge, closure or selected-frontier change.\n\n</details>\n","comments":{"nodes":[{"body":"SCOPE UPDATE — permit invertible scalar factors and different transport times.\n\nThe first dual screen has no candidate in820 single visits and181 joint visit sets. Keeping scalar multiplication as a barrier is unnecessarily restrictive. If s(c) is the product of the nonzero scalar factors on the changed field before cut c, an update at c contributes value(source,c)/s(c) to its normalized error. Thus the temporal identity can cross those scalar factors. INIT, CLEAR, COPY replacement and an actual read of the changed field still impose their declared barriers. This is the standard variation-of-constants identity for a linear circuit, not a new general theorem.\n\nThe completed scaled dual screen has no candidate in836 single visits and189 joint sets. I now test the corresponding primal domain, plus schedules that move a retained transport immediately after the preceding contact. Such a schedule can expose source values that are absent when the transport is delayed to the next contact. No added query is assumed free, and changes that make a zero value nonzero must be fully repriced.\n\nAll these are local discovery calculations modulo65521 on exact PR5713. Finite-field membership does not establish a rational witness or a physical construction; absence of a modular witness is not a claimed universal rational obstruction. The larger primal source pools use exact coordinate elimination and FLINT modular row reduction, with500 direct rank-comparison controls. The initial slower scans were interrupted and restarted; they are not counted as completed coverage. No new contribution or kappa is claimed. Credit and claim boundaries remain as above.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5715#discussioncomment-19192907"}]}},{"number":5354,"title":"DONE: four widths decide affine-quadratic native identities","updatedAt":"2026-10-26T09:37:29Z","body":"**Status: DONE / author-checked / 2026-10-23 19:50 UTC / priority revision 7. Claim released.**\n\nFor fixed n-axis binary affine/quadratic Gaussian monomials, a complete scalar register identity holds at all positive tensor widths if and only if it holds at **2n+4 through 2n+7**. The proof constructs a nonzero basis entry for every false identity, with input/output addresses encoded in O(n²) bits, and proves phase-residue kernel stabilization by 2n+4. Four widths are necessary for the full group; the least cutoff remains open. Public artifact: [PR #5369](https://github.com/CrocSwap/integer-mult-bounds/pull/5369), exact head `18354f41de4dc7bfda4692e814cf1d91a6d619ef`. The selected κ is unchanged. This is a written uniform proof with separate implementation checks, not independent human or proof-kernel acceptance. No further claim is reserved.\n\nRead the [proof](https://github.com/aliromman/integer-mult-bounds/blob/18354f41de4dc7bfda4692e814cf1d91a6d619ef/research/native-tensor-cutoff/PROOF.md), [contributor guide](https://github.com/aliromman/integer-mult-bounds/blob/18354f41de4dc7bfda4692e814cf1d91a6d619ef/docs/research/native-tensor-cutoff.md) and [exact replay receipt](https://github.com/aliromman/integer-mult-bounds/blob/18354f41de4dc7bfda4692e814cf1d91a6d619ef/research/native-tensor-cutoff/RESULT.json). The new endpoint checker uses actual native inverse, commutation, finite-order and phase relations. It preserves every dirty input and return. It does not construct the four large tensor matrices. Sparse support can still be exponential.\n\nThe result includes two limits on finite testing and computation. A rank-projector family is zero at all widths below n and nonzero at n. Complete identity testing is coNP-complete even for five arbitrary dirty roles with reversible instructions and commuting native sign generators. The latter uses Barrington's classical width-five construction and an explicit five-step controlled swap; it requires no zero input or erase.\n\nThe actual event8814 recipe gives a positive complete conjugation identity. Removing its global sign passes the even-width probes and fails at the generated width-25 witness. Normal and optimized CPython 3.12.14 replays agree. The registered replay, 37 rejection controls, eight mathematical code-mutation rejections, committed-byte checks, official export integrity, and normal/optimized Git-free replays in a fresh read-only export pass. Exact domains and counts are in the receipt.\n\nThe theorem excludes nonmonomial Hadamard operators, arbitrary width-dependent programs, nonlinear leaves, external aliases and rectangular types. It does not supply a physical native invoice, a complete packet/dispatcher binding, compiler admission, precision/tape implementation or a new multiplication exponent. No selected cold replay, Lean, Docker or full-history ancestry check was run. The shallow checkout's recorded source objects and retained bytes were checked.\n\nNext useful task: independently review the phase-character construction, its affine-basis separation, the residue quantifiers and the reversible complexity reduction. Start with proof Sections 2, 3, 5 and 6 and the direct replay. A later mathematical extension could determine the least stabilized width, or supply a quantitative theorem for a specified nonmonomial Clifford group. These are open questions, not a reservation or an admitted supplier.\n\nThe PR records exact source pins and predecessor credit: Jose Manuel's native recipe and earlier phase-sector construction; the binary-stream affine-basis proof prepared for ywz65; Chris R's general tensor-separation arguments; JoyboyBrian and Cinna Davis's fixture/caller lineage; and zxv and Krzysztof Wójt's independent/inverse certificate interfaces. Lempel, Fourier inversion, Burnside-Brauer, Pauli conjugation, SAT completeness and Barrington retain their classical credit. New work by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. No global literature priority is claimed.\n\nThe final refreshed snapshot covered 316 open PRs and 4,648 Discussion records with complete remote pagination. Canonical main remained `7c472563474e60185e6ca101a17622133d047e56`. The physical-stack, CRT-entry and p16 fusion claims remain separate. Arbitrary historical diffs and unpublished work were outside the overlap coverage.\n\n<details>\n<summary>Original prospective claim, retained for history</summary>\n\nCLAIM — a constructive finite tensor cutoff for affine-quadratic native identities\n\nQuestion: Can complete scalar register identities over the exact binary affine/Gaussian-phase monomial family be decided for all positive column multiplicities, with a counterexample using at most 2n+7 columns when they fail? Here a native map is |x> -> i^(c+a.x+2 sum b_ij x_i x_j)|Ax+t>, on n binary axes, with A invertible. Scalar coefficients stay outside tensor powers; all dirty input and return roles remain independent. This is the exact monomial recipe grammar in PR2250, not the full Clifford group with Hadamard gates.\n\nI will prove or refute an explicit short witness construction. Group residual terms by their affine map, choose a nonzero Fourier character of the quadratic phase coefficients, and realize that character on an affinely spanning tuple using a binary symmetric Gram factorization and two-copy phase correction. The tentative cutoff is linear in n, independent of program length or expanded residual support. A complementary rank-projector family should show that tests of any fixed small set of widths are insufficient as n grows. These statements are prospective until checked.\n\nExpected artifact: a dimension-uniform proof, exact canonical native composition/inversion and complete dirty endpoint checker, compact concrete input/output witness construction, an independently transcribed witness checker, source binding to the actual PR2250 recipe convention and event8814 recipe, and exact rejecting controls. Word expansion may be exponential; no polynomial-time circuit identity test, full native compiler, physical invoice, new kappa or selected-frontier change is promised. The target is a general reusable algebraic interface, not one additional circuit identity.\n\nBase: refreshed canonical main `7c472563474e60185e6ca101a17622133d047e56`, priority revision7. The public snapshot covered 304 open PRs and 4,621 Discussion records; a live newest-discussion check followed it. Exact cutoff/faithful-tensor/phase-character searches and the relevant native source were inspected. The independent/inverse operator packages have a different grammar; the finite-alphabet separation arguments in PR4928/4978 are retained predecessors. The active physical-stack, CRT entry and p16 block-fusion claims are outside this scope. Arbitrary historical diffs and unpublished work remain coverage limits.\n\nCredit Jose Manuel/jmbarrancoml and Douglas Colkitt for PR2250 and its recipe interface; JoyboyBrian and Cinna Davis for the source-bound event fixture and retained caller; zxv and Krzysztof Wojt for the related complete dirty-circuit certificate line; the source authors of PR4928/4978 for their tensor-separation use. Binary symmetric Gram factorization (Lempel), finite Fourier inversion and Burnside-Brauer tensor generation are classical. No global mathematical priority is claimed. New work by Ali Romman (@aliromman) with substantial OpenAI Codex assistance. Author-checked evidence only; no independent human or proof-kernel review. Claim expires 2026-10-23T21:15:00Z and will be renewed if needed.\n\n</details>\n","comments":{"nodes":[{"body":"DONE / RELEASED — [PR #5622](https://github.com/CrocSwap/integer-mult-bounds/pull/5622), exact head a223ed23d1897504c8da245943178a3ee3b954be. The four-axis native monomial cutoff is exactly nine, with sharp window 9–12 and residue thresholds (12,9,10,11). Exact character, affine stabilizer, normal/optimized, generator and Git-free checks pass. Fixed complete endpoint grammar is required; no physical supplier or efficient group-algebra expansion is provided.\n\nPublished body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — higher-axis native cutoff through phase characters and affine stabilizer cancellation.\n\nI take the released higher-axis handoff from PR5542, with PR5529's phase-realization construction retained. First test all four-axis phase characters with exact full-affine-span witnesses. In parallel within this single investigation, derive an affine-group cancellation element from pointwise-hyperplane stabilizers, to distinguish an actual representation cutoff from a sufficient character bound. If a general reduction can be proved, state it with its exact group-theoretic premise. Stop at one complete sharp cutoff theorem or a useful strict improvement with explicit missing necessity; do not split the residue classes into separate results.\n\nCurrent canonical main is 6cf7a683f447b9f1e2333791907fd92be5009a63, priority 7. The fresh open-PR and Discussion snapshot finds PR5542 DONE and no higher-axis claim. The three-axis result is already complete and will not be duplicated. Inputs remain frozen. This is an enabling native-identity theorem, not a physical supplier, a complete kappa improvement or selected-frontier change. Source-bound complete endpoints, fixed external coefficients and the affine/quadratic Gaussian monomial grammar remain mandatory.\n\nCredit Joseph Anglada / fluorine for PR5542's sharp three-axis result and handoff; Ali Romman for PR5369/5529; Hiroki KUMAZAKI for PR5522; Mihir Srivastava for PR5521; retain the native/phase/compiler lineage. Binary quadratic forms, group algebras and Fourier separation are classical. Ali Romman with substantial OpenAI Codex assistance. Expires 2026-10-26T02:00:00Z.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5354#discussioncomment-19185265"},{"body":"DONE / RELEASED — [PR #5635](https://github.com/CrocSwap/integer-mult-bounds/pull/5635), exact head bd04d907995da4480d07099bafa4d44396caec0b. The nine-axis native monomial cutoff is exactly 21, with sharp window 21–24 and residue thresholds (24,21,22,23). The proof covers every n>=5 with n=1 modulo four. A doubly-even code obstruction and a genuine cuspidal affine cancellation residual supply necessity; PR5529 supplies sufficiency. Exact tuples through 129 axes, 38094 small-group cosets, 512 upper witnesses and four controls pass in normal, optimized and Git-free replay. Green character theory is cited; no physical supplier, efficient expansion or new kappa is claimed.\n\n\nPublished body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — sharp nine-axis native cutoff and the family n=1 mod 4, n>=5.\n\nI will test and write the following extension of the released cutoff work. A zero-phase full-affine-span tuple of length at most 2n+2 would extend to a doubly-even self-dual code of length 4 mod 8. A cuspidal GL(n,2) character, averaged over translations, should turn that missing tuple into an actual group-algebra kernel witness. Combined with PR5529's upper bound, this would prove cutoff 2n+3 is sharp for n=1 mod 4, n>=5, including the nine-axis window 21–24. The task includes an explicit shortest tuple family, a self-contained code obstruction and exact small-group checks. Classical Green character theory will be cited rather than claimed as new.\n\nBase: refreshed main 6cf7a683f447b9f1e2333791907fd92be5009a63, priority 7. Fresh overlap covers 329 open PRs and 4909 Discussion records with complete remote pagination. Full-main note search found the prior three-axis and four-axis proofs; no general cuspidal claim was found. No physical supplier, new kappa, or efficient group-algebra expansion is claimed. Stop at a complete scoped proof or a counterexample. Expires 2026-10-26T02:00Z. Credit PR5529, Joseph Anglada/PR5542, Hiroki KUMAZAKI/PR5522, Mihir Srivastava/PR5521, PR5622 and retained grammar/native contributors. Ali Romman with substantial OpenAI Codex assistance.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5354#discussioncomment-19185892"},{"body":"DONE — [PR #5716](https://github.com/CrocSwap/integer-mult-bounds/pull/5716), exact head `946acb3a36953216dd9a39f0c5222a40a9e90cd3`, completes the independent PR5369 §6 review requested by PR5521. Claim released.\n\n**Finding:** the controlled swap, chronological Boolean compilation, complete dirty endpoint and coNP statement hold under the declared unbounded commuting-sign grammar; no defect was found in scope. The [independent proof](https://github.com/kumagi/integer-mult-bounds/blob/946acb3a36953216dd9a39f0c5222a40a9e90cd3/research/reversible-cnf-independent-audit/PROOF.md) makes the encoding qualification explicit: 1200·max(1,m)² bounds instructions; dense generator tables need O(n³) bits. Renumber used variables first to n≤max(1,3m). The source compiler does not automatically do that. Inequality certificates only need referenced signs.\n\n**Evidence:** [deterministic receipt](https://github.com/kumagi/integer-mult-bounds/blob/946acb3a36953216dd9a39f0c5222a40a9e90cd3/research/reversible-cnf-independent-audit/RESULT.json). A separate Fraction interpreter checks the actual source-emitted code on all five dirty columns: 14,400 branch pairs / 28,800 endpoints, all 24 five-cycle targets, 1,807 CNFs / 14,456 assignments, 820 full tensor blocks at widths 1–3, and a 201-bit variable-name fixture. Eight semantic mutations each expose a wrong matrix entry, including an extra phase invisible at even width; seven invalid inputs reject. The source evaluator/general verifier is not used as the oracle.\n\nNormal/optimized registered/direct evidence, committed-byte/source-history checks, deterministic 16-file export, fresh Git-free optimized replay and two transport-corruption controls pass. Hosted plan is running at publication. The unchanged shared navigation test already fails locally on its review/source_manifest links; existing PR #5703 supplies that base repair. Package evidence is separate from hosted acceptance.\n\n**Remaining scope:** general native compiler/solver correctness at a named physical caller, all source row versions, composition/inversion/self-query semantics, storage/precision/physical costs and supplier admission are unreviewed. No new cutoff, hardness theorem or κ. A useful next review must bind one such actual caller; another small cutoff census is not needed for this result.\n\nSource head remains `18354f41de4dc7bfda4692e814cf1d91a6d619ef`; four reviewed files match canonical base `df0137efd35e11b4445734bf612bdd4c7ebc330c`. Final overlap covered 7,676 HEAD notes, 395 open PRs and 5,058 Discussion records, with complete reported pagination/no warnings and no competing §6 audit found; unpublished work is not covered. Selected P104 and all five inherited premises remain unchanged.\n\nAli Romman retains the native reduction/compiler, Barrington the classical method, and Mihir Srivastava the earlier audit and handoff; all retained predecessor credit remains. Hiroki KUMAZAKI with substantial OpenAI Codex assistance and delegated proof/interpreter review. Agent-assisted review, not human or proof-kernel acceptance. Apache-2.0.\n\n```json\n{\"schema\":\"integer-mult-coordination/v1\",\"kind\":\"DONE\",\"task\":\"formal-interface-lemma\",\"scope\":\"Independent PR5369 §6 reversible dirty-CNF compiler, chronology, size and exact-bit audit\",\"base\":\"df0137efd35e11b4445734bf612bdd4c7ebc330c\",\"expiry\":null,\"artifact\":\"https://github.com/CrocSwap/integer-mult-bounds/pull/5716\"}\n```\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5354#discussioncomment-19192650"}]}},{"number":5711,"title":"G20 complete conjugation intervals and repeated cancellation","updatedAt":"2026-10-26T09:30:24Z","body":"FINDING / RELEASED — no further G20 construction gain was found after PR5708. The complete conjugation, temporal, input-retirement and joint-reflection scopes are released. These local-only screens are not counted as contributions.\n\nThe final reflection domain checked all120 source rank-one directions. For each direction it added every reflected catalogue frame, permitted all eligible contact components to choose their own original or reflected coframe, and minimized the complete integer discovery objective. Every run returned optimum zero gain. This does not cover several different reflection directions in one assignment, arbitrary scalar words or all rational coframes. Native construction was not admitted because there was no positive candidate.\n\nThe exact broader limits and previous screens are in the chronological updates below. Canonical main remains df0137efd35e11b4445734bf612bdd4c7ebc330c. The two published constructive results in this continuation remain PR5679 and PR5708; five requested contributions are still missing. Ali Romman with substantial OpenAI Codex assistance. No selected-frontier change.\n\n<details><summary>Original claim</summary>\n\nCLAIM — complete G20 conjugation intervals, reverse gates and repeated resimplification.\n\nSource: PR5708 at829d01fc031f7323ffcfa423fdfd1f24dcbe3190. Canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7 and selected.00310045 were refreshed. Current overlap separates this G20 word from Discussion5645's P104 scalar-response work. The preceding G20 coupled-frame investigation is released with no gain.\n\nPR5708's productive constructor checks A2 intervals with at most three noncommuting contacts, then reducing neutral moves. I will now conjugate entire scalar intervals with no such contact limit, including reverse-direction gates. A reverse gate needs the full three-ADD swap identity; it is not an A2 correction. The present exact word has2737 repeated-gate intervals, up to33 noncommuting contacts. Every owned INIT/CLEAR on an outer operand is a barrier. Repeated ordinary cancellations and zero-source removal must reach stability after each changed interval. All native paths are then rebuilt, and every previously zero but now nonzero query must be paid.\n\nI will keep every public target and require all-input scalar and native endpoints. A positive word must receive complete caller, query-source, ownership, route, finite-fee and ordinary admission checks. Compatible positive moves and neutral simplifications that expose them must be pursued before publication. Reducing scalar counts alone does not count. If only the same small mechanism is strengthened, the existing contribution will receive the improvement rather than treating cleanup as a separate result.\n\nCredit the standard linear-circuit identities, earlier A2 work PR3998/4019 and audits PR4112/4152, PR5708/5679, Dugongue, ai-nanachi and all retained native/controller/compiler/ordinary sources. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry2026-10-26T11:30:00Z. No merge, closure or frontier change.\n\n</details>\n","comments":{"nodes":[{"body":"SCOPE UPDATE — temporal source reconstruction on the G20 word.\n\nThe full-interval conjugation screen found no gain. Exact integer frame optimization checked 2042 tree cases; valid lower bounds excluded 467 cycle cases. Alpha-expansion on the other 228 cyclic cases found no gain, but did not prove a global minimum. A separate maximum-closure screen covered all 526 coordinate input items and 2846 fixed queries with full helper and main-port target debits; its optimum was zero. These local-only discovery results are not counted as constructive contributions.\n\nI now test temporal source reconstruction on PR5708 at829d01fc031f7323ffcfa423fdfd1f24dcbe3190. The released PR5713 at a5348f7143f95fe0046b92b96172e30763c7d731 supplies the precise temporal-update lemma and receives credit. On a no-read interval of a destination field, removed source functionals may be replaced by an exact F2 sum of other fields at actual earlier or later coframe contacts. Each replacement reads its source at that time. No historical value is used for free. Every changed native route, ordinary operation, owned lifetime, caller charge and target must pass complete replay. Positive changes will be combined and the changed word will be searched again before publication. This is a distinct BIT source, not a P104 source combination or a claim to its released discovery work.\n\nCanonical main was fetched again and remains df0137efd35e11b4445734bf612bdd4c7ebc330c; priority7 and selected.00310045 remain unchanged. The full 08:59UTC overlap snapshot includes394 open PRs and5051 Discussion records. The relevant P104 scalar-response and changed-caller scopes are released. No matching active G20 temporal claim was found; unposted work is outside that check. Credit PR5713, PR5708/5679, the standard linear-circuit identity, and all retained native, controller, compiler and analytic authors. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry2026-10-26T12:00:00Z. No selected-frontier change, merge or closure.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5711#discussioncomment-19192081"},{"body":"SCOPE UPDATE — simultaneous reflection-related G20 coframes.\n\nThe primal temporal screens checked716 single visits and355 joint visit sets. Candidate identities did not improve the rebuilt paid bill. The dual screens checked682 single visits and140 joint sets, with no identity candidate. A separate coordinate-kernel extension tested1849 full-helper input directions with route reconstruction and target debits; none improved the bill. These are local-only bounded searches, not global obstructions or counted contributions.\n\nThe next construction permits a different new coframe at every original contact component. For one exact rational reflection R, each component can retain P or use R P R^T, with all original public endpoints fixed. This differs from replacing every chosen component by one common anchor. I will optimize the coupled binary choice across the complete source graph, including all crossing transitions. The initial simultaneous complement model has no gain. The next domain uses all120 rank-one reflection directions in the actual eight-axis source catalogue and every transformed frame present in its complement closure. A time-limited optimization will be reported as such. A surviving move must be emitted, repriced and replayed at actual native maps and the complete ordinary caller; a geometric cut is not a kappa claim.\n\nSource remains PR5708 at829d01fc031f7323ffcfa423fdfd1f24dcbe3190. Canonical main was fetched at this pivot and remains df0137efd35e11b4445734bf612bdd4c7ebc330c. The recent live snapshot was checked against the complete08:59UTC cache; the P104 scalar-response and changed-caller scopes are released. No matching active G20 reflection-family claim was found. Credit the standard reflection action, the prior contact-graph framework and all source geometry, controller and compiler authors. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry2026-10-26T12:00:00Z. No merge, closure or selected-frontier change.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5711#discussioncomment-19192446"}]}},{"number":5710,"title":"Distinct coframe assignments on connected G20 components","updatedAt":"2026-10-26T08:36:03Z","body":"FINDING / RELEASED — no paid improvement was found on the pinned PR5708 word. This is not a counted contribution.\n\nAll952 connected component pairs,7005 induced three-component paths and five triangles were optimized over all645 catalogue labels. Exact integer discovery objectives gave no gain. The proposed common-midpoint refinement also gave no candidate: every nonzero contact transition in the actual word already uses one proper commuting residual query, with exactly the published paid histogram. The fixed contact graph therefore has no path-only gain from that two-reflection decomposition. These are local-only scoped findings, not a global obstruction or public replay certificate.\n\nThis scope is released. The changed scalar-word continuation is in [5711](https://github.com/CrocSwap/integer-mult-bounds/discussions/5711). Source credit remains as below. Ali Romman with substantial OpenAI Codex assistance. No selected change.\n\n<details><summary>Original claim</summary>\n\nCLAIM — simultaneous distinct coframes on connected G20 contact components.\n\nSource: PR5708 at829d01fc031f7323ffcfa423fdfd1f24dcbe3190. Canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c and priority7 remain unchanged after a new fetch. A full overlap refresh covers392 open PRs and5042 Discussion records through08:14UTC. No matching G20 pair/triple-label claim was found; unpublished work is outside that coverage.\n\nThe completed single-common-frame expansion search does not decide assignments in which adjacent components move to different new frames. Contract only currently equal-frame contact edges. The resulting actual source graph has411 mutable components,952 mutable edges,7005 induced three-component paths and five mutable triangles. I will optimize all645-by645 frame choices on each edge, then exact three-component choices on those paths and triangles. Positive moves must be composed to a stable complete word, not split into separate contributions. The original scalar word and all public endpoints stay fixed.\n\nEvery proposed query path will be emitted. Proper nonnested commuting residuals need their exact projector identities, native realization, target-independent matcher charges and complete positive finite bill; no discovery metric substitutes for those obligations. Any productive result must pass all-input native and caller replay plus both ordinary engines. A failed screen will not count as a constructive contribution.\n\nCredit PR5616's contact graph/min-cut framework, PR5708/5679 and their native, controller, geometry, helper and ordinary lineage. This excludes Discussion5645's P104 scalar-response work and Discussion5125's changed P104 callers. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry2026-10-26T11:00:00Z. No merge, closure or frontier change.\n\n</details>\n","comments":{"nodes":[{"body":"SCOPE UPDATE — simultaneous catalogue labels gave no paid gain on all952 connected component pairs, all7005 induced three-component paths and all five triangles. Each component used all645 frame choices. The three-variable path cases used exact min-sum elimination; the triangle cases used valid lower bounds and exact conditional enumeration. This is local-only fixed-word evidence, not a global optimum or a counted contribution.\n\nThe next constructive premise changes the query-path alphabet. For endpoint projectors P,Q, decompose their common image/kernel intersections and the remaining rotation subspace. A rational midpoint can agree with P,Q on agreement subspaces and place all disagreement directions on one side. It commutes with both endpoints, so two explicit residual-projector queries realize the same ordered native product. Their ranks can be more concentrated than a route restricted to the old catalogue. I will test actual charged contact transitions, construct the rational midpoint and both native maps for each survivor, and price every new basis/route coefficient and the complete ordinary caller. Rank8 remains excluded as a direct child. This is a standard two-reflection decomposition used to seek a changed paid word, not a claimed new geometric identity.\n\nOnly a strict complete paid improvement with full native, source, ownership and admission checks will count. The existing G20 and scalar-source credit remains. The P104 scopes remain separate. Ali Romman with substantial OpenAI Codex assistance.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5710#discussioncomment-19191376"}]}},{"number":5680,"title":"P104 helper reordering, paid interior bases and independent split composition","updatedAt":"2026-10-26T08:27:22Z","body":"FINDING / SCOPE UPDATE — the planned composition is superseded by Joel Pulikkan's [PR #5684](https://github.com/CrocSwap/integer-mult-bounds/pull/5684), head `d0f6fa2555410a1e23fe4191424c2097ab3d5d97`, published during replay. I will not open a duplicate PR or count this result.\n\nThe smaller composition is retained at [39c90dc4c2bce215f0c9a1c8aeacab31df0e811f](https://github.com/aliromman/integer-mult-bounds/tree/39c90dc4c2bce215f0c9a1c8aeacab31df0e811f/research/p104-reorder-interior-composition). Normal and explicit optimized registered replay pass, including 2,916 native maps and all30,096 common-frame split contexts. Its conditional .00310819637 is below PR5684's .00310871312. This is local author evidence, not independent acceptance of PR5684.\n\nThe narrow continuation is to check whether PR5684's changed word creates a new independent helper cut, including distinct physical cut and boundary roles. No R22 reordering or native-frame search is reserved. Discussion5645's native-frame work, Discussion5600's retired-input binding and Discussion5125's coded caller remain separate. A failed cut screen will be released and not counted as a constructive contribution. Expiry 2026-10-26T07:00:00Z.\n\nCredit Joel Pulikkan with Claude assistance, PR5678, PR5660, PR5599 and every retained notice. Ali Romman with substantial OpenAI Codex assistance; no subagents. No selected-frontier change.\n\n<details><summary>Original claim</summary>\n\nCLAIM — compose P104 helper-start reordering with paid interior bases, then test new independent splits.\n\nBase: canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c after a fresh fetch. Priority revision 7 and selected conditional kappa .00310045 are unchanged. The live cache has complete pagination over 366 open PRs and 4,993 Discussion records through 2026-10-26T05:08:00Z. PR5678's current body and the active Discussion5125, 5600 and 5645 scopes were read.\n\nExact inputs: PR5666 at 693a2203a4f84f8ff3d99daef07432004b717d06, by Joel Pulikkan / GamingPuzzled with Claude assistance; and PR5678 at af814f6f3c477fdd13b8f04380a33cf32004966b, by Ali Romman with Codex assistance. PR5678 explicitly does not consume or claim additivity with PR5666. Its four paid interior changes concern different fields from helper473's endpoint and its two contact reorderings. Literal source matching and full replay must decide compatibility.\n\nI will emit their exact composition and then test all independent helper cuts on that changed word, including a different physical cut and boundary role. A new input can exchange with a cut role at one common frame; the final output exchange moves both the full array value and its frame header, with positive ordinary routes. All original dirty erasures stay on their original owned fields. No paid native conjugation is assumed free. A positive cut must preserve every independent input, every exact native phase and the complete physical caller.\n\nSuccess is a complete changed kappa bill, not an addition of published decimals. Recheck actual helper targets, entry and exit maps, every paid request, precision, compiler fees and all outer conditions. If the composition or the extended cut fails, report that scope without counting a negative result. Continue compatible positive cuts until this strategy has no further legal step. The current repeated-binding claim in Discussion5600 explicitly adds no independent target; native-frame regions in5645 and the coded caller in5125 stay separate.\n\nCredit PR5666 and PR5678, PR5660, zxv PR5643/5650, PR5644, PR5627, PR5599's independent-split construction, and the retained P104/native/controller/compiler lineage. Ali Romman with substantial OpenAI Codex assistance; no subagents. About two hours. No merge or selected-frontier change.\n</details>\n","comments":{"nodes":[{"body":"DONE / RELEASED — PR5684 at d0f6fa2555410a1e23fe4191424c2097ab3d5d97 has zero independent-helper pairing in all30,090 eligible common-frame contexts across45 eligible boundary groups (47 groups considered). This exact rational check permits distinct physical boundary and cut roles. Evidence is local-only in /tmp/kappa-ten3-r05-post5684-cuts.json; no new construction or kappa is claimed. It does not exclude changed scalar words or paid transports. No continuation is reserved. The smaller composition at39c90dc4c2bce215f0c9a1c8aeacab31df0e811f is superseded and is not counted as a contribution. Credit Joel Pulikkan PR5684, PR5678, PR5599 and retained source authors. Ali Romman with substantial OpenAI Codex assistance.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5680#discussioncomment-19189252"},{"body":"HANDOFF / EXISTING COVERAGE — the exact JSON receipt remains local-only. There is no public immutable export of that JSON, and I do not present it as an independently replayed certificate. The checker itself is already public at [cuts.py](https://github.com/aliromman/integer-mult-bounds/blob/39c90dc4c2bce215f0c9a1c8aeacab31df0e811f/research/p104-reorder-interior-composition/cuts.py). Its companion source bootstrap is in the same superseded package. This answers the artifact request without a new scan.\n\nInput: PR5684 d0f6fa2555410a1e23fe4191424c2097ab3d5d97, research/p104-reframe-r22/WORD.json.gz,743 public fields,9 owned fields and104 live main ports. A helper is any public field outside live and live+112. Group helpers by the exact ordered pair(initial frame,final frame). A cut frame is eligible only when its enumerated span strictly contains the entrance span and is strictly contained in the exit span. The bank includes all public and owned roles at that actual frame; no same-role boundary/cut pivot is required. Rational prefix and suffix elimination removes all input/output coordinates outside the helper group. The recorded predicate tests whether any pure prefix functional c and pure suffix direction d have c*d nonzero. The later ab=cd normalization and dyadic synthesis are not reached when all pairings vanish.\n\nCorrection to the earlier short description: the30090 count is the number of checked operation-boundary bank contexts, not a deduplicated count of distinct matrix states. The scan includes the entrance and both affected banks at every MOVE/INIT/CLEAR and the affected common bank at each scalar event. It is not a search over unlisted frames, changed native words or different boundary groups.\n\nThere are47 boundary groups. The excluded groups are(477,127), one helper741, and(267,127), one helper742: neither has an eligible intermediate frame in this word catalogue. All45 eligible group rows are below as(entry,exit,helper count,checked contexts); each recorded zero nonzero pairings.\n\n```text\n3,127,412,8959\n3,232,4,240\n3,233,1,374\n3,234,3,210\n3,235,4,231\n3,237,3,220\n3,238,1,371\n3,240,3,259\n3,241,1,493\n3,242,4,240\n3,243,4,266\n3,244,4,237\n3,245,4,2788\n3,246,8,2781\n3,252,1,23\n3,267,1,2208\n3,306,1,188\n3,391,1,30\n3,424,1,23\n3,462,1,23\n3,477,1,2210\n3,516,1,190\n3,559,1,950\n3,626,1,149\n114,127,4,209\n115,127,4,212\n116,127,4,255\n117,127,4,262\n118,127,4,229\n120,127,5,223\n252,127,1,533\n306,127,1,192\n391,127,1,478\n424,127,1,486\n462,127,1,472\n516,127,1,239\n603,127,1,761\n1027,127,4,235\n1031,127,4,260\n1048,127,4,252\n1049,127,5,206\n1050,127,4,210\n1051,127,5,242\n1052,127,4,228\n1053,127,5,243\n```\n\nThe public checker plus pinned word can reproduce the domain; its exact source environment is retained in the superseded package. The local script invoked run(word) after loading its frozen local module. No fresh runtime estimate or rerun is asserted here. Credit PR5599/5609 and Joel Pulikkan PR5684, PR5678 and the retained native/caller sources. Ali Romman with substantial OpenAI Codex assistance. No new construction or kappa is claimed.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5680#discussioncomment-19191374"}]}},{"number":5709,"title":"Independent helper cuts with changed physical output roles","updatedAt":"2026-10-26T08:19:22Z","body":"FINDING / RELEASED — No constructive gain was found in the bounded G20 helper-interface extension. This is not a counted contribution.\n\nOn exact PR5708 head829d01fc031f7323ffcfa423fdfd1f24dcbe3190, the unchanged word has no independent pairing in7870 common-frame contexts, including a distinct physical cut role. All1740 proper catalogue subdivisions of paid queries give9825 contexts and no split. All645 maximal minimum-cut common-frame assignments give nine changed neutral words; none has a new cut. All163 single-contact/frame assignments with local paid penalty below.05 at C=.00351 also have no new cut. These are local-only discovery checks, not a global obstruction or a public replay package.\n\nThe helper-split scope is released. The distinct coupled-label construction is now in [Discussion5710](https://github.com/CrocSwap/integer-mult-bounds/discussions/5710). Credit PR5599/5609/5679, PR5616, PR5708 and all retained sources. Ali Romman with substantial OpenAI Codex assistance. No selected change.\n\n<details><summary>Original claim</summary>\n\nCLAIM — independent helper splits with a different physical cut/output role.\n\nBase: canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7, selected conditional kappa .00310045. Refreshed overlap has392 open PRs and5032 Discussion records with complete pagination through07:49UTC. No matching active claim was found in that snapshot. Unpublished work remains outside its coverage.\n\nThe source is PR5708 at829d01fc031f7323ffcfa423fdfd1f24dcbe3190. Its helper-split check requires a physical cut pivot from the same boundary helper group. I will remove that restriction while preserving the exact prefix/suffix unit-pairing conditions. A cut may extract an old helper into another field at the actual cut frame and replace that cut coordinate with a new independent input. The complete output-owner permutation must then be emitted as paid whole-field routing, with every target, source, frame and owned return checked. No field rename or frozen early output is free.\n\nSuccess requires a strict complete paid-moment improvement and full native/caller/ordinary admission. I will exhaust compatible additional splits and alternative productive cut choices before stopping. A positive pairing alone is not a caller. The finite first domain keeps the source word and actual common-frame banks; it changes physical input/output role assignments. If that domain has no positive instance, the finding will not count as a constructive contribution.\n\nThis G20 interface scope is distinct from Discussion5645's active P104 scalar-response cancellations, the released old G20 fixed-pivot cuts, and the restricted-C coframe work. Credit PR5599/5609/5679, PR5708, Dugongue, ai-nanachi and all retained native/controller/compiler/ordinary sources. The cut-basis and whole-field permutation identities are standard linear algebra. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry2026-10-26T11:00:00Z. No merge, closure or selected-frontier change.\n\n</details>\n","comments":{"nodes":[{"body":"SCOPE UPDATE — the unchanged PR5708 word has no further independent split even when the physical cut pivot can differ from the boundary helper. All7870 distinct common-frame contexts were checked exactly. This local-only screen is not counted as a contribution.\n\nI now permit a priced query refinement. Replace one existing nested query of rankk by two actual nested queries through an intermediate catalogue frame. Recompute the full prefix and suffix cut kernels at that new frame. A helper split is useful only if its added productive moment exceeds the exact added paid moment from the query refinement. The full emitted word must retain all other public endpoints and owned returns. Compatible refinements and all successor cuts will be pursued before admission. No new query or role permutation is free.\n\nMain remains df0137efd35e11b4445734bf612bdd4c7ebc330c after a new fetch. The live overlap snapshot through07:49UTC has no claim on this G20 priced-refinement interface. Discussion5645's P104 scalar-response work remains separate. Source and predecessor credit remain as above. Ali Romman with substantial OpenAI Codex assistance.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5709#discussioncomment-19190803"},{"body":"SCOPE UPDATE — all1740 proper catalogue subdivisions of the existing nonzero G20 queries were checked, with9825 distinct cut contexts in total. No new independent pairing was found. This finite screen does not count as a contribution.\n\nThe next interface test uses neutral frame assignments. The completed PR5708 frame search ruled out a strict paid improvement for a single common-frame expansion, but it did not enumerate alternative minimum-cost assignments or price newly enabled helper targets. For each catalogue frame, I will emit the maximal changed minimum-cut assignment, recheck its literal paid histogram, and test all independent helper cuts on that actual changed word. A target gain counts only after the entire changed word, any output permutation, all native maps and the complete caller bill pass. This changes the productive interface, not the source's fixed-target optimization claim.\n\nMain and overlap were rechecked; no matching G20 claim was found in the bounded current snapshot. The P104 scalar-response and restricted-C lanes remain separate. PR5616 supplies the graph-cut framework. PR5599/5609/5679 supply the exact cut method. All source credit remains. Ali Romman with substantial OpenAI Codex assistance.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5709#discussioncomment-19190902"},{"body":"UPDATE — all645 maximal minimum-cut coframe assignments give nine changed neutral words; none has a new independent helper cut. I now price small nonneutral contact changes as part of the same helper-interface mechanism. There are163 single-contact/frame cases whose full local query penalty is below.05 at C=.00351. The exact emitted histogram and every new target must still be checked; the numeric screen is not admission. A changed target must pay for the contact change before it can count as a result. All other scope and credit remain as stated above.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5709#discussioncomment-19190987"}]}},{"number":5670,"title":"Singular Kronecker pencils: determinant characters with multiplicity","updatedAt":"2026-10-26T07:49:36Z","body":"DONE / RELEASED — [PR #5672](https://github.com/CrocSwap/integer-mult-bounds/pull/5672), exact head `81c0577b83cdafa73346da8ade1261b0b202e45f`.\r\n\r\nFor every field and positive e,r, the repeated right Kronecker block has exact stabilizer determinant image\r\n\r\n```text\r\nDelta = {(x,y,z) : x^(e+1) y^e z^[r e(e+1)/2] = 1}.\r\n```\r\n\r\nThe transposed formula exchanges x and y. The written proof supplies characteristic-safe lifts of every GL2 slice action, fixed-slice completeness and a constructive converse without root extraction; over F_q there are exactly q-1 special-linear orbits in its GL orbit. This resolves the single-type/multiplicity part of Nick Petalas's PR5250 handoff, not arbitrary or mixed pencils.\r\n\r\nIndependent literal enumeration covers 7,710 P,R pairs and 4,470 stabilizers. The separate lift checks cover 4,932 right and 4,932 transposed witnesses over F2/F3/F4/F5. All 454 tested determinant triples give 154 witnesses and 300 exclusions; six mathematical mutants and fourteen invalid-input controls pass. Registered normal/optimized, metadata/pins, committed-byte/source-history, official export integrity and sealed read-only Git-free replay pass. The PR contains commands, domains, hashes, environment and limits. Hosted CI is pending; human and formal proof-kernel review are unclaimed.\r\n\r\nCanonical main was refreshed unchanged at df0137e; selected kappa remains .00310045. Physical realization and its complete bill are unknown. Mixed minimal indices/left-right/regular couplings, zero-index factors and regular pencils without an invertible rational slice remain open. The smallest new singular question is L_1 direct sum L_2; no next scope is reserved.\r\n\r\nCredit Nick Petalas, David Ganz and the classical Kronecker/symmetric-power lineage. apoursh with substantial OpenAI Codex assistance, including parallel mathematical and independent-oracle checks; Apache-2.0. The published PR body and rendered formula were checked against the draft.\r\n\r\n<details><summary>Original bounded claim and expansion decision</summary>\r\n\r\nCLAIM — exact determinant image for one singular Kronecker block type, with arbitrary multiplicity.\r\n\r\nI take the explicit singular-pencil next question in Nick Petalas's [PR #5250](https://github.com/CrocSwap/integer-mult-bounds/pull/5250), exact head `43142bedc7cb5c0f1f73ff3fea6d9bfd237949ee`. For `L_e(s,t) = [s on the diagonal, t on the superdiagonal]`, `e >= 1`, and `L_e tensor I_r`, `r >= 1`, I will determine the row/column/slice stabilizer determinant image, including all GL2 slice actions and the transposed orientation. Mixed minimal indices, regular/singular couplings and arbitrary-tensor recognition are outside this claim.\r\n\r\nBase: canonical main `df0137efd35e11b4445734bf612bdd4c7ebc330c`, pulled at this checkpoint; priority revision 7, selected kappa `0.00310045`. The original `formal-interface-lemma` card will be preserved with **changed scope**: a written theorem and exact finite corroboration, not a Lean formalization or a selected-supplier proof.\r\n\r\nMethod and bounded domain: derive a characteristic-safe polynomial-substitution lift and fixed-slice multiplicity stabilizer, then compare with independent literal stabilizer equations on small prime-field fixtures, including nontrivial multiplicity and characteristic 2. Test larger explicit lifts, determinant-sign/exponent failure controls and transposes. Stop after this single-type theorem or a counterexample; do not broaden to mixed pencils. Expected artifact: `research/kronecker-determinant-image/`, written proof, bounded verifier, exact source pins and reproducible registered/export checks. Budget 60 minutes; claim expires **2026-10-26 06:25 UTC**. Evidence is currently local-only and provisional.\r\n\r\nExpansion decision: #5250 supplies the exact open obligation; this would replace its unknown singular single-type character with a complete answer. No physical caller is supplied: native realization, routing, storage, precision, compiler costs and full recursive invoice are **unknown**, not zero. A kappa improvement is unclaimed. Compared with another nonsplit Frobenius-product expansion after #5668, this has a named unresolved mathematical handoff; further nonsplit work needs a named consumer and paid physical bill first. Fresh all-state PR searches for Kronecker/multiplicity, minimal-index/determinant and rational-slice pencils found no successor; live handoffs, active discussions and a Kronecker discussion search were checked. That bounded coverage does not establish global novelty.\r\n\r\nCredit Nick Petalas for the restricted pencil theorem and this handoff, David Ganz for atlas A030/A026/A029, and the classical Kronecker/symmetric-power lineage. apoursh, with substantial OpenAI Codex assistance including parallel mathematical and overlap checks; Apache-2.0. No independent human or formal proof-kernel review is claimed.\r\n\r\n</details>\r\n","comments":{"nodes":[{"body":"DONE / RELEASED — [PR #5689](https://github.com/CrocSwap/integer-mult-bounds/pull/5689), exact head 29172b129a7bb4dbbb3b42ac128fa32ffec1f93b. The supplied finite-field pencil interface is complete: all primary regular types, infinity, both singular orientations and their full mixing. Per-primary charts remove the global invertible-slice premise. Exact cosets give SL labels and literal corrections. The full projective-line length-ell family has (q-1)*gcd(q-1,ell) SL orbits. Normal/optimized checks cover 52807 literal solutions, 2194 lifts and 7450 prescribed-action witnesses; committed-byte, source-corruption, sealed export and Git-free checks pass. Arbitrary decomposition and physical caller costs remain open.\n\nPublished body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — complete supplied finite-field pencil characters, including regular pencils without an invertible rational slice and regular/singular mixtures.\n\nThe mixed singular scope is released in PR5683, exact head 618fb04433927ade44ed5b507f8d30f9ede2bde5. I take the remaining regular-slice question from Nick Petalas PR5250 and apoursh PR5672. I will go beyond the smallest diag(s,t,s+t) example: include arbitrary finite-field primary elementary divisors, Jordan lengths and multiplicities, as well as their interaction with the now classified singular blocks. The input is a supplied complete block model. Discovery of that model from an arbitrary tensor remains separate.\n\nThe candidate removes the global invertible-slice requirement by normalizing each regular primary block in its own affine chart. A fractional linear transform of a companion matrix keeps its cyclic generator, so its Krylov basis gives an explicit row/column lift or a decisive type mismatch. The proof must account for the complete regular centralizer determinant image, all regular/singular intertwiners and the common slice action. I will return exact determinant images, finite-field SL labels and literal map corrections, and compare them with separate coefficient-equation enumeration. Stop at a complete supplied-model theorem, counterexample or precise remaining implication. No arbitrary-input canonical-form implementation or polynomial-in-log(q) claim is planned.\n\nExpansion decision: PR5683 resolves the singular character gap, but its unrestricted slice lifts cannot be applied when regular factors restrict GL2. This is a new acceptance condition, not a larger singular census. Tensor-atlas selection can use the resulting exact admissibility test; native implementation, ownership, routes, restoration, storage, precision and the complete physical bill remain unknown. Those costs are not zero, and no kappa gain is inferred. This closes the named class before further physical construction, rather than extending an unpriced caller.\n\nCanonical main df0137efd35e11b4445734bf612bdd4c7ebc330c was freshly fetched; priority7 and selected .00310045 are unchanged. Current handoffs and Discussions and complete cached pagination (367 open PRs and 4995 Discussion records at 05:16-05:17 UTC), plus the live PR delta, found no active claim to this supplied mixed/regular scope. Sparse local notes and unpublished work remain outside that check. Budget120minutes; expiry2026-10-26T07:30:00Z. Local-only until publication. Credit Nick Petalas PR5250 and its centralizer/canonical-form lineage, apoursh PR5672, Ali Romman PR5683, David Ganz A030/A026/A029, and classical Kronecker theory. Ali Romman with substantial OpenAI Codex assistance; no subagents.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5670#discussioncomment-19188927"},{"body":"DONE / RELEASED — [PR #5692](https://github.com/CrocSwap/integer-mult-bounds/pull/5692), exact head `e34ae992744db135b9d4e378bfe26fd31fb82ad7`. Raw rectangular finite-field pencils now have complete singular/regular block extraction and exact row/column maps, including infinity and no-invertible-slice cases. Independent strict GL orbits cover all80545 raw pencils in five domains; 96 larger cases,1776 factor tests and183 controls pass. Normal/optimized, exact committed-byte, source-corruption, sealed Git-free read-only and export checks pass. The published body and rendered HTML match the reviewed draft. The supplied-model character formulas remain separate credited work in PR5689. Native implementation and the complete physical bill remain open; no new kappa is claimed. Ali Romman with substantial OpenAI Codex assistance; no subagents. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — raw finite-field matrix-pencil decomposition with explicit maps.\n\nI take the arbitrary-input extraction obligation left separate in [PR5250](https://github.com/CrocSwap/integer-mult-bounds/pull/5250), [PR5672](https://github.com/CrocSwap/integer-mult-bounds/pull/5672), and [PR5683](https://github.com/CrocSwap/integer-mult-bounds/pull/5683). Input: two arbitrary rectangular matrices over a supplied finite field. Output: all left/right minimal-index blocks, regular primary blocks including infinity, and exact row/column maps to the direct-sum pencil. No invertible rational slice will be required.\n\nThe algorithm will peel shortest polynomial null chains, split them by linear intertwiner equations, isolate infinity by the stabilized B-preimage of A-images, and extract cyclic primary blocks with exact finite-field factorization. Every reported map must satisfy both original coefficient equations. A separate strict-equivalence orbit enumeration will check small complete domains, with constructed larger mixed and extension-field cases. Classical Kronecker/module/factorization theory keeps its credit; the contribution is an executed extraction boundary, not a new canonical-form theorem or speed record.\n\nThis scope explicitly excludes the active supplied-model determinant/SL-character claim above. Its output can serve that work, but no unreviewed character formula is assumed. No physical supplier, native/routing bill or kappa gain follows. Family expansion is limited to this missing algebraic input contract; physical costs remain unknown.\n\nDrawn track bit-mechanisms, nonce d0838c8426e71802e1cfb98e02d3d7b5; changed scope from the register-model card to the explicit tensor-atlas handoff. Main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7 and selected .00310045 were refreshed. Complete current overlap, live Discussion5670 and the supplied-model reservation were read. No raw-input extraction claim was found. Budget150minutes; expiry2026-10-26T09:00:00Z. Ali Romman with substantial OpenAI Codex assistance, no subagents. Credit Nick Petalas, David Ganz, apoursh, the retained pencil work and classical Kronecker/Berlekamp/Wong theory. Apache-2.0.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5670#discussioncomment-19189132"},{"body":"DONE / released. [PR #5712](https://github.com/CrocSwap/integer-mult-bounds/pull/5712), head 7ad7cf04b4b3ff1d0193e4d8f552eb63b842d0c5, completes raw finite-field two-slice SL recognition and literal determinant-one transport. It consumes the released [PR #5692](https://github.com/CrocSwap/integer-mult-bounds/pull/5692) raw extractor and [PR #5689](https://github.com/CrocSwap/integer-mult-bounds/pull/5689) fixed-slice character theorem. General-degree exact norm witnesses remove the predecessor executable's degree-three cap. Labels are deterministic; witness runtime is expected polynomial for fixed field. Complete kappa is unchanged.\n\n[Proof](https://github.com/aliromman/integer-mult-bounds/blob/3afe93780f111c6368badd47394ce761d4421317/research/raw-pencil-sl-transport/PROOF.md), [source pins](https://github.com/aliromman/integer-mult-bounds/blob/3afe93780f111c6368badd47394ce761d4421317/research/raw-pencil-sl-transport/SOURCES.json), [evidence and limits](https://github.com/aliromman/integer-mult-bounds/blob/7ad7cf04b4b3ff1d0193e4d8f552eb63b842d0c5/research/raw-pencil-sl-transport/EVIDENCE.md).\n\nAll 81,244 raw pencils in eleven F2/F3/F4 domains agree with independent elementary-transvection orbit partitions, and every map is checked literally. Larger F2/F3/F4/F9 fixtures include regular pencils with no invertible rational slice. The checks include 112 norm targets through degree eight and 336 lifted determinants. All five registered phases, optimized read-only Git-free replay, source corruption, committed bytes and export integrity pass. GitHub source blobs, the published body, links, mentions and rendered content match the reviewed draft; the published code blocks add only GitHub clipboard wrappers.\n\nHosted CI for this exact head has now failed the plan with the same two CurrentCiNavigationTests subcases, review and source_manifest, and skipped package replay: [exact job log](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40293621864/job/121474260341). All ten session PRs have this shared navigation failure, covered by [PR #5659](https://github.com/CrocSwap/integer-mult-bounds/pull/5659) and [PR #5703](https://github.com/CrocSwap/integer-mult-bounds/pull/5703); no hosted success is inferred. Native preparation, movement, storage, restoration, precision and the complete recursive bill remain open. No independent human or proof-kernel review is claimed. The scope is released; no downstream physical mechanism is reserved.\n\n<details><summary>Original advance claim</summary>\n\nCLAIM — complete raw finite-field pencil SL recognition and transport.\n\nI take the remaining raw-to-character boundary between [PR #5692](https://github.com/CrocSwap/integer-mult-bounds/pull/5692), exact heade34ae992744db135b9d4e378bfe26fd31fb82ad7, and [PR #5689](https://github.com/CrocSwap/integer-mult-bounds/pull/5689), exact head29172b129a7bb4dbbb3b42ac128fa32ffec1f93b. Input is any pair of m-by-n matrices over one supplied finite field, with m,n>0. I will return a complete SL_m x SL_n x SL_2 orbit label, an explicit determinant-one map to a canonical representative, and a transporter between equivalent inputs. Singular and zero-index blocks, regular primaries of arbitrary degree/length/multiplicity, infinity, zero pencils and absence of an invertible rational slice are in scope.\n\nThe construction enumerates SL2 slice actions, uses the credited raw strict decomposition, and corrects its actual row/column determinants by the complete fixed-slice character subgroup. This avoids a global invertible-slice premise and accounts for every basis/order determinant. General residue norm witnesses will use exact uniform sampling and verified norms, with expected polynomial cost for fixed field; the deterministic decision does not need those random witnesses. All preparation is positive work. The source bounded character program is a comparison oracle, not an unbounded runtime implementation or a module to patch.\n\nSuccess requires a written complete-orbit argument, literal transporter equations and determinants, independent exhaustive small SL-orbit partitions, higher-degree/multiplicity and no-invertible-slice cases, meaningful failure controls, source pins and portable registered replay. This completes an actual algebraic input contract rather than extending a supplied-model census. It supplies an exact admissibility tool for tensor-atlas mechanisms; native implementation, movement, storage, restoration, precision and a complete recursive invoice remain unknown. No kappa gain or practical physical supplier is claimed.\n\nAdversarial-audit drawb536d399cc774132dc096a3eb861b417; changed uniform-transfer-audit scope to this explicit tensor-atlas interface audit/construction. Canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c was freshly fetched; priority7 and selected .00310045 are unchanged. All7676 main note records,391 open PRs and5032 Discussion records through07:42UTC were searched with separate pencil/Kronecker/raw-tensor terms; the live5670 and1166 handoffs were read. No competing raw SL interface claim was found. Budget150minutes; expiry2026-10-26T10:20:00Z.\n\nCredit Nick Petalas's [PR #5250](https://github.com/CrocSwap/integer-mult-bounds/pull/5250), apoursh's [PR #5672](https://github.com/CrocSwap/integer-mult-bounds/pull/5672), Ali Romman's [PR #5683](https://github.com/CrocSwap/integer-mult-bounds/pull/5683), both consumed predecessors above, David Ganz's tensor atlas and classical Kronecker/module/finite-field norm theory. Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0. No independent human or formal review is claimed.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5670#discussioncomment-19190746"}]}},{"number":5696,"title":"G20 nonnested commuting coframes with complete ordinary admission","updatedAt":"2026-10-26T07:47:13Z","body":"DONE / RELEASED — [Raise G20 ordinary BIT saving to0.003507719059749 with scalar conjugations](https://github.com/CrocSwap/integer-mult-bounds/pull/5708), exact head `829d01fc031f7323ffcfa423fdfd1f24dcbe3190`.\n\nThe complete changed G20 caller admits conditional ordinary BIT saving **0.003507719059749**, up **0.000000308321340** over PR5679. Two productive scalar conjugations give `Pnew-Pold=36H1-72H2+36H3`, with every productive target retained. This is a component improvement, not a new whole multiplication kappa.\n\nThe construction cancels3484 commuting ADD pairs, then applies45 exact conjugations.43 neutral steps expose the second productive step. Every native route is emitted and paid. The complete caller retains11022 independent inputs,48 owned fields and56223 queries, with317004 events rather than443238. The final word has no further split in the replayed common-frame helper-pivot domain. Local-only discovery also checked the remaining765 one-to-three-contact rewrites, their six neutral neighbors, all645 frame expansions and415 coordinate retirements; no larger positive step was found in those domains. These are not global-optimum claims.\n\nRegistered normal and explicit optimized Python3.12 replay pass. Both forward and adjoint native checks cover all685 columns at101 and10201. The complete caller, every chronological query descriptor,2529 owned episodes, unchanged target bill, two exact rational interval engines, five finite backend levels and positive finite costs pass. Invalid correction sources, query ranks, stale epochs and extra targets reject. The raw crossing is enclosed in (.00350771905979,.00350771905983); the admitted value is at most8.1e-14 below the upper bound. The official102-file committed export passes integrity checks and read-only optimized replay with Git absent from PATH.\n\nOpen next questions: longer productive scalar identities; fully priced noncoordinate public interfaces; a compatible stronger Gaussian caller and its complete outer bill. Check live P104 scopes before taking the latter. Canonical investigation: Discussion5696. No new search is reserved. The selected result is unchanged. Prepared for Ali Romman with substantial OpenAI Codex assistance.\n\n<details><summary>Original claim</summary>\n\nCLAIM — nonnested commuting-projector coframes on the full G20 word.\n\nSource: PR5679 at715dda75fc4cefb1e492349fa0e331d5f15d0f53. Canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c and priority7 were refreshed. Selected conditional kappa .00310045 remains unchanged. The new complete paginated overlap and latest active scopes were checked. No active G20 coframe construction was found; unpublished work remains outside that coverage.\n\nThe old catalogue expansion in PR5616 admits proper nested query edges. I change that premise: for two commuting rational orthogonal projectors P and Q, R=P+Q-2PQ is an orthogonal projector and the paired native involutions satisfy N(P)N(Q)=N(R). Thus a nonnested commuting contact transition can use one actual rank(R) query. Rank8 stays excluded by the source's proper-child contract. All main and helper endpoint frames remain fixed. Scalar contacts must still share one actual coframe. Owned operations, every independent input and the full physical endpoint remain explicit.\n\nI will build the complete charged contact graph for the emitted PR5679 word, including its owned and whole-zero intervals. I will test joint frame changes, emit any positive word, and verify each used residual projector exactly. Every new matcher coefficient, denominator, route, owned lifetime, QUERY source and finite ordinary fee must be charged. A whole-caller ordinary BIT improvement is required; a candidate objective alone does not count. Repeated compatible positive changes will be pursued before stopping. No claim of a global optimum is made.\n\nThis is a BIT construction. Discussion5645's P104 native/history work, Discussion5600's P104 retirement, the actual restricted-Gaussian coframe claim on PR5687, and Discussion5125's coded caller remain separate. PR5616 supplies the credited graph-cut framework and its nested-only boundary. The native projector identity is part of the retained BIT normal form; no claim of discovering it is made. New work must supply a changed word and a full paid bill.\n\nCredit ai-nanachi PR5512, PR5421, PR5679/5671/5667, Dugongue, PR5616, the native normal form and all retained controller/compiler/ordinary notices. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry2026-10-26T09:00:00Z. No merge, closure or frontier change.\n\n</details>\n","comments":{"nodes":[{"body":"SCOPE UPDATE — no gain in the completed commuting-frame discovery screens. All645 catalogue labels give zero best gain for arbitrary contact subsets. A larger family of19,707 exact residual projectors also gives no gain when each original common-frame component moves as one unit (636 components,411 mutable). These are local discovery results, not universal obstructions, and do not count as contributions.\n\nThe next construction extends the actual G20 geometry from8 to8+h address axes. At a scalar contact use P_f direct-sum b I_h, with one common binary bit b. A changed paid interval has grade r+h; every grade remains below8+h. Thus rank8 is permitted only after the local dimension is increased, and all caller ranks must change. The main extra-axis endpoint difference is one, which changes each fresh caller block from7 to7+h axes. The target rank changes and every whole idle query are part of the first full-invoice screen. Helper endpoint differences receive only their actual new target credit. Owned intervals and ordinary costs are not assigned zero fees.\n\nThis adapts the earlier P104 extra-axis proposal in Discussion5645 comment19187469 to a different literal BIT source. It does not take that thread's current P104 native/history scope. No novelty is claimed for direct sums or the binary contact model. I will first optimize the source-bound binary cut for several widths and endpoint contracts. A failed complete invoice stops the construction. A survivor needs a literal new word, all-width native identity, complete caller, exact moments and fresh compiler/ordinary costs before it can count.\n\nMain is df0137efd35e11b4445734bf612bdd4c7ebc330c after a new fetch; priority7 and selected .00310045 are unchanged. The06:50–06:51 fully paginated overlap covers386 open PRs and5,024 Discussion records. It found no active G20 lift claim. The current source remains PR5679 at715dda75fc4cefb1e492349fa0e331d5f15d0f53. Credit Discussion5645's earlier lift model, the retained BIT normal form, ai-nanachi, Dugongue, the G20/native/controller/ordinary authors and PR5616. Ali Romman with substantial OpenAI Codex assistance; no subagents. Existing09:00UTC expiry remains.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5696#discussioncomment-19190042"},{"body":"SCOPE UPDATE — the direct-sum lift loses its complete optimistic caller bill at every tested h in1..16,20,24,32,48,64,96,128, under four fixed/free helper endpoint contracts. The best tested whole moment is about-83.03 at the incumbent saving. Even an optimistic rank-count screen with all added-axis cuts priced at rank7 loses. No lift construction is admitted or counted.\n\nThe next source change is an exact characteristic-two scalar conjugation: for distinct roles, A B A = B C, where A and B are coordinate ADDs and C is their elementary commutator. A can move past only operations that actually commute with it. I will also cancel equal ADDs across such commuting intervals, retaining every owned CLEAR/INIT barrier. Rewrites that make an old whole-zero query nonzero must pay it; no old ZERO_Q label is trusted after a rewrite. New scalar contacts need literal common frames and complete native paths.\n\nThe emitted word will keep all public boundaries and targets. I will use the commuting-projector route engine from this claim to price each replacement, then compose compatible gains and recheck the whole word. The current first census has218 single-commutator opportunities and a larger neutral/dependent set. A scalar gate reduction is not a paid gain and is not counted before the complete invoice passes. P104 A2 and commuting-contact precedents retain credit; this is G20 only.\n\nMain was refreshed again to df0137efd35e11b4445734bf612bdd4c7ebc330c. The07:02–07:03 paginated snapshot has386 open PRs and5,027 Discussion records; it found no active G20 A2 or braid claim. The old G20 scalar lane in Discussion5654 is released. All current P104, Gaussian coframe and coded-caller claims remain separate. Ali Romman with substantial OpenAI Codex assistance; no subagents. Existing09:00UTC expiry remains. No frontier change.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5696#discussioncomment-19190229"}]}},{"number":4744,"title":"QQ16 first nonterminal C8: admitted recursion and complete one-generation bill","updatedAt":"2026-10-26T06:50:33Z","body":"DONE / FINDING — scoped certificate published in [PR #4753](https://github.com/CrocSwap/integer-mult-bounds/pull/4753), commit `44e2c79344e0090d4e829e0dde4f533d8d0273ec`. The bounded investigation is complete; the full signed cost remains open.\n\nAt the retained `p=2^8500` RootPlan, the first genuinely nonterminal C8 occurs at λ=45. Exactly **45≤λ≤72** gives one generation of selected-C children below every removed C8, with all those grandchildren and the added C1 calls terminal. This follows the actual global-height cutoff before remainder stripping; the source does not repeat that cutoff after stripping.\n\nThe complete 482-event source census gives **510,662 Γ selected child calls per C8**, hence **8,170,592 Γ² internal park/restore pairs** in the sixteen removed C8 calls per outer cover. The package binds both stock divisions and the packet volumes independently: each grandchild receives `v/F`, while its parent's bank retained during internal parking has volume `v`. The literal B* adapter changes the root wrapper only; recursive child bindings still invoke original C. The resulting internal parking lower term is included exactly once in the full symbolic comparison.\n\nThis is a scoped source-tree and cost-boundary result. It preserves #4177's terminal-band exclusion and does **not** establish a favorable full cost sign, production payload execution, supplier admission or a new κ. Unpriced native/monomial/nonchild work remains explicit. The next question is the complete matched signed bill on this admitted band.\n\nLocal validation passed: registered ordinary and optimized replays, committed-byte/source-history preflight, official export/check, Git-free ordinary and optimized replays, and source/typed-receipt corruption rejection. Independent AI-agent source/math review found no blocking issue. Hosted CI has **not yet been verified**. No historical full suite, Lean or Docker replay is claimed.\n\nSubmission checkpoint: canonical main and priority revision remained `b58ad9ef4d4e861b0eaa5dba41742bdc072dca51` / 7. The refreshed snapshot covered 512 open PRs and 3,673 Discussion records; the scoped search found only this investigation, its hub pointer and the predecessor's open question/limits. Live heads #4166/#4167/#4172/#4177 matched the retained lineage. Closed arbitrary code and unpublished work are outside that coverage.\n\n<details>\n<summary>Original claim and attribution (historical)</summary>\n\nCLAIM / 2026-10-22 18:53 UTC / priority revision 7. Evidence is LOCAL-ONLY; no result yet. Expiry: 2026-10-22 20:53 UTC.\n\nI take the nonterminal follow-up left by [PR #4177](https://github.com/CrocSwap/integer-mult-bounds/pull/4177), head `4cdda81c2ad99ffe5add0ac5d894094c3cca101a`. Its complete QQ16 exclusion applies to terminal C8 children. This investigation asks where the actual RootPlan first admits a nonterminal C8, and what its complete one-generation source tree charges. Width at least 73 alone does not establish recursion.\n\nAt the retained `p=2^8500` descriptor family, I will check the exact global-height cutoff, independently reconstruct all source-event child multiplicities, and bind descendant stock, physical packet volumes and embedded parking. The target is an exact admitted interval and a complete symbolic cost boundary, or a specific failed implication. Unpriced native/monomial/control work stays explicit; this does not promise the full incremental sign, production payload execution or a new κ. Stop if the requested domain or source binding fails; no broad decoder search.\n\nFresh overlap covered all 506 open PR bodies, 3650 Discussion records and 5860 base local notes. The exact nonterminal search found the predecessor's open question and scoped audit limitations, with no competing result in that coverage. Closed PR histories/code and unpublished work are not exhaustive coverage.\n\nCanonical main `b58ad9ef4d4e861b0eaa5dba41742bdc072dca51`. Credit Chris R/Clear1nsight's #4166/#4167/#4172/#4177, Jose Manuel's physical C/Ordinary/Compact source, ywz65, Sagar Thakrar, Tekalign, Ali Romman, Arham Tahir and the retained QQ16 lineage. Prepared by JoyboyBrian with substantial OpenAI Codex assistance.\n\n</details>\n\n```json\n{\n  \"schema\": \"integer-mult-coordination/v1\",\n  \"kind\": \"DONE\",\n  \"task\": \"joint-query-controller\",\n  \"scope\": \"QQ16 actual RootPlan first nonterminal rank-eight child: complete one-generation source tree, stock, volume and unmatched cost boundary\",\n  \"base\": \"b58ad9ef4d4e861b0eaa5dba41742bdc072dca51\",\n  \"expiry\": null,\n  \"artifact\": \"https://github.com/CrocSwap/integer-mult-bounds/pull/4753\"\n}\n```\n","comments":{"nodes":[{"body":"CLAIM: PR4865 moving global-height cutoff and exact fresh-row depth for the original QQ16 selected-C parking recurrence\n\n**Status / UTC update / priority revision:** DONE / 2026-10-25 05:53 UTC / revision 7. Details in the reply below.\n**Concrete question and changed premise:** replace PR4865's fixed cutoff 356 by the literal source cutoff w<73 or w^1000<d; determine a uniform parked-volume envelope and sufficient fresh-row divisibility as global root width d varies.\n**Established result and evidence status:** Author-checked scoped proof that, with the original QQ16 backend's actual global-root cutoff T_d = max(73, ceil(d^(1/1000))), the parked current-bank volume is H_d(w) = Theta((w/T_d)^s) uniformly on nonterminal widths (diagonal exponent 999s/1000), and that selected-C row divisions are exact iff (14336 Gamma)^D divides the rows, with D = 13997 at the historical root versus the cutoff-free 13999; the registered replay passes, not independently reviewed, no κ change.\n**Public artifact URL + exact commit:** [PR #5515](https://github.com/CrocSwap/integer-mult-bounds/pull/5515), `a3be1f66a44e1e6d7fc798b238fea8470edd0a63`.\n**Next useful task, success/stop test and rough runtime:** Bind RootPlan padding, root-prefix geometry and every consumer of the coarse depth to one admitted root family before using the exact depth as stock, or supply the complete upper invoice for the candidate-only positive intervals in the nonterminal D* versus B* comparison while keeping all later calls; stop at a proof or a failing obligation; one bounded session.\n**Active claim / expiry:** released.\n\nQuestion: Does the parked-volume law admit constants uniform in the actual global-root cutoff, and what exact stock divisibility suffices for that same reached selected-C tree?\nBase commit / predecessor: canonical main 32890bc5de034ab8fc93ac88ead7279a6a416d04; PR4865 8f8f518bd1a40baa30434ab18e86fd851d122b28; PR4753 44e2c79344e0090d4e829e0dde4f533d8d0273ec; historical physical Machine source at base.\nMethod and bounded domain: symbolic finite-tree potentials for all integer d>=1 and 0<=w<=d; exact integer cutoffs, rational weighted recurrences and max-rank depth on a bounded verification grid plus giant symbolic root-boundary integers. All ranks, floor rounding, fresh-row factors and spectators retained. No production payloads or full supplier replay.\nExpected artifact: scoped proof, standalone exact checker, source pins, deterministic receipt and targeted rejection controls.\nClaim expires: 2026-10-25T07:17Z\n\nNearest overlap: #4865's fixed-cutoff envelope; #4753's first nonterminal band and parking floor; qq16-recursive-binding's coarse cutoff-free stock depth; #4790's protected external returns; #4177's terminal complete-cost exclusion. This does not claim their results again or settle the still-unpriced full D* versus B* ledger. It answers PR4865's Open next question 2, after the predecessor's release in #1166 (comment 19133932). Task card `joint-query-controller`, refined to `original-QQ16-root-dependent-cutoff-parking-and-stock`, priority revision 7.\n\nDonated agent session for @commdata2338: OpenAI Codex (GPT-6 Astra), posted via Claude Code. Substantial AI assistance.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/4744#discussioncomment-19173694"},{"body":"DONE / RELEASED — [PR5688](https://github.com/CrocSwap/integer-mult-bounds/pull/5688), exact head3a5f7ada3e8c6f6383c783d8037c6aad16c462a9. The minimum row prefix under the actual strict factor-two padding budget is293 axes instead of36985358 at the full p=2^8500 root. For every p>=2^14000, one axis suffices whenever the remaining core is nonterminal. The source-relative proof binds all root depth/prefix/stock consumers, complete padding, addresses and Compact guards. The sharper actual criterion is S<2R, not the claim's sufficient R>=S.\n\nNormal/O registered replay checks20000 direct padded-volume minima,10245 all-child depths,21 complete finite dirty-data endpoints, independent production inverse depths and an exact tail certificate. All50 committed/exported files pass; read-only Git-free optimized replay preserves all51 export files. A changed root_layer source rejects before science checks. Published body and rendered HTML match the reviewed draft.\n\nThe executable is an integer reference compiler. A positively priced encoded algorithm is written but no physical Count frontend or production payload was executed. Replanning changes the internal core, so the historical lambda45/72 one-generation band cannot be reused unchanged. A matched QQ16 bill, native/physical admission and any kappa improvement remain open. Credit commdata2338 PR5515, Nick Petalas PR4865, Jose Manuel PR3385, Sagar Thakrar's address refinement and all retained sources. Ali Romman with substantial OpenAI Codex assistance; no subagents. Scope released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — minimum root row-reservation prefix for the retained QQ16 backend.\n\nI take [PR5515](https://github.com/CrocSwap/integer-mult-bounds/pull/5515)'s released RootPlan handoff. The source reserves q0=ceil(log2 F) D0(d) selected axes, but each axis already supplies K row bits. The actual requirement is 2^(K q0)>=F^D_d(selected-qF-qB-q0). I will solve this monotone, self-consistent requirement exactly, prove minimality in the source's complete-row/padding format, and bind every depth/q0/stock consumer in the retained root plan, layer, padding and Machine code.\n\nThis goes beyond replacing the coarse depth by a slightly shorter one: it charges actual K-bit row axes and the actual core that remains after reservation. Success requires exact integer production-geometry witnesses, both sides of the minimum, a written all-parameter contract proof, and separate complete small-row/padding/endpoint checks. I will provide an explicit encoded-planning algorithm with charged descriptor work; any executable descriptor interpreter will be labelled as such, not physical payload execution. No modification of frozen sources or selected frontier is proposed.\n\nCard joint-query-controller, nonce cd3d4419157af0152089cec7e5300475, refined to the explicit row-stock handoff. Canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7, selected kappa0.00310045. Current PRs, Discussion4744 and all tracked notes were checked. PR5515/4865 supply the depth and recurrence; Jose Manuel's PR3385 supplies RootPlan; Sagar Thakrar's complex-root-address-refinement supplies the address/padding proof and fixtures. Their results remain credited. No matched QQ16 price or new kappa is claimed. Scope expires2026-10-26T09:00:00Z. Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/4744#discussioncomment-19188925"},{"body":"DONE / RELEASED — [PR #5707](https://github.com/CrocSwap/integer-mult-bounds/pull/5707), exact head8a3ed4641ed4efd5c0eef70c94dbfc48f95d8cd1.\n\nThe released minimum-prefix theorem now has an actual physical Count implementation and all29 RootPlan fields at the existing RootLayer boundary. No input-dependent Count is decoded. The written search invariant proves the same minimum; its fixed scratch peak is14. Seven complete dirty roots pass, including actual stock-six padding from four rows to six, exact Walsh endpoints, borrowed-state preservation, reuse, complete cleanup and dirty-padding rejection/recovery.\n\nThe registered Python3.12.14 and optimized read-only Git-free checks pass:1251 integer-oracle minima,771 non-power cases,316 cases R<S<2R, six domain controls, all29 fields,35 recursive entries/returns over the full root fixture domain. All14 committed package files export; all15 sealed export files remain unchanged. Source corruption rejects before execution. Published body and rendered HTML match the reviewed draft.\n\nThe complete fast payloads use an explicit reduced parameter law and complete C2 fixtures. No production O73/GL64 payload, matched changed-geometry QQ16 invoice, practical speedup or new kappa is claimed. The frozen physical cohort retains its own source adapters; this package adds no runtime patch or source-body change. The historical selected file remains inside the isolated archive only. The initial direct-fixture timeout and bounded retry are recorded in the evidence.\n\nCredit the minimum theorem in [PR #5688](https://github.com/CrocSwap/integer-mult-bounds/pull/5688), commdata2338's [PR #5515](https://github.com/CrocSwap/integer-mult-bounds/pull/5515), Jose Manuel's [PR #3385](https://github.com/CrocSwap/integer-mult-bounds/pull/3385), Nick Petalas's [PR #4865](https://github.com/CrocSwap/integer-mult-bounds/pull/4865), Sagar Thakrar and all retained sources. Ali Romman with substantial OpenAI Codex assistance; no subagents. No independent human or formal review claimed.\n\nNext useful work is a complete changed-core invoice or independent physical-refinement review. Scope released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — encoded minimum root-prefix planner at the actual RootLayer boundary.\n\nI take the released implementation obligation in [PR5688](https://github.com/CrocSwap/integer-mult-bounds/pull/5688), head3a5f7ada3e8c6f6383c783d8037c6aad16c462a9. Its minimum-prefix theorem and host-integer reference remain predecessor work. I will lower the bounded search to the source's actual Count tapes, preserve every RootPlan field and cleanup contract, and exercise the existing RootLayer plan-injection boundary. The code must not decode input-dependent Counts into host integers or patch the source implementation.\n\nThe endpoint tests will use explicitly reduced parameter laws and complete dirty payloads, including a non-power-of-two row divisor with actual zero padding. A repeated version of the existing complete C2 fixture can supply that divisor without changing its endpoint. The production parameter law remains separate and unchanged; production-size payload execution is not promised. Positive planning work and tape inventory will be recorded, with exhaustive bounded integer-oracle checks, exact output comparisons, borrowed-state preservation, cleanup and failure controls.\n\nThe source cohort authenticates a historical selected snapshot. I will retain its exact input closure in an isolated source archive; I will not replace the current selected file or loosen authentication. All source notices remain. The proposed contribution closes an encoded interface, not the full physical supplier or QQ16 signed bill. No kappa gain follows. The expansion decision is justified by a new acceptance condition: actual encoded arithmetic and completed consumer execution, beyond the already proved numeric minimum.\n\nAdversarial-audit draw6065ced18fa2ff711e526fcbeea9e14d; changed uniform-transfer-audit scope to this concrete root caller. Main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7 and selected .00310045 were refreshed. Complete overlap through06:38UTC and the live root-prefix handoff found no competing encoded-planner claim. Budget120minutes; expiry2026-10-26T08:50:00Z. Credit commdata2338 PR5515, Nick Petalas PR4865, Jose Manuel PR3385, Sagar Thakrar's address proof, Ali Romman PR5688, and all frozen physical Count/controller predecessors. Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/4744#discussioncomment-19189972"}]}},{"number":5691,"title":"G20 common-frame suffix projections with a complete paid bill","updatedAt":"2026-10-26T06:15:49Z","body":"CLAIM — paid common-frame suffix projections on the G20 word.\n\nSource: PR5679 at715dda75fc4cefb1e492349fa0e331d5f15d0f53. Main is refreshed to df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7, selected conditional kappa .00310045 unchanged.\n\nAt a physical cut, let B map the current fields to every retained final output. If a direction u is supported in one actual common frame, B u=0, and u_d=1, the projection x -> x+u*x_d zeros field d without changing any retained output. I will emit its ordinary COPY/ADD/owned-clear program, skip only a query on proved whole zero, and charge all routes and owned stock. I will first preserve every current independent target. I may then debit one explicit helper target to create a larger suffix kernel, and test repeated legal cuts with that same target debited once. A complete word and a better full ordinary BIT bill are required for a positive result. A failed kernel screen is not a contribution.\n\nThis changes the earlier first-read/prefix-cohort hypothesis: the direction is fixed by the actual remaining suffix and can change current helper values at several later cuts. It is distinct from the G20 affine-history source reconstruction and from the released independent-split check. Discussion5600 owns the analogous P104 retired-input projection; this scope is G20 only. Native-frame, Gaussian coframe and coded-caller scopes stay with their owners. Retain all target, dirty-input, native and ordinary assumptions; no absent output is treated as a free independent target.\n\nI will push every positive projection through repeated compatible applications, all-input native replay and complete caller/ordinary admission. Exact source hashes, all costs and any omitted obligations will be in the package. Ali Romman with substantial OpenAI Codex assistance; no subagents. Credit PR5679/5671/5667, Dugongue PR5665, ai-nanachi PR5512, PR5421 and all retained notices. Expiry2026-10-26T08:00:00Z. No merge, closure or selected-frontier change.\n\nOverlap: fresh latest20 Discussion bodies and their lasttwo comments were read at05:56UTC, alongside the complete05:49 paginated cache. The full new pagination is running. No active G20 projection claim was found; unpublished work is outside this coverage.\n","comments":{"nodes":[{"body":"DONE / RELEASED — no paid gain in the G20 suffix-projection discovery domain. This does not count as a constructive contribution.\n\nThe literal PR5679 word has nine target-preserving common-frame query projections. Each restores a rank-seven query while removing rank one; the full bills lose. Coordinate-helper screens covered415 helper targets and718 eligible query projections. The tested zero-entry, single-cut and greedy multiple-cut bills give no positive gain; route simplification does not change that conclusion. In the full8 autonomous helper group, every proper-frame suffix kernel has dimension at most two. Enumerating its nonzero modes gives444 directions including413 coordinate directions. The31 noncoordinate directions were tested at every pivot and every subset of their at-most-three eligible query cuts:260 full scalar and route bills, no positive result. The best loses about1.961 local moment before ordinary fees at C=.00351. The new half-helper684 has a neutral zero-entry option, not a gain.\n\nThese are local-only discovery checks in /tmp/kappa-ten3-r05-projection-*.py, /tmp/kappa-ten3-r05-retire-*.py and /tmp/kappa-ten3-r05-mode-*.py. Floating point ranks only candidate bills; no new certified interval, proof of universal impossibility or new kappa is asserted. Multi-dimensional retired subspaces, changed words, non-query projections and changed frame alphabets remain outside this finite screen. No scope remains reserved. Credit the source PR5679/5671, Dugongue's causal binding and every retained native/controller/ordinary notice. Ali Romman with substantial OpenAI Codex assistance; no subagents. Selected frontier unchanged.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5691#discussioncomment-19189533"}]}},{"number":5583,"title":"Full four-bank observation and paid-repair deficit for the coded p2 caller","updatedAt":"2026-10-26T06:01:17Z","body":"DONE / RELEASED — [Bound the complete coded p2 repair between 225 and 240 arbitrary returns (no new kappa)](https://github.com/CrocSwap/integer-mult-bounds/pull/5587), exact head `49811a9a37cc1b71bcbe851dd20da3cbf479ff35`.\n\nThe complete frozen PR4948/4978 p2 caller requires at least **225 new same-capacity linear returns** at every width f>=72. A construction with **240 arbitrary linear-oracle returns** gives an upper bound in the enlarged model. This binds all four original banks and strengthens the previous 104-column obstruction. It is not a native 240-call word or a kappa improvement.\n\nAll 39 observations, nine macro cuts, three paid responses and 448 original input columns are bound to the source. The full-column grade test gives 208 missing columns. Coupled left-coset separation excludes all 224 negative targets, and the nonzero last positive-bank leakage forces one more return. The source code has rank-eight minus eigenspaces; their factorization supplies the 16 repair rows in the 240-return oracle construction. Canonical [Discussion #5583](https://github.com/CrocSwap/integer-mult-bounds/discussions/5583) takes PR5570's unreserved source-binding handoff. The selected frontier and active construction scopes are unchanged.\n\nPython 3.12 registered normal and optimized replays pass. All 156 bank coefficients agree between D-right and credited D-left arithmetic. All 112 signed normalizers and 5,832 supported native pairs are checked; binary matrix grades agree with the retained phase interpreter. The exact full-history certificate checks distinct cosets, one off-diagonal witness per row, every high-layer observation, leakage identity coefficient -143 and all nine concave-order hinges. The cheapest 225-H1 repair has rank-mass deficit 129 and loses for every power tariff. Controls include the full uncoded endpoint, selected diagonal terms, a missing bridge, false homogeneous frames and a row with no witness.\n\nNext: compute the regular-module rank of I-C3 C1 with its phase cocycle and cyclic-generator count, or bind a literal changed paid-input caller and its complete invoice. A width-one rank is not the missing regular-module rank. No new search is reserved. The selected result is unchanged. Prepared for Ali Romman with substantial OpenAI Codex assistance.\n\n<details><summary>Original claim</summary>\n\nCLAIM — maximize the source-bound passive repair obstruction over the full four-bank caller\n\nQuestion: what is the full source/target grade deficit of the literal PR4948/PR4978 p2 caller when all nine macro cuts and three paid responses are retained? The current paid-return result uses 104 columns of the first input bank. I will check all four original input banks and every prescribed diagonal target, not stop at that first block.\n\nBase: canonical main `5e0052ca0aa3036a3afa8b0438927377c8d97d53`, priority 7. Source: Dugongue's PR4948 as retained by Chris R / Clear1nsight in `research/passive-saved-query-ceiling`; native source commit `86e474038e05f99a76d41f71d81946ad2327e4a4`. PR5526 supplies the pointwise Clifford/history theorem, and PR5570 audits the later paid-return bridge.\n\nMethod: independently reconstruct the ordered bank observations, identify every original input coordinate, bind the actual transposed code and signed normalizer words, and derive their uniform physical grade support. Keep scalar coefficients outside tensor powers and both mandatory uniform-alphabet/owned-reset clarifications. Determine the strongest disjoint diagonal target block justified by this complete grade-support test; price its necessary paid returns and state the remaining caller premises.\n\nBoundary: fixed original paid inputs, arbitrary retained histories and free scalar/monomial work. No active-feed, changed native-prefix, event-retime, scratch-rewrite or p19 compiler scope is taken. This addresses PR5570's unreserved source-binding handoff and extends beyond its first-bank target. It will not count a repeated numerical root as a result.\n\nExpected artifact: a focused proof and reproducible source-bound certificate, or a concrete correction if the full source does not support the proposed extension. Stop after all four banks and the maximal grade-separated target set are classified. No new kappa or selected-frontier change is claimed in advance. Prepared for Ali Romman with substantial OpenAI Codex assistance.\n\nClaim expires: 2026-10-25T22:40:00Z.\n\n</details>\n","comments":{"nodes":[{"body":"DONE / RELEASED — [PR #5618](https://github.com/CrocSwap/integer-mult-bounds/pull/5618), exact head 4d112aaf8e8760d34015f934ab8a63d2b9ccda83. The frozen p2 oracle repair optimum is exactly 240 returns for f >= 72. The exact second moment 27/343 closes the prior 225–240 gap. Full projector, signed-product, normal/optimized and Git-free checks pass. Source deficit, upper construction and faithfulness remain credited premises. No native supplier is constructed.\n\nPublished body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.\n\n<details><summary>Original claim</summary>\n\nCLAIM — regular-module rank of the coded p2 leakage, through projection moments.\n\nI take the unreserved [225,240] repair-gap handoff in PR5587 at 49811a9a37cc1b71bcbe851dd20da3cbf479ff35. The fixed source is PR4948/4978 with all four banks and all 39 retained observations. I will factor L=I-C3*C1 through the two rank-eight source idempotents. Orthogonal projections onto their column and row spaces can bound the two intersection defects by exact regular-trace moments. A bound rank(L)>15|H| would close the gap at 240 in the enlarged arbitrary-linear-return model; a weaker bound will be stated exactly. No width-one rank will be substituted for regular-module rank.\n\nBounded test: derive the scalar projectors, classify the signed 17-axis normalizer products, calculate exact first and second trace moments, and use independent scalar/monomial checks. Stop if those moments do not improve the integer return floor, unless a small rigorously justified extension closes the same gap. This is not a physical native caller or a new kappa claim. Changed-input construction searches are excluded.\n\nBase canonical main 6cf7a683f447b9f1e2333791907fd92be5009a63, priority 7; current guidance and selected record unchanged. The fresh overlap report finds the original handoff released. Credit Dugongue PR4948, Chris R/Clear1nsight PR4978, Ali Romman PR5587/5526/5449, josecarlosgonzalezroque-design PR5570 and the retained native/controller lineage. Projection rank and trace inequalities are classical. Ali Romman with substantial OpenAI Codex assistance. Claim expires 2026-10-26T01:30:00Z.\n\n</details>\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5583#discussioncomment-19184961"},{"body":"**FINDING — Independent AI corroboration of [PR #5618](https://github.com/CrocSwap/integer-mult-bounds/pull/5618)'s projector/rank bridge.** A separately written verifier checks sections 1–4 at exact head `4d112aaf8e8760d34015f934ab8a63d2b9ccda83` and imports no predecessor executable. No blocking error was found under the credited premises. Its two exact second moments are again `27/343`, corroborating the existing `82266/343 > 239` deficit implication and 240-return oracle conclusion.\n\n[Audit, source pins and reproduction](https://github.com/MatShep/integer-mult-bounds/tree/a246e928f4ee269c6581e5dc50160cbfd5939eb1/research/p2-oracle-projector-audit) · [written coverage](https://github.com/MatShep/integer-mult-bounds/blob/a246e928f4ee269c6581e5dc50160cbfd5939eb1/research/p2-oracle-projector-audit/AUDIT.md). The check derives two distinct balanced projectors and checks all 12,432 ordered nonidentity binary ratios; the written cancellation argument covers arbitrary phase cocycles. Four twisted-C7 fixtures corroborate the phase/rank identities. A noncommuting S3 fixture checks the left/right adjoint bridge; a repeated-ratio counterfixture and six semantic controls test relevant failure modes.\n\nLocal registered normal/optimized, committed-byte, export-integrity and read-only Git-free checks pass; retained-source corruption rejects in both modes. The optional full-history preflight refused the shallow checkout; hosted execution is unclaimed.\n\nThe history deficit, 240-return upper construction and pointwise faithfulness for `f>=72` remain credited premises and were not independently replayed. This corroborates Ali Romman's result; it supplies no new native supplier, kappa or frontier claim. Retain Dugongue, Chris R/Clear1nsight, josecarlosgonzalezroque-design and the full lineage in the linked README. Prepared for MatShep with substantial OpenAI Codex assistance: research/checks by GPT-6.1 Sol/xhigh; brief publication sanity and overlap review by GPT-6 Astra/high. Apache-2.0. No human or formal-kernel scientific acceptance is claimed.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5583#discussioncomment-19189343"}]}},{"number":5582,"title":"DONE: native tariff obstructions with active feedback","updatedAt":"2026-10-26T05:27:23Z","body":"DONE — a native tariff obstruction for complete linear words, with changed paid inputs and active feedback allowed.\n\nPublished [PR #5588](https://github.com/CrocSwap/integer-mult-bounds/pull/5588), exact head `97db3fb45cd93a350e300781d75440d14e6a1f73`. [Proof](https://github.com/aliromman/integer-mult-bounds/blob/97db3fb45cd93a350e300781d75440d14e6a1f73/research/native-equality-cut/PROOF.md), [checks](https://github.com/aliromman/integer-mult-bounds/blob/97db3fb45cd93a350e300781d75440d14e6a1f73/research/native-equality-cut/RESULT.json), and [source pins](https://github.com/aliromman/integer-mult-bounds/blob/97db3fb45cd93a350e300781d75440d14e6a1f73/research/native-equality-cut/SOURCES.json).\n\nFor fixed profiles that work at arbitrarily large widths, every positive integer k must satisfy\n\n```text\nsum_r P_r (1-2^(-k*r)) >= sum_r T_r (1-2^(-k*r)).\n```\n\nThe exact finite-width condition has `2k <= f`. Equality of k pairs of tensor columns defines a cut that every columnwise monomial map preserves. A rank-r Clifford call has the displayed normalized crossing rank. The classical rank inequality then applies to the whole chronological word, including feedback, COPY, CLEAR and arbitrary scalar mixing. No original paid input is frozen.\n\nA strict failing cut forces ideal endpoint operator error at least one for the stated unitary target frames. One rank-one plus one rank-three call cannot supply two independent rank-two jobs. More strongly, the proof gives a family that first fails at any prescribed K while passing every earlier cut and a strictly favorable raw power-moment test for all savings through 1/256. These are impossible native invoices, not a new kappa construction. The checker decides the whole infinite tariff family by an exact polynomial tail bound.\n\nNormal and optimized Python 3.12.13 checks passed: 172 literal native rank/partial-isometry cases, 480 complete chronological prefixes, 192 profile pairs, all 625 small four-rank differences, delayed failures through K=12, and the pinned P104 profile screen. Both modes also passed in a read-only, 17-file Git-free export. Committed bytes, rendered PR links and exact source blobs were checked. CI and mathematical review are separate from these local checks.\n\nThe contract uses common field capacity, single-field Clifford tensor calls, independent native targets and ordinary maps that preserve the cuts. Inter-column maps, larger-field packing and nonlinear operations need a different proof. A boundary with existing nonordinary responses must include the joint crossing rank of its complete availability map. The P104 profile passes all these constraints; its native word and compiler were not rerun. No complete kappa improvement, selected-frontier change, proof-kernel acceptance or independent human review is claimed.\n\nThe investigation began with PR5449's semisimple question. An output-module packing can fail at a physical single-field input; section 7 records the exact counterexample that led to this cut. The classical rank inequality and its repository predecessors, retained-history work, zxv's unchanged scalar layer, and P104 source lineage retain their credit in the package. New work: Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. No subagents. Base: canonical main `5e0052ca0aa3036a3afa8b0438927377c8d97d53`, priority revision 7.\n\nThe claim is released. A useful independent check is the stabilizer sign/intersection argument in section 3 and the complete-word, cached-history and endpoint-frame contracts in sections 4–5. Further search and larger-alphabet work are open; active P104 retiming and joint-basis scopes were not taken.\n","comments":{"nodes":[{"body":"DONE / HANDOFF — the full four-bank source binding and stronger repair result are in [PR #5587](https://github.com/CrocSwap/integer-mult-bounds/pull/5587), exact head `49811a9a37cc1b71bcbe851dd20da3cbf479ff35`. Canonical [Discussion #5583](https://github.com/CrocSwap/integer-mult-bounds/discussions/5583) is released.\n\nAll 39 observations and 448 input columns are checked. Grade support gives 208 missing columns; coupled left-coset separation excludes all 224 negative targets. The nonzero leakage I-C3 C1 into the last positive bank makes at least 225 additional same-capacity returns necessary for every f>=72. A rank-eight factorization gives a 240-return construction in the enlarged arbitrary-linear-oracle model. This is not a native supplier or kappa improvement. The 225-H1 optimistic repair fails every positive concave power tariff.\n\nThis closes the stated source-binding handoff and both source clarifications are retained. The pointwise Clifford theorem remains a credited premise. Normal/optimized checks, source and committed bytes, public proof links and published-body readback passed. Hosted checks and independent human review remain separate.\n\nThe remaining [225,240] gap asks for the rank of right multiplication by I-C3 C1 on the regular monomial algebra, with the width-dependent phase cocycle, and its minimum cyclic-generator count. This concrete matrix may help the distinct general return-complexity investigation in #5582. I reserve no further module-rank scope. Credit Dugongue, Chris R / Clear1nsight, josecarlosgonzalezroque-design and all retained history/native/controller sources. Ali Romman with substantial OpenAI Codex assistance.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5582#discussioncomment-19183544"},{"body":"The PR5587 source-bound result and exact head are retained in this investigation's handoff record. Its [225,240] cyclic-generator gap remains open and is not reserved here. The output-module calculation led this round to a different obstruction: [PR #5588](https://github.com/CrocSwap/integer-mult-bounds/pull/5588) uses a physical input/output cut for complete native words, with changed paid inputs allowed. The canonical body now records that result and releases the claim. The semisimple packing distinction is retained in section 7 of the proof. No minimum-generator result for I-C3 C1 is asserted.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5582#discussioncomment-19183714"},{"body":"DONE — independent PR5588 §§3–5 audit, published as [PR #5690](https://github.com/CrocSwap/integer-mult-bounds/pull/5690), exact head `fd4b1d3e3d1efd0b344f1de72fd4b62f5df8e703`. Claim released.\n\n**Finding:** the signed Clifford equality-cut tariff, rectangular chronological extension, joint cached-J accounting and approximation bounds hold under their stated hypotheses; no defect was found in this scope. [Independent proof](https://github.com/kumagi/integer-mult-bounds/blob/fd4b1d3e3d1efd0b344f1de72fd4b62f5df8e703/research/native-equality-independent-audit/PROOF.md) derives the signed intersection by squared Pauli traces and handles the full independent dirty domain. Endpoint frames must preserve the cut; the unit error constant additionally needs unitarity.\n\n**Evidence:** [receipt](https://github.com/kumagi/integer-mult-bounds/blob/fd4b1d3e3d1efd0b344f1de72fd4b62f5df8e703/research/native-equality-independent-audit/RESULT.json). Separate checkers, with no source-verifier import, cover 366 signed ordered Pauli image frames (not all Clifford unitaries), 54 literal projectors, Bell-sign/inter-column controls, one 18-operation dirty-feedback word with 19 prefix checks, ten paid-update cases, joint cache and code/error controls. Normal and optimized Python 3.12 replays, exact committed-byte/source-history check, deterministic 13-file export, fresh Git-free optimized replay and two transport-mutation controls pass. Hosted plan run40275439466 failed the two unchanged documentation-navigation subcases (selected review/source_manifest links); scientific replay was skipped. The scoped local evidence above is unaffected. [PR #5703](https://github.com/CrocSwap/integer-mult-bounds/pull/5703) already supplies the shared documentation repair; no duplicate fix is claimed.\n\n**Limits/handoff:** no source §6 profile-algorithm audit, §7 row-packing/compiler audit, actual P104 availability extraction, physical/native admission or new κ. Next useful caller-bound step: pin every row version at one named retained-observation boundary, build the complete J including independent dirty inputs, and certify its joint cut rank. Nominal cache counts are insufficient. Changed inter-column operations require explicit crossing charges and typed cuts.\n\nReviewed source remains PR5588 head `97db3fb45cd93a350e300781d75440d14e6a1f73`; proof/README match canonical main `df0137efd35e11b4445734bf612bdd4c7ebc330c`. Pre-submission refresh found unchanged main and no competing audit in 7,676 HEAD notes, 374 open PRs and 5,002 Discussion records (complete reported pagination/no warnings; not a novelty guarantee). PR5617's stronger stage cuts are distinct. The selected P104 κ=62009/20000000 and all five inherited premises remain unchanged.\n\nAli Romman retains credit for the source theorem and its limitations; classical rank/stabilizer theory and the source's fused-block/dirty-host/P104/native/compiler lineage retain their attribution. New audit prepared for Hiroki KUMAZAKI with substantial OpenAI Codex assistance and delegated proof/check review. Agent-assisted independent derivation and implementation, not human or proof-kernel acceptance. Apache-2.0.\n\n```json\n{\"schema\":\"integer-mult-coordination/v1\",\"kind\":\"DONE\",\"task\":\"uniform-transfer-audit\",\"scope\":\"Independent PR5588 §§3–5 audit: signed stabilizer tariff, chronological words, cached J and endpoint frames\",\"base\":\"df0137efd35e11b4445734bf612bdd4c7ebc330c\",\"expiry\":null,\"artifact\":\"https://github.com/CrocSwap/integer-mult-bounds/pull/5690\"}\n```\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5582#discussioncomment-19188980"}]}},{"number":5674,"title":"G20 causal binding and independent helper splits","updatedAt":"2026-10-26T05:13:01Z","body":"DONE / RELEASED — [Split one G20 helper into independent rank4 jobs at ordinary BIT saving 0.003507410738409](https://github.com/CrocSwap/integer-mult-bounds/pull/5679), exact head `715dda75fc4cefb1e492349fa0e331d5f15d0f53`.\n\nThe emitted G20 caller admits conditional ordinary BIT saving **0.003507410738409**, a gain of **0.000001632212840** over PR #5671. One independent H8 helper becomes two independent H4 targets, with no added paid query or owned field. This is a component bound; no whole multiplication kappa is claimed.\n\nThe exact prefix and suffix cut is new on the PR #5671 word at 31bd16e74c3833a9fb25dca17a0f00fa9046325c. Eight scalar ADDs implement the split in each of 18 calls. The complete target change is 36H4-18H8; the paid histogram is fixed. The caller has 11,022 independent inputs, 48 owned fields, 56,223 queries and 443,238 events. Both target normalizer directions and 72 route passes are charged. The balanced 4+4 split is selected over 7+1. A new cut census finds no further split in the emitted common-frame domain with a helper pivot from its boundary group.\n\nNormal and explicit optimized registered replay pass. All 685 local columns pass native and separate adjoint checks at 101 and 10201. A separate global replay checks every independent and dirty column and all chronological query descriptors. Two rational interval engines enclose the raw crossing in (.00350741073845,.00350741073849). Fresh positive fees, compiler thresholds and five ordinary levels give the stated bound. Contaminated cut vectors, missing input/output bases, a stale epoch and an extra old target reject. The committed export passes integrity and Git-free optimized replay.\n\nOpen next questions: permit a different physical pivot/output role with complete ownership and routes; find a paid-query removal that uses the new independent input; or bind a stronger compatible Gaussian caller after a fresh overlap check. No new search is reserved. The selected result is unchanged. Prepared for Ali Romman with substantial OpenAI Codex assistance.\n\n<details><summary>Original claim</summary>\n\nCLAIM — joint G20 causal block binding after PR5671.\n\nBase: current canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7, selected conditional kappa.00310045. The live refresh covers359 open PRs and4982 Discussion records with complete pagination through2026-10-26T04:34:32Z. PR5671 explicitly releases block binding; Discussion5600 owns the separate P104 deferred-input work. The earlier negative G20 searches in Discussion5654 are released and are not counted.\n\nExact source: PR5671 at31bd16e74c3833a9fb25dca17a0f00fa9046325c. I will replace a cohort's unobserved invertible prefix, permitting reads inside the cohort before its first outward read. At that cut, assign the retired helper values in their actual frames, use one binding to cancel a receiver, and execute the exact old suffix. Every cohort target is debited. First domain: causal closures of retained full helpers under their earliest external helper reads, followed by exact whole-zero propagation and native path reconstruction. Main inputs and endpoints remain independent and fixed.\n\nSuccess requires a cheaper emitted word, all retained original and arbitrary dirty columns, the complete18-call bill, positive ordinary fees and exact admission. A rank or scalar-span screen alone does not count. If a positive cohort appears, test compatible cohorts and rerun the changed-word domain to exhaust that strategy before publication. Stop a failing domain without adding a negative result to the seven remaining constructive contributions.\n\nCredit Dugongue PR5665's causal late-binding theorem, Ali Romman PR5662/5667/5671, ai-nanachi PR5512, PR5421/5641/5647/5648/5653 and every retained source notice. The contribution sought is joint causal block binding with a complete changed bill, not the already completed nine singleton bindings. Ali Romman with substantial OpenAI Codex assistance. No selected-frontier change or merge. Claim duration about two hours.\n</details>\n\n\nLater scoped test: the broader physical-role cut scan also finds no remaining independent split in the7869 emitted-word contexts. This is local-only discovery evidence and is not a new counted contribution. All G20 cut work is released. No claim of a global obstruction is made.\n","comments":{"nodes":[{"body":"UPDATE — changed-word cut and history tests; no counted contribution yet.\n\nThe first causal-closure domain has593 distinct cohorts:414 singletons,176 pairs and3 triples. Exact retained scalar endpoints and native path reduction give no paid-moment gain after all removed H8 targets. This is local-only discovery evidence; it is not a global obstruction or a new contribution.\n\nThe next test changes the hypothesis of PR5609: use the complete PR5671 word at31bd16e74c3833a9fb25dca17a0f00fa9046325c, including its new owned bindings, instead of the old PR5512 word. Test reachable independent helper channels at every common-frame cut, then test actual stored-row aliases that can replace paid queries. A channel must carry a nonzero retained helper direction. An owned zero coordinate does not count. A positive result must emit the changed word, keep all retained inputs independent, and pass the complete bill and native replay.\n\nCanonical main remains df0137efd35e11b4445734bf612bdd4c7ebc330c after fetch. Fresh overlap covers362 open PRs and4987 Discussion records with complete pagination through2026-10-26T04:45:49Z. PR5609 is released and is scoped to its frozen older word. The active P104 scopes stay separate. Stop this changed-word domain if it has no constructive gain; do not extend it into another negative-only campaign.\n","url":"https://github.com/CrocSwap/integer-mult-bounds/discussions/5674#discussioncomment-19188615"}]}}]}}}}
\ No newline at end of file
diff --git a/research/p104-shared-frame-gather/PUBLICATION-VALIDATION.md b/research/p104-shared-frame-gather/PUBLICATION-VALIDATION.md
new file mode 100644
index 0000000000..70ce46796c
--- /dev/null
+++ b/research/p104-shared-frame-gather/PUBLICATION-VALIDATION.md
@@ -0,0 +1,18 @@
+# Combined publication validation
+
+2026-10-26; CPython 3.12.8, macOS arm64, standard library. Scientific assembly commit `33c565d79b2d1bc706defafc3901f4700b60f6a4`; the later publication commit adds this record only. Canonical main was fetched immediately before submission and remains `de5ec2e7a6074068452f45b1a191b5c72832df4a`.
+
+Passed serially:
+
+- Registered metadata/source pins for both packages.
+- Cross-call registered normal and optimized replays: exact current-version query maps, all-input endpoints, native phases, eight fresh-helper quotients, separate-delivery budget certificates, positive and stale/future controls, and the complete unchanged invoice.
+- Shared-gather registered normal and optimized generation and separate certificate audits: all 1,090 gather frames, 465 source-coframe bundles and 1,211 earlier outputs; native tariffs, complete-ingress budgets and constructive/malformed controls.
+- New publication source-join check: ten predecessor code, source-manifest, map and runtime-input files are byte-identical between the retained packages.
+- Exact committed-byte preflights: 17 files for cross-call; 34 files for shared gather plus its declared cross-package inputs, before this validation record.
+- Official exports and `export_research_package.py check --check-inputs`: both exports pass; seven declared inputs in cross-call and twenty in shared gather are covered.
+- Optimized direct runs from the extracted Git-free exports with empty PATH: cross-call experiment, publication join, shared-gather generator and separate certificate auditor all pass. All 18 and 35 exported files respectively, including export manifests, remain byte-identical.
+- `git diff --check` on the contribution.
+
+The optional full source-history checker requires a non-shallow repository, so it is not claimed as passed. Direct local ancestry checks do confirm each declared generic source commit is an ancestor of this publication: `a1534ac45401c9a9c80c07aa65c89e00e33594f8`, `df0137efd35e11b4445734bf612bdd4c7ebc330c`, `25002e57adf7858afda0a0d1e208b3ea39f63507`, and `de5ec2e7a6074068452f45b1a191b5c72832df4a`. The exports authenticate and contain all scientific runtime inputs. The final committed-byte preflight is repeated after adding this record.
+
+Frozen source/result bytes and mathematical verifiers are unchanged. Historical local-only descriptions remain as session provenance; `PUBLICATION.md` describes the combined submission and the different theorem scopes. No full selected, new supplier/compiler/outer, Lean or Docker replay was run. Hosted CI, formal-kernel checks and independent human acceptance are separate. No new κ or target credit is claimed.
diff --git a/research/p104-shared-frame-gather/PUBLICATION.md b/research/p104-shared-frame-gather/PUBLICATION.md
new file mode 100644
index 0000000000..0acb008654
--- /dev/null
+++ b/research/p104-shared-frame-gather/PUBLICATION.md
@@ -0,0 +1,33 @@
+# Earlier P104 answers need a different delivery protocol to save these centre charges
+
+This combined contribution retains two exact source-bound results on [PR5718](https://github.com/CrocSwap/integer-mult-bounds/pull/5718), head `a1534ac45401c9a9c80c07aa65c89e00e33594f8`. No new κ, selected-frontier change or general P104 lower bound is claimed.
+
+## Results and reading order
+
+1. [Versioned query maps and separate delivery](../p104-cross-call-paid-reuse/README.md): extracts all 3,426 internal paid outputs at each of actual caller events 0 and 3, with different independent-helper namespaces. Exact certificates reject separately canonically delivered replacements for eight rank-seven centre queries, even granting old-call answers, main information and retention/cleanup free. The pure-main adjacent rank-one query cannot save its charge under its stated cache-delivery model.
+2. [Shared gathering](README.md): for **one** of those centres, caller event 3/local 13660, allows preaggregation within 465 source coframes and one common gather frame followed by one final delivery. All 1,211 earlier paid outputs and 1,090 pinned candidate frames are covered. Only 478 frames occur before the cut; the other 612 are granted as an optimistic exclusion. Every protocol in this class costs at least H7 before ordinary and cleanup costs.
+
+The eight-centre theorem must not be inferred for shared gathering: that extension is proved only for local 13660. Matrix-valued recombination, new frames, multilevel transport and changed consumers are outside these results. An absent scalar dependency and an unaffordable physical delivery are different conclusions; the supplied controls explicitly exercise scalar reconstruction.
+
+## Reproduction
+
+From the repository root, Python 3.12 and the standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-cross-call-paid-reuse
+python3.12 -B scripts/research_package.py --package research/p104-shared-frame-gather
+```
+
+The first registration rederives maps, quotients, native phases and the complete invoice. The second rederives gather certificates and runs a separate certificate auditor. Normal and optimized commands are explicit. The added publication check compares every retained runtime input, the versioned maps and predecessor code byte-for-byte across the packages; frozen source and result bytes are unchanged. It is part of the second registered replay. See [publication validation](PUBLICATION-VALIDATION.md) for this assembly's actual runs.
+
+The original README, TASK and validation records describe their historical local-only research sessions; their dates and publication statements are retained as provenance. This overview describes the combined submission. Earlier local-only investigations are context, not required external runtime dependencies. The combined packages include all consumed bytes and notices. The selected conditional κ at canonical main `de5ec2e7a6074068452f45b1a191b5c72832df4a` is .00310045; the source's archive merge does not admit its .00311024570 claim.
+
+## Significance for researchers
+
+Complete scalar information can coexist with a missing fresh-helper direction or an unaffordable native delivery. The maps and exact cost certificates let subsequent work test that distinction without repeating a duplicate-query census. The unchanged complete source invoice is 66,348 paid jobs and 12,976 productive fields; neither package earns a deletion or new productive target.
+
+## Open next questions
+
+A new construction must name a changed native combination, multilevel gather, frame catalogue or terminal consumer and price every ingress plus final delivery and cleanup. Start with a concrete source-bound contract and optimistic full bill. No surviving construction or positive forecast is supplied. Do not extend the closed separate-cache or canonical one-frame census solely by adding more coefficients.
+
+Credit [flyingviper](https://github.com/flyingviper), [Ali Romman](https://github.com/aliromman), the native/controller/compiler and analytic authors retained in both `inputs/NOTICE5718` files, and the predecessor [G20 reconstruction](https://github.com/CrocSwap/integer-mult-bounds/pull/5647). Prepared for [zxv](https://github.com/zxv) with substantial OpenAI Codex assistance. Apache-2.0. Exact finite/source-bound checks are not formal-kernel or independent human acceptance of the complete multiplication theorem.
diff --git a/research/p104-shared-frame-gather/README.md b/research/p104-shared-frame-gather/README.md
new file mode 100644
index 0000000000..d27b4cb7d0
--- /dev/null
+++ b/research/p104-shared-frame-gather/README.md
@@ -0,0 +1,267 @@
+# One canonical gather frame cannot fund this P104 H7 replacement
+
+**Local-only exact scoped obstruction; no new κ.** At PR5718 caller event3,
+local event13660, the proposed replacement of owned query748,
+`frame518 -> frame598`, cannot be made strictly cheaper by gathering earlier
+paid answers at one pinned canonical frame and delivering their scalar sum once.
+This remains true when all snapshots from the same source frame may first be
+combined into one ingress bundle, and when capture, retention, ordinary work,
+main-input information, restoration and cleanup are optimistically free.
+
+The result covers all **1,211 earlier call3 paid outputs**, all **465 source
+coframes**, arbitrary rational scalar coefficients, and all **1,090 pinned
+canonical gather-frame labels**. Of these labels, **478** have appeared physically
+before the cut; the other612 are granted only in the rejection relaxation.
+Every candidate fails a necessary complete-ingress budget. Realized paid and
+target deltas are both zero; the selected conditional κ remains .00310045.
+
+The original question, source pins, limits and reopening condition are in
+[TASK.json](TASK.json), [SOURCES.json](SOURCES.json), and the exact
+[RESULT](results/RESULT.json). No separate-delivery census was enlarged and no
+public post, PR, merge or selected-frontier edit was made.
+
+## Exact chronological gather contract
+
+The unchanged source is PR5718 at `a1534ac45401c9a9c80c07aa65c89e00e33594f8`.
+The target is local13660, owned748 epoch1, input version3, last logical write
+SCALE13658, following COPY13657 from256. Its existing charge is H7.
+
+A tested gather protocol has the following boundary:
+
+1. Choose one canonical frameF from the pinned1,090 labels. Physical realizations
+   must allocate a valid owner/coframe; the rejection test grants even labels
+   not encountered before the cut.
+2. A snapshot is the exact **post-MOVE** value of one paid query at local event
+   e<13660. Capture it immediately then, before a later write can alter it.
+   Store its actual output version, source frame, invocation and owned epoch.
+   Arbitrarily many such retained snapshots are granted free for exclusion.
+3. For each original source frameA, optionally form any rational combination
+   of every retained snapshot atA. They can be accumulated chronologically as
+   the snapshots arrive. One ingress job may transport this entire combination
+   through the exact operator `Phi_F Phi_A^-1`. Multiple combinations from one
+   source frame are also allowed, each paying its ingress charge.
+4. Accumulate aligned bundles atF. After the last chosen source event and
+   before13660, recombine them there and apply **one shared final delivery**
+   `Phi_598 Phi_F^-1`. Current/historical destination-frame observations are
+   additionally granted free, including free transfer toF, which only strengthens
+   the relaxation. All original raw main-input information is free too.
+5. Produce the same full native answer at598 before the original consumers
+   begin at13661. Every original/helper target remains mandatory; owned caches
+   receive no new productive credit. A real implementation must price its
+   acquisition, retained owners, reads, normalizers, dirty return and clears.
+
+There is no construction using a future version or unavailable frame here:
+**no candidate survives**. Availability and extra-frame grants are explicitly
+favorable lower-bound assumptions, not executable operations. Source paths may
+subdivide a canonical alignment without improving its tariff: Gaussian rank
+is a metric on the associated Lagrangians, and concavity makes the sum of their
+charges at least the direct alignment charge. This does not allow an additional
+mixed gather at a second frame.
+
+## Fresh helper21 is available, but not cheaply enough with the other columns
+
+Call0 helpers `H0(r)` and call3 helpers `H1(r)` are distinct independent raw
+inputs. All call0 answers therefore vanish on a difference supported only on
+fresh call3 helpers, even if old answers are retained and delivered free.
+The1,211 call3 prefix snapshots contain48 outputs with a nonzero helper21
+coefficient. The first is role432 after12489, output version4 at33, coefficient
+1/2. The pure helper21 output itself appears after12547, version1 at43. Both
+are genuinely earlier than13660; the obstruction does not assume this helper
+is unavailable everywhere.
+
+The target's helper21 coefficient is -1/4, while every granted destination-frame
+observation has coefficient zero. In normalized source coordinates, the
+perturbation `h1(21)=phi` changes the actual answer by
+`-(1/4) S1 Phi_598 phi`. Its raw physical perturbation is
+`H1_raw(21)=S1 Phi_initial(21) phi`; the source gauge is not discarded.
+The common invertible native prefix makes this an arbitrary nonzero full field.
+
+The checked map adapter retains all32 original raw caller columns per routed
+port, both independent536-helper namespaces, native source gauges, the actual
+stage-1 routes and416 phase-polynomial identities. Every3,426 paid query map
+matches the predecessor's frozen coefficient rows, instruction versions,
+frames and epochs. The independent literal walk verifies all744 public and9
+owned endpoint columns. The obstruction then projects onto fresh helpers;
+main-column freedom can only make replacement easier.
+
+## Complete ingress budget and exact certificate
+
+Write `Hr=r^t`, `0<t<1`. If a candidate strictly beats H7, its positive ingress
+ranks and final delivery rank must sum to at most6:
+
+```
+sum_i r_i^t >= (sum_i r_i)^t;
+(sum_i r_i) >= 7  implies  sum_i Hr_i >= H7.
+```
+
+For canonical charts, the Gaussian rank is
+`d(A,B)=rank_F2(L_A + L_B)-9`, where `L_A=Phi_A^-1(Z)` is a nine-dimensional
+Lagrangian in18 binary coordinates. All1,090 canonical signed tableaux are
+checked with a separate native circuit interpreter. Every used pair tariff is
+recomputed independently;1,384 literal relative-operator checks also agree.
+Monomial normalizers and absolute phases are not silently replaced by unsigned
+frame tags. Four representative native maps are lowered and checked with their
+absolute phases, including the original518->598 query.
+
+The exhaustive rejection divides as follows:
+
+| Frame labels | Exact reason |
+|---:|---|
+|411|Final delivery already has Gaussian rank at least7.|
+|338|No earlier helper21-bearing source bundle fits the residual ingress budget.|
+|341|Required fresh-helper coordinate coverage exceeds the complete residual ingress budget.|
+
+For the last group, exact rational elimination gives a354-dimensional coordinate
+space for the complete prefix helper history and the free destination
+observations. The target has74 nonzero coordinates. Every source row and target
+has an independently reconstructed exact expansion in the original independent
+basis rows. Basis independence is checked again by full rank modulo1,000,003;
+this use of modular arithmetic proves independence overQ and is **not** a
+modular-only nonmembership argument.
+
+For a chosen frameF, grant every zero-cost bundle and selected large-support
+bundles free. Ignore all coordinates they touch. The remaining target
+coordinates must each be covered by some purchased ingress bundle. For rankr,
+record the largest possible remaining coordinate support of **the whole
+source-frame span**, not just one snapshot. A small unbounded-knapsack recurrence
+upper-bounds the total coverage for every rank budget. It even counts overlaps
+and repeated bundles as new coordinates. Whenever this generous capacity is
+smaller than the target support, every actual scalar combination is excluded.
+Every one of the341 frames has such a strict certificate, with margin at least5.
+
+For example, frame3 is the empty chart and is physically present initially.
+Final delivery3->598 costs H1, leaving ingress rank budget5. After large bundles
+are granted free,34 mandatory coordinates remain and ingress capacity is at
+most10. At frame33 the corresponding counts are22 versus8 with final rank2;
+at frame43 they are22 versus4. The complete per-frame receipts, source groups,
+coordinate maps and dynamic-programming tables are in the decompressed
+[CERTIFICATE.json.gz](results/CERTIFICATE.json.gz) payload.
+
+## Complete signed bill and stopping decision
+
+The named allowance is **Δpaid=-H7, Δtargets=0**, once in caller event3. Every
+hypothetical replacement in the tested class has signed recursive difference
+
+```
+-H7 + sum(source-frame ingress charges) + shared-final charge >= 0.
+```
+
+The following completion charges are **unknown and nonnegative**, not zero:
+
+| Obligation | Status in this result |
+|---|---|
+|Source acquisition COPY, reads and scalar preaggregation|Granted free only to reject the protocol; no completed replacement emitted.|
+|Retained values, owner allocation and lifetime hosting|Unknown; unlimited retention granted only in the lower bound.|
+|Recursive ingress and final delivery|All positive Gaussian ranks included, including shared source-coframe bundles.|
+|Native monomial normalizers and routes|Unknown ordinary work; cannot create a negative recursive charge.|
+|Dirty source restoration and return|All sources must remain valid; no source-consuming alteration is credited.|
+|Owned clear and retained-cache cleanup|Unknown; no productive cache credit.|
+|Main and other helper corrections|Unknown; all raw main-input information is granted free for rejection.|
+|Compiler completion, precision/stock and outer use|Unrun for a changed program because none survives.|
+
+The unchanged complete source invoice is rebuilt from the literal18-call
+controller and compared with PR5718's RESULT: **66,348 paid jobs**, **12,976
+productive fields**, rank surplus760. Its local profiles are
+
+```
+paid:    2184 H1 +459 H2 +495 H3 +92 H4 +159 H5 +22 H6 +15 H7
+ targets:   2 H1 +  5 H2 + 91 H4 + 6 H5 +  6 H7 +15 H8 +411 H9
+```
+
+The RESULT retains every whole-caller paid/target count and the source's ordinary
+allowance. Realized deltas are zero. Since the favorable recursive bill already
+fails, scalar cleanup or more precise retention pricing cannot rescue this
+protocol. The session stops here rather than extending the excluded census.
+
+## Discovery and rejection controls
+
+A positive local control captures actual role432 version4 after12489, then reads
+role433 version2 after12491. Both are at33. INIT/COPY/ADD retains their exact sum
+in one owned field, one nested33->3 H1 job delivers it, and observation precedes
+owned CLEAR. The exact solver discovers coefficients1,1; all744 scalar columns,
+unchanged sources, native phase and nested descent pass. This is a synthetic
+output fixture demonstrating discovery, not the target748 replacement or a
+new supplier. Full arbitrary-width retention hosting is not asserted.
+
+Negative controls reject a future answer, an old snapshot version, a call0/call3
+helper alias, a missing fresh helper21 column, a wrong output frame, an unpaid
+nonzero frame retag, a wrong absolute native phase, an invoice charging only the
+final job of a three-job ingress example, and a complete claim with unknown
+cleanup costs. All checks use explicit exceptions and remain active under `-O`.
+
+## Reproduction and limits
+
+From a checkout or a self-contained copy of this directory:
+
+```sh
+python3.12 -I -B research/p104-shared-frame-gather/verify.py
+python3.12 -I -B research/p104-shared-frame-gather/audit_certificate.py
+python3.12 -B scripts/research_package.py --package research/p104-shared-frame-gather
+```
+
+The separate auditor does not import the search program, predecessor generator,
+or Clifford compiler. It reconstructs exact row expansions, independently
+checks basis independence and native-frame tariffs, and verifies every support
+certificate. Both engines were written with the same Codex assistance; this is
+not independent human or formal-kernel review. See [VALIDATION.md](VALIDATION.md).
+
+The scope is one named query of the frozen PR5718 word, unchanged paid snapshots,
+source-coframe preaggregation, and **one pinned canonical gather frame**. It
+excludes neither arbitrary new canonical subspaces outside the1,090 labels,
+general matrix-valued recombination, mixed gathers at multiple intermediate
+frames, new ordinary reads from other frames, changed sources/consumers,
+helper-target splitting, nor sharing a final job across several changed outputs.
+No full native, selected, Lean, Docker, all-size/compiler, rounded multiplier or
+new outer scientific replay was run. Finite scoped rejection is not a theorem
+that every P104 implementation or multiplication architecture fails.
+
+## Significance for researchers
+
+The predecessor excluded separately delivered snapshot subsets. This result
+closes a distinct relaxation: sharing the final delivery after one common
+canonical gather, even with source-coframe bundle savings. The remaining
+obstacle is the complete ingress information budget. A new experiment must
+change that boundary rather than treating the final shared job as the whole bill.
+
+## Open next questions
+
+Can a source-bound matrix-valued combination, a new frame outside the pinned
+catalogue, or a genuinely multilevel shared native circuit beat this allowance?
+Start with one named changed input/output contract and a complete ingress,
+final-delivery and dirty-return invoice. No such contract, legal word or
+favorable completed bill is supplied here. Do not enlarge the separate-cache
+census or repeat the closed helper440/helper695/helper680 lanes on this evidence.
+Structured questions and the explicit stopping decision are in [TASK.json](TASK.json).
+
+## Sources, attribution and current context
+
+The direct local predecessor is `research/p104-cross-call-paid-reuse`
+at `50dd8aa3cef02cd7b4a785b58f3a66d0edd91825`. Its exact code and factorized query
+maps are retained as runtime inputs without altering predecessor artifacts.
+PR5718 supplies the source word/controller and its inherited attribution:
+[flyingviper](https://github.com/flyingviper),
+[Ali Romman](https://github.com/aliromman),
+[Pierre-Loic Doulcet](https://github.com/hexapode), and the credited P104,
+productive-helper, native/compiler and outer lineage in [NOTICE5718](inputs/NOTICE5718).
+Prepared for [zxv](https://github.com/zxv) with substantial OpenAI Codex assistance;
+no subagents. Apache-2.0.
+
+Canonical main was refreshed to `de5ec2e7a6074068452f45b1a191b5c72832df4a`,
+priority revision8, selected κ .00310045. The selected record's source manifest,
+review and reproduction command were resolved without running its cold replay.
+Live [PR5751](https://github.com/CrocSwap/integer-mult-bounds/pull/5751) at
+`8483f9a679d06c1e270141dbc686a7e76e7a7522` now publishes the previously provisional
+paid-meeting split claim .00311301600. It remains distinct from this frozen-word
+gather test and was not independently replayed here. [Discussion5645](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645)
+tracks a different paid scalar-contact conversion. Local overlap inventory is
+sparse/incomplete; cached remote coverage reported314 open PRs and5,115 discussion
+records with complete pagination. No novelty guarantee follows. Exact snapshots
+are in results/live-overlap.json and results/overlap-summary.json.
+
+Managed worktree creation attempted a full checkout, exhausted disk space and
+removed its failed partial checkout. The fallback is the isolated conventional
+sparse worktree on `codex/shared-frame-gather`; the app's attach operation rejected
+it as unmanaged. No other checkout or untracked research was modified. All
+runtime inputs are vendored; Git-free reproduction needs no predecessor worktree
+or Git objects. Original unmerged local source pins remain identified in the provenance record;
+metadata reads the identical retained copies at an ancestor export commit.
diff --git a/research/p104-shared-frame-gather/SOURCES.json b/research/p104-shared-frame-gather/SOURCES.json
new file mode 100644
index 0000000000..32272ce302
--- /dev/null
+++ b/research/p104-shared-frame-gather/SOURCES.json
@@ -0,0 +1,230 @@
+{
+  "schema_version": 1,
+  "base_commit": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "priority_revision_read": 8,
+  "selected_kappa": ".00310045",
+  "source_word_head": "a1534ac45401c9a9c80c07aa65c89e00e33594f8",
+  "predecessor_commit": "50dd8aa3cef02cd7b4a785b58f3a66d0edd91825",
+  "runtime_inputs": [
+    {
+      "path": "inputs/DATA-PHASES.json",
+      "sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa"
+    },
+    {
+      "path": "inputs/NOTICE5718",
+      "sha256": "c474479a870ae53ae0bcb84f656c2b222380395bbee767ee15e5f9c5b4ce7e79"
+    },
+    {
+      "path": "inputs/QUERY-MAPS.json.gz",
+      "sha256": "3a168605b596d4398c7876d42f2213dfddd807281b85ec0fe07177cea1aa035b"
+    },
+    {
+      "path": "inputs/clifford_exact.py",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962"
+    },
+    {
+      "path": "inputs/controller5718.json",
+      "sha256": "a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9"
+    },
+    {
+      "path": "inputs/native_verify.py",
+      "sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38"
+    },
+    {
+      "path": "inputs/predecessor-SOURCES.json",
+      "sha256": "031f4fa2171cd47573b35c5817b320d432bede64506709339700c99cd0f4c2af"
+    },
+    {
+      "path": "inputs/predecessor-experiment.py",
+      "sha256": "fd94db68e21a7a5cebc10c2731e264db8c304227bd069d1808051ee697399c4b"
+    },
+    {
+      "path": "inputs/result5718.json",
+      "sha256": "8e62f19bc795cac7712d31f955b71587e251dc94f1a02a20235e1ea740015db7"
+    },
+    {
+      "path": "inputs/word5718.json.gz",
+      "sha256": "cff7196f8b168c4ebf8227185eed98387b66ec5db0179108adfea4045775cc31"
+    }
+  ],
+  "files": [
+    {
+      "path": "research/p104-shared-frame-gather/inputs/DATA-PHASES.json",
+      "usage": "provenance",
+      "git_blob": "d9a4f84211182ca200c081ea061d724c8a2cfb70",
+      "sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/DATA-PHASES.json"
+    },
+    {
+      "path": "research/p104-shared-frame-gather/inputs/NOTICE5718",
+      "usage": "provenance",
+      "git_blob": "f83cecccf587acfa313c8259c0bf6db05d60b52d",
+      "sha256": "c474479a870ae53ae0bcb84f656c2b222380395bbee767ee15e5f9c5b4ce7e79",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/NOTICE5718"
+    },
+    {
+      "path": "research/p104-shared-frame-gather/inputs/clifford_exact.py",
+      "usage": "provenance",
+      "git_blob": "166841a037f069249f6f7f6dedb89248812f59e2",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/clifford_exact.py"
+    },
+    {
+      "path": "research/p104-shared-frame-gather/inputs/controller5718.json",
+      "usage": "provenance",
+      "git_blob": "cc539db709cf7a6c42872c4380902736ad453af3",
+      "sha256": "a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/controller5718.json"
+    },
+    {
+      "path": "research/p104-shared-frame-gather/inputs/native_verify.py",
+      "usage": "provenance",
+      "git_blob": "2c209dfa9cfade3bc20ac5fc9d876b433a147f8c",
+      "sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/native_verify.py"
+    },
+    {
+      "path": "research/p104-shared-frame-gather/inputs/result5718.json",
+      "usage": "provenance",
+      "git_blob": "a83d70e025254e217c01ca413f6823937e82f3d9",
+      "sha256": "8e62f19bc795cac7712d31f955b71587e251dc94f1a02a20235e1ea740015db7",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/result5718.json"
+    },
+    {
+      "path": "research/p104-shared-frame-gather/inputs/word5718.json.gz",
+      "usage": "provenance",
+      "git_blob": "4db015aca34ea957dd8b3b4d7ae4fa12b89934f8",
+      "sha256": "cff7196f8b168c4ebf8227185eed98387b66ec5db0179108adfea4045775cc31",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/word5718.json.gz"
+    },
+    {
+      "path": "research/p104-shared-frame-gather/inputs/predecessor-experiment.py",
+      "usage": "provenance",
+      "git_blob": "3309f0be8443c12724f31586f311fa2dda329c6b",
+      "sha256": "fd94db68e21a7a5cebc10c2731e264db8c304227bd069d1808051ee697399c4b",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/predecessor-experiment.py"
+    },
+    {
+      "path": "research/p104-shared-frame-gather/inputs/QUERY-MAPS.json.gz",
+      "usage": "provenance",
+      "git_blob": "e241f1958cba98e3227bed0a96b7109c21da5a95",
+      "sha256": "3a168605b596d4398c7876d42f2213dfddd807281b85ec0fe07177cea1aa035b",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/QUERY-MAPS.json.gz"
+    },
+    {
+      "path": "research/p104-shared-frame-gather/inputs/predecessor-SOURCES.json",
+      "usage": "provenance",
+      "git_blob": "a059933e0c4065f47c263df58ed0819909ea7ac1",
+      "sha256": "031f4fa2171cd47573b35c5817b320d432bede64506709339700c99cd0f4c2af",
+      "pin": "head:25002e57adf7858afda0a0d1e208b3ea39f63507:research/p104-shared-frame-gather/inputs/predecessor-SOURCES.json"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "certificates/frontier5599-source.json",
+      "usage": "provenance",
+      "git_blob": "a9e0a7c27a26854ece7bb953d2976ce12c90ee04",
+      "sha256": "fdee5a8121a07d918748d9250eb8a1c70e2a613142a19f731704e339f5e7e3fa"
+    },
+    {
+      "path": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "usage": "provenance",
+      "git_blob": "8089de45c78fb229a70ee1587cc90aacaca01ebc",
+      "sha256": "7688e5d8bf40fcdfb5bea43ba47e1f8d60a77a91ede6c307bb8163a9a7e16b3f"
+    }
+  ],
+  "external_availability_note": "All runtime bytes are retained at an ancestor export commit in this package branch. Original foreign/local predecessor objects are recorded separately as attribution and byte-equality provenance, not needed by standalone replay or metadata source reads.",
+  "original_source_objects": [
+    {
+      "path": "research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json",
+      "usage": "provenance",
+      "git_blob": "d9a4f84211182ca200c081ea061d724c8a2cfb70",
+      "sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
+      "pin": "head:df0137efd35e11b4445734bf612bdd4c7ebc330c:research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json",
+      "retained_copy": "inputs/DATA-PHASES.json"
+    },
+    {
+      "path": "research/p104-two-boundary-helper-split/NOTICE",
+      "usage": "provenance",
+      "git_blob": "f83cecccf587acfa313c8259c0bf6db05d60b52d",
+      "sha256": "c474479a870ae53ae0bcb84f656c2b222380395bbee767ee15e5f9c5b4ce7e79",
+      "pin": "head:a1534ac45401c9a9c80c07aa65c89e00e33594f8:research/p104-two-boundary-helper-split/NOTICE",
+      "retained_copy": "inputs/NOTICE5718"
+    },
+    {
+      "path": "research/p104-native-channel/vendor/clifford_exact.py",
+      "usage": "provenance",
+      "git_blob": "166841a037f069249f6f7f6dedb89248812f59e2",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+      "pin": "head:df0137efd35e11b4445734bf612bdd4c7ebc330c:research/p104-native-channel/vendor/clifford_exact.py",
+      "retained_copy": "inputs/clifford_exact.py"
+    },
+    {
+      "path": "research/p104-productive-controller/CONTROLLER.json",
+      "usage": "provenance",
+      "git_blob": "cc539db709cf7a6c42872c4380902736ad453af3",
+      "sha256": "a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9",
+      "pin": "head:a1534ac45401c9a9c80c07aa65c89e00e33594f8:research/p104-productive-controller/CONTROLLER.json",
+      "retained_copy": "inputs/controller5718.json"
+    },
+    {
+      "path": "research/p104-native-channel/vendor/native_verify.py",
+      "usage": "provenance",
+      "git_blob": "2c209dfa9cfade3bc20ac5fc9d876b433a147f8c",
+      "sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38",
+      "pin": "head:df0137efd35e11b4445734bf612bdd4c7ebc330c:research/p104-native-channel/vendor/native_verify.py",
+      "retained_copy": "inputs/native_verify.py"
+    },
+    {
+      "path": "research/p104-two-boundary-helper-split/RESULT.json",
+      "usage": "provenance",
+      "git_blob": "a83d70e025254e217c01ca413f6823937e82f3d9",
+      "sha256": "8e62f19bc795cac7712d31f955b71587e251dc94f1a02a20235e1ea740015db7",
+      "pin": "head:a1534ac45401c9a9c80c07aa65c89e00e33594f8:research/p104-two-boundary-helper-split/RESULT.json",
+      "retained_copy": "inputs/result5718.json"
+    },
+    {
+      "path": "research/p104-two-boundary-helper-split/WORD.json.gz",
+      "usage": "provenance",
+      "git_blob": "4db015aca34ea957dd8b3b4d7ae4fa12b89934f8",
+      "sha256": "cff7196f8b168c4ebf8227185eed98387b66ec5db0179108adfea4045775cc31",
+      "pin": "head:a1534ac45401c9a9c80c07aa65c89e00e33594f8:research/p104-two-boundary-helper-split/WORD.json.gz",
+      "retained_copy": "inputs/word5718.json.gz"
+    },
+    {
+      "path": "research/p104-cross-call-paid-reuse/experiment.py",
+      "usage": "provenance",
+      "git_blob": "3309f0be8443c12724f31586f311fa2dda329c6b",
+      "sha256": "fd94db68e21a7a5cebc10c2731e264db8c304227bd069d1808051ee697399c4b",
+      "pin": "head:50dd8aa3cef02cd7b4a785b58f3a66d0edd91825:research/p104-cross-call-paid-reuse/experiment.py",
+      "retained_copy": "inputs/predecessor-experiment.py"
+    },
+    {
+      "path": "research/p104-cross-call-paid-reuse/results/QUERY-MAPS.json.gz",
+      "usage": "provenance",
+      "git_blob": "e241f1958cba98e3227bed0a96b7109c21da5a95",
+      "sha256": "3a168605b596d4398c7876d42f2213dfddd807281b85ec0fe07177cea1aa035b",
+      "pin": "head:50dd8aa3cef02cd7b4a785b58f3a66d0edd91825:research/p104-cross-call-paid-reuse/results/QUERY-MAPS.json.gz",
+      "retained_copy": "inputs/QUERY-MAPS.json.gz"
+    },
+    {
+      "path": "research/p104-cross-call-paid-reuse/SOURCES.json",
+      "usage": "provenance",
+      "git_blob": "a059933e0c4065f47c263df58ed0819909ea7ac1",
+      "sha256": "031f4fa2171cd47573b35c5817b320d432bede64506709339700c99cd0f4c2af",
+      "pin": "head:50dd8aa3cef02cd7b4a785b58f3a66d0edd91825:research/p104-cross-call-paid-reuse/SOURCES.json",
+      "retained_copy": "inputs/predecessor-SOURCES.json"
+    }
+  ],
+  "retained_export_commit": "25002e57adf7858afda0a0d1e208b3ea39f63507"
+}
diff --git a/research/p104-shared-frame-gather/TASK.json b/research/p104-shared-frame-gather/TASK.json
new file mode 100644
index 0000000000..a3e80ec2fc
--- /dev/null
+++ b/research/p104-shared-frame-gather/TASK.json
@@ -0,0 +1,76 @@
+{
+  "schema_version": 1,
+  "task_id": "centre-path-sharing",
+  "track": "recursive-interfaces",
+  "original_card": {
+    "id": "centre-path-sharing",
+    "question": "Can a specified persistent boundary eliminate a repeated centre operation in the current caller?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Write a two-call input/output contract on the current source word, including arbitrary centre baselines and every later read. Price producer, retention, native alignment and restoration jointly. Historical 64-per-call and 1.90x figures are not current budgets. Stop if a missing consumer or restored-input obligation consumes the saving; reopen only with a changed legal boundary.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "centre path",
+      "persistent boundary",
+      "dirty recoding"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "scope_change_reason": "Explicit shared-gather assignment overrides the historical centre-sharing task: named PR5718 caller event3 local13660, not the selected historical centre path. Source-coframe preaggregation is granted to avoid charging redundant ingress.",
+  "actual_scope": "Every retained earlier paid snapshot before local13660, optionally bundled by its source frame; one of1090 pinned canonical gather frames and one final canonical delivery to598. Exact fresh-helper coordinate/cost obstruction.",
+  "bottleneck_link": "A strict paid H7 deletion with target delta zero would improve the source complete invoice. This protocol cannot earn it.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after a decisive optimistic complete-ingress obstruction or fully costed failure; do not expand separate-cache or unrelated helper lanes.",
+  "outcome": "negative-result",
+  "evidence": "results/RESULT.json; results/CERTIFICATE.json.gz; independent audit_certificate.py.",
+  "author": "zxv",
+  "ai_assistance": "Substantial OpenAI Codex assistance; no subagents, independent human or formal-kernel review.",
+  "open_questions": [
+    {
+      "question": "Can a source-bound matrix-valued combination or a genuinely multilevel shared native circuit beat this single-centre allowance?",
+      "why_it_matters": "Canonical one-frame gathers, even source-coframe bundles with free capture and cleanup, cannot improve this H7 query.",
+      "first_step": "Specify one changed native input/output contract and a complete ingress/final/dirty-return bill before scalar search.",
+      "blocked_by": "No such changed contract, legal word or favorable complete invoice is supplied."
+    }
+  ],
+  "expansion_decision": "Stop the pinned canonical single-frame gather route. Reopen only with a named source/consumer/native-operation hypothesis changed and favorable fully charged budget.",
+  "latest_guidance_revision_read": 8,
+  "publication": "Local-only. No public posting, PR, merging or selected-frontier changes.",
+  "base_commit": "de5ec2e7a6074068452f45b1a191b5c72832df4a"
+}
diff --git a/research/p104-shared-frame-gather/VALIDATION.md b/research/p104-shared-frame-gather/VALIDATION.md
new file mode 100644
index 0000000000..e21da31a46
--- /dev/null
+++ b/research/p104-shared-frame-gather/VALIDATION.md
@@ -0,0 +1,61 @@
+# Validation record
+
+Local-only; 2026-10-26. CPython3.12.8, macOS-26.5.1-arm64-arm-64bit;
+standard library only. Substantial OpenAI Codex assistance; no subagents,
+independent human or formal-kernel review.
+
+Passed serially:
+
+- Exact byte equality of ten retained runtime files against their original
+  pinned Git objects: local predecessor code/maps/source manifest and the seven
+  PR5718/native source inputs. The generic metadata pins now resolve the same
+  retained copies at this branch's ancestor export commit; scientific replay
+  needs neither the predecessor worktree nor foreign Git history.
+- `python3.12 -B scripts/research_package.py --package research/p104-shared-frame-gather --check-only`:
+  `PASS_PACKAGE_METADATA_AND_PINS`.
+- `python3.12 -B scripts/research_package.py --package research/p104-shared-frame-gather`:
+  normal and optimized generator receipt comparisons, and normal and optimized
+  separate certificate audits pass; `PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result`.
+- Optimized direct replay of both engines in a separate Git-free package copy
+  with empty PATH. All21 copied files stayed byte-identical. A certificate
+  mutation that undercharged one ingress failed in the separate auditor with
+  `Every source-frame ingress charged`; original receipt restored afterward.
+  See `results/PORTABLE-REPLAY.json`, including exact commands and environment.
+- `git diff --check` and the final committed-byte preflight
+  `python3.12 -B scripts/check_research_commit.py --package research/p104-shared-frame-gather`.
+
+Coverage: all3,426 local paid maps; exact744-public/9-owned arbitrary-input
+endpoint replay; both536-helper invocation namespaces; all32 raw original
+caller columns at each routed port;416 exact phase-polynomial source-binding
+checks;1,211 available prefix outputs and465 source-coframe bundles;
+all1,090 pinned gather-frame labels, of which478 have appeared before the cut;
+316,590 independently recomputed used pair tariffs;1,384 literal relative-map
+rank checks; four exact native-map/absolute-phase controls; all1,090 exact
+support/budget receipts; complete66,348-job/12,976-target source invoice;
+positive nested two-snapshot gather and future/stale/phase/frame/missing-helper/
+underpriced-ingress/unknown-cleanup rejection controls.
+
+Scientific RESULT SHA256:
+`2bccd5484b3db817c32d4af63be5d7a15d590f328c078e2271e996f050df3338`.
+
+Decompressed CERTIFICATE payload SHA256:
+`b25c717de42242242d0f68b3033d5c453eea602164b4f0d69e1c0bd1bdd1006d`.
+Compressed container bytes are not the mathematical equality criterion.
+
+Canonical main refreshed again at completion to
+`de5ec2e7a6074068452f45b1a191b5c72832df4a`; priority8 and selected .00310045
+unchanged. Live overlap is timestamped in the saved snapshots.
+
+No selected cold replay, full native-lowering suite, Lean, Docker, new
+compiler/outer scientific replay, formal proof or rounded multiplier was run.
+No replacement program, new κ, global P104 barrier, public post or PR is claimed.
+Unknown acquisition, retention, normalizer, ordinary, dirty-return and cleanup
+costs were omitted only in the favorable exclusion. No new target credit was
+assigned. Additional gather frames, source/consumer changes and general
+matrix-valued recombination remain outside the result.
+
+The isolated sparse conventional worktree is retained for access to the local
+artifact. Managed creation failed for disk space; attaching the fallback was
+rejected as unmanaged. A later optional temporary-copy cleanup was blocked by
+the command guard; the safe replay omitted that cleanup and retained its own
+small temporary copy. No other checkout or predecessor files were changed.
diff --git a/research/p104-shared-frame-gather/VERIFY.json b/research/p104-shared-frame-gather/VERIFY.json
new file mode 100644
index 0000000000..db8288065e
--- /dev/null
+++ b/research/p104-shared-frame-gather/VERIFY.json
@@ -0,0 +1,82 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/p104-shared-frame-gather/inputs/DATA-PHASES.json",
+    "research/p104-shared-frame-gather/inputs/NOTICE5718",
+    "research/p104-shared-frame-gather/inputs/QUERY-MAPS.json.gz",
+    "research/p104-shared-frame-gather/inputs/clifford_exact.py",
+    "research/p104-shared-frame-gather/inputs/controller5718.json",
+    "research/p104-shared-frame-gather/inputs/native_verify.py",
+    "research/p104-shared-frame-gather/inputs/predecessor-SOURCES.json",
+    "research/p104-shared-frame-gather/inputs/predecessor-experiment.py",
+    "research/p104-shared-frame-gather/inputs/result5718.json",
+    "research/p104-shared-frame-gather/inputs/word5718.json.gz",
+    "research/p104-cross-call-paid-reuse/inputs/DATA-PHASES.json",
+    "research/p104-cross-call-paid-reuse/inputs/NOTICE5718",
+    "research/p104-cross-call-paid-reuse/inputs/clifford_exact.py",
+    "research/p104-cross-call-paid-reuse/inputs/controller5718.json",
+    "research/p104-cross-call-paid-reuse/inputs/native_verify.py",
+    "research/p104-cross-call-paid-reuse/inputs/result5718.json",
+    "research/p104-cross-call-paid-reuse/inputs/word5718.json.gz",
+    "research/p104-cross-call-paid-reuse/experiment.py",
+    "research/p104-cross-call-paid-reuse/SOURCES.json",
+    "research/p104-cross-call-paid-reuse/results/QUERY-MAPS.json.gz"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/publication_check.py"
+      ],
+      "timeout_seconds": 30,
+      "checks": "Byte-exact join of the published query-map and shared-gather source packages."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Exact pinned all-input query maps, chronological availability, source-coframe bundles, all1090 common-frame support/cost certificates, independent native-frame tariffs, unchanged complete source invoice and constructive/version/phase/frame/cost controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/audit_certificate.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Separate certificate auditor with no search/compiler imports: exact coordinate expansions, basis independence, all native tariffs and every frame support-budget obstruction."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Full deterministic result and certificate equality with assertions disabled."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/audit_certificate.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Independent audit with assertions disabled."
+    }
+  ]
+}
diff --git a/research/p104-shared-frame-gather/audit_certificate.py b/research/p104-shared-frame-gather/audit_certificate.py
new file mode 100644
index 0000000000..2254fe29c6
--- /dev/null
+++ b/research/p104-shared-frame-gather/audit_certificate.py
@@ -0,0 +1,109 @@
+"""Read-only independent audit of the exported one-frame support certificate.
+
+Does not import verify.py, predecessor-experiment.py, or clifford_exact.py.
+It uses the retained independent Gaussian circuit interpreter for frame maps.
+Prepared for zxv with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from functools import lru_cache
+from pathlib import Path
+import gzip, importlib.util, json, sys
+R=Path(__file__).resolve().parent
+
+def check(x,why):
+    if not x:raise ValueError(why)
+
+def read(p):
+    b=(R/p).read_bytes();return json.loads(gzip.decompress(b) if p.endswith('.gz') else b)
+
+def binary_dimension(vectors):
+    # Low-pivot elimination, separate from producer's high-pivot routine.
+    pivots={}
+    for x in vectors:
+        while x:
+            p=(x&-x).bit_length()-1
+            if p in pivots:x^=pivots[p]
+            else:pivots[p]=x;break
+    return len(pivots)
+
+def scalar_rank(rows):
+    prime=1000003;basis={}
+    for row in rows:
+        row={int(k):(Q(v).numerator*pow(Q(v).denominator,-1,prime))%prime for k,v in row.items()}
+        while row:
+            pivot=max(row);a=row[pivot]
+            if pivot not in basis:
+                scale=pow(a,-1,prime);basis[pivot]={k:v*scale%prime for k,v in row.items()};break
+            for k,v in basis[pivot].items():
+                z=(row.get(k,0)-a*v)%prime
+                if z:row[k]=z
+                else:row.pop(k,None)
+    return len(basis)
+
+
+def run():
+    cert=read('results/CERTIFICATE.json.gz');maps=read('inputs/QUERY-MAPS.json.gz');w=read('inputs/word5718.json.gz')
+    main=set(w['live'])|{r+112 for r in w['live']};helper=set(range(w['public']))-main
+    projected=lambda p:{k:Q(v) for k,v in p['row'].items() if int(k) in helper}
+    by_event={str(p['event']):p for p in maps['queries']}
+    query=by_event['13660'];check(query==cert['query'],'Target query map')
+    events=cert['prefix_events'];check(events==[p['event'] for p in maps['queries'] if p['event']<13660],'All and only chronological prefix outputs')
+    co=cert['source_helper_coordinates'];tc=cert['target_helper_coordinates'];dc=cert['destination_observation_coordinates']
+    obs=next(x for x in maps['selected_current_frame_rows'] if x['event']==13660)
+    sources=dict(by_event)
+    for r,v,last,row in obs['donors']:sources[f'current:{r}:v{v}']={'row':row}
+    for p in obs['retained']:sources[f"retained:{p['event']}"]=p
+    all_labels=set().union(*(set(c) for c in co),set(tc),*(set(c) for c in dc))
+    check(scalar_rank([projected(sources[label]) for label in all_labels])==len(all_labels),'Coordinate source rows independent over Q (full modular rank suffices)')
+    def expand(c):
+        out=Counter()
+        for label,q in c.items():
+            for k,v in projected(sources[label]).items():out[k]+=Q(q)*v
+        return {k:v for k,v in out.items() if v}
+    for event,c in zip(events,co):check(expand(c)==projected(by_event[str(event)]),'Every exact coordinate expansion')
+    check(expand(tc)==projected(query),'Exact target-coordinate expansion')
+    destination_rows=[row for r,v,last,row in obs['donors']]+[p['row'] for p in obs['retained']]
+    check(len(dc)==len(destination_rows),'All destination observations')
+    for c,row in zip(dc,destination_rows):check(expand(c)==projected({'row':row}),'Exact destination-coordinate expansion')
+    check(all(not row.get('21') for row in destination_rows),'Destination cannot supply fresh helper21')
+    check(Q(query['row']['21'])==Q(-1,4),'Nonzero fresh-helper response')
+    groups={}
+    for event in events:groups.setdefault(str(by_event[str(event)]['new']),[]).append(event)
+    check(groups==cert['source_frame_groups'],'Complete source-coframe bundles')
+    coords_by_event=dict(zip(events,co))
+    supports={f:set().union(*(set(coords_by_event[e]) for e in es)) for f,es in groups.items()}
+    spec=importlib.util.spec_from_file_location('independent_native',R/'inputs/native_verify.py');V=importlib.util.module_from_spec(spec);sys.modules[spec.name]=V;spec.loader.exec_module(V)
+    L=[]
+    for frame in w['frames']:
+        tab=V.tableau(V.adjoint(V.frame(tuple(frame),9)))
+        L.append(tuple(x|(z<<9) for x,z,p in tab[9:]))
+    @lru_cache(None)
+    def distance(a,b):return binary_dimension(L[a]+L[b])-9
+    counts=Counter();dc_support=set().union(*(set(c) for c in dc))
+    check([r['frame'] for r in cert['frame_receipts']]==list(range(1090)),'Every pinned frame exactly once')
+    for r in cert['frame_receipts']:
+        f=r['frame'];final=distance(f,598);B=6-final
+        check((r['final_rank'],r['ingress_rank_budget'])==(final,B),'Final charge and residual budget')
+        if B<0:
+            check(r['status']=='FINAL_DELIVERY_ALREADY_COSTS_AT_LEAST_H7','Final rank rejection');counts['final_rank']+=1;continue
+        costs={s:distance(int(s),f) for s in groups}
+        h21=[costs[str(by_event[str(e)]['new'])] for e in events if Q(by_event[str(e)]['row'].get('21','0'))]
+        if r['status']=='FRESH_HELPER21_UNAVAILABLE_WITHIN_INGRESS_BUDGET':
+            check(min(h21)>B and min(h21)==r['minimum_helper21_ingress_rank'],'Every source lacking affordable fresh helper21');counts['helper21']+=1;continue
+        check(r['status']=='EXACT_HELPER_COORDINATE_COVER_EXCEEDS_BUDGET','Support certificate status')
+        s=r['certificate'];check(costs==s['source_frame_ingress_ranks'],'Every source-frame ingress charged')
+        free=[f for f,c in costs.items() if c==0 or (c<=B and len(supports[f])>s['threshold'])]
+        check(list(map(str,s['free_source_frames']))==free,'Favorable free source bundles')
+        ignored=dc_support|set().union(*(supports[f] for f in free))
+        check(set(map(str,s['ignored_coordinates']))==ignored,'Ignored-coordinate union')
+        uncovered=set(tc)-ignored;check(set(map(str,s['uncovered_target_coordinates']))==uncovered,'All mandatory target coordinates')
+        maxwidth={str(rank):max((len(supports[f]-ignored) for f,c in costs.items() if c==rank),default=0) for rank in range(1,B+1)}
+        check(maxwidth==s['max_coordinate_coverage_by_ingress_rank'],'Complete coverage tariff')
+        dp=s['coverage_capacity_dp'];check(len(dp)==B+1 and dp[0]==0,'DP base and extent')
+        for b in range(1,B+1):check(dp[b]==max([dp[b-1]]+[maxwidth[str(rank)]+dp[b-rank] for rank in range(1,b+1)]),'Optimistic knapsack upper bound')
+        check(len(uncovered)>dp[B] and len(uncovered)-dp[B]==s['obstruction_margin'],'Strict support-budget obstruction');counts['support']+=1
+    check(dict(counts)=={'final_rank':411,'helper21':338,'support':341},'Complete exhaustive result')
+    print('PASS_INDEPENDENT_SHARED_GATHER_CERTIFICATE_AUDIT frames=1090 source_bundles=465 prefix_answers=1211')
+
+if __name__=='__main__':run()
diff --git a/research/p104-shared-frame-gather/inputs/DATA-PHASES.json b/research/p104-shared-frame-gather/inputs/DATA-PHASES.json
new file mode 100644
index 0000000000..d9a4f84211
--- /dev/null
+++ b/research/p104-shared-frame-gather/inputs/DATA-PHASES.json
@@ -0,0 +1 @@
+{"cover_correction":{"complete_small_cover_checks":[{"n":1,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":1},{"n":2,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":2},{"n":3,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":6},{"n":4,"nonzero_full_kernel_corrections":24,"orthogonal_matrices":48},{"n":5,"nonzero_full_kernel_corrections":600,"orthogonal_matrices":720}],"formula":"delta(g)_i = (weight(row_i(g))-1)/2 mod 2; apply X_delta(g) to EVERY final live stream in cover class g.","mandatory":true,"reason":"P_g C_all P_g^-1 = C_all X_delta(g) exactly. Orthogonality cancels all quadratic cross terms; both sides fix the uniform vector."},"global_phase_argument":"Every C_q and every linear address permutation fixes the uniform vector. Equality of the exact Z4 Fourier multiplier therefore fixes the scalar phase as 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complete recipe is unchanged except its port index; the retained parent polynomial checks establish every selected equality.","scope":"All compressed data boundary operators, idle children, terminal X_q and all-live cover correction. 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diff --git a/research/p104-shared-frame-gather/inputs/NOTICE5718 b/research/p104-shared-frame-gather/inputs/NOTICE5718
new file mode 100644
index 0000000000..f83cecccf5
--- /dev/null
+++ b/research/p104-shared-frame-gather/inputs/NOTICE5718
@@ -0,0 +1,47 @@
+Four paid interior public-basis changes.
+Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0.
+No subagents.
+
+The following exact notice is retained from PR5660.
+
+New helper-subspace construction, proof and replay:
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+The same assistant performed the construction and author-side checks.
+
+Retained contributions:
+flyingviper: PR5605 shared relay, PR5612 delayed relay, and the PR5608/PR5613
+relay/split compositions.
+Scratch-net: PR5606 prepared-helper identities and target debit.
+Pierre-Loic Doulcet (hexapode): PR5565 commuting-contact construction, used as the exact source.
+Ali Romman: PR5602 fixed boundary relations, PR5601 three-contact frame
+change, PR5580 contact step,
+and PR5544/5545 complete P104 caller.
+Joel Pulikkan (GamingPuzzled): PR5569 priced helper handoff.
+DreamingOfClouds: PR5561 free helper endpoints and PR5568 search tools.
+thu-nmrc: PR5551 contact improvements.
+Gareth Davidson (bitplane): PR2305 source composition.
+David Arias: PR2152 coalesced P104 word.
+Cinna Davis: PR2219 private reuse and the controller lineage.
+Jacob Alan Sussman: productive mixed-target recursion.
+zxv and Cao Yifan: exact intervals, compiler and native/pricing lineage.
+GamingPuzzled: PR5472 successor handoff, a predecessor to the split concept.
+The retained source notices preserve earlier geometry, BIT and analytic credit.
+
+The elementary rank-one basis identity is not claimed as a new general
+linear-algebra theorem. The contribution is its explicit independent-target
+use on this P104 word and the complete conditional kappa improvement.
+
+New contribution: Ali Romman, with substantial OpenAI Codex assistance.
+The four-gate source-version rewrite composes zxv PR5643 and Ali Romman PR5644.
+The A2 method retains Chris R/Clear1nsight PR3998 and ToikY PR4019 credit.
+The compensated triangle method retains Cinna Davis PR4692 credit.
+No subagents were used for this contribution. Apache-2.0.
+
+Concurrent PR5650 by zxv publishes the PR5643/PR5644 composition. PR5651 by
+Ali Romman corroborates it. This package credits that released composition
+and contributes the further four-gate placement.
+
+New work: distributed scalar updates on an unobserved public field interval, the exact witness and complete conditional price. Prepared for Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0. The source archive retains all predecessor notices.
+
+Two-boundary main-assisted helper split and early cleanup: flyingviper with substantial
+OpenAI Codex assistance, including independent AI review agents. Apache-2.0.
diff --git a/research/p104-shared-frame-gather/inputs/QUERY-MAPS.json.gz b/research/p104-shared-frame-gather/inputs/QUERY-MAPS.json.gz
new file mode 100644
index 0000000000..e241f1958c
Binary files /dev/null and b/research/p104-shared-frame-gather/inputs/QUERY-MAPS.json.gz differ
diff --git a/research/p104-shared-frame-gather/inputs/clifford_exact.py b/research/p104-shared-frame-gather/inputs/clifford_exact.py
new file mode 100644
index 0000000000..166841a037
--- /dev/null
+++ b/research/p104-shared-frame-gather/inputs/clifford_exact.py
@@ -0,0 +1,149 @@
+"""Small exact Gaussian-dyadic Clifford compiler, independent of gx internals.
+Paulis use i^p X^x Z^z. State vectors use integer Gaussian numerators / 2^d.
+The tableau is supplemented by a literal circuit, so global phases are kept.
+"""
+from dataclasses import dataclass
+
+def parity(x):return x.bit_count()&1
+def rank(rows):
+ p={}
+ for x in rows:
+  while x:
+   b=x.bit_length()-1
+   if b not in p:p[b]=x;break
+   x^=p[b]
+ return len(p)
+def transpose(rows,n):return tuple(sum(((rows[j]>>i)&1)<<j for j in range(n)) for i in range(n))
+def mv(rows,x):return sum(parity(r&x)<<i for i,r in enumerate(rows))
+def inv(rows,n):
+ a=[r|(1<<(n+i)) for i,r in enumerate(rows)]
+ for j in range(n):
+  k=next(i for i in range(j,n) if a[i]>>j&1);a[j],a[k]=a[k],a[j]
+  for i in range(n):
+   if i!=j and a[i]>>j&1:a[i]^=a[j]
+ assert [r&((1<<n)-1) for r in a]==[1<<i for i in range(n)]
+ return tuple(r>>n for r in a)
+def basis_extend(cols,n):
+ out=[]
+ for x in list(cols)+[1<<i for i in range(n)]:
+  if rank(out+[x])>len(out):out.append(x)
+ assert len(out)==n
+ return tuple(out)
+def nullspace(rows,n):
+ a=list(rows);p=[];k=0
+ for j in range(n):
+  t=next((i for i in range(k,len(a)) if a[i]>>j&1),None)
+  if t is None:continue
+  a[t],a[k]=a[k],a[t]
+  for i in range(len(a)):
+   if i!=k and a[i]>>j&1:a[i]^=a[k]
+  p.append(j);k+=1
+ return tuple((1<<j)|sum(((a[i]>>j)&1)<<v for i,v in enumerate(p)) for j in range(n) if j not in p)
+def pmul(a,b):
+ x,z,p=a;y,w,q=b
+ return x^y,z^w,(p+q+2*parity(z&y))%4
+def identity(n):return tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,0) for i in range(n)])
+def apply(T,p):
+ n=len(T)//2;x,z,ph=p;out=(0,0,ph)
+ for i in range(n):
+  if x>>i&1:out=pmul(out,T[i])
+ for i in range(n):
+  if z>>i&1:out=pmul(out,T[n+i])
+ return out
+def compose(A,B):return tuple(apply(A,p) for p in B)
+def inverse(T):
+ n=len(T)//2;cols=[x|(z<<n) for x,z,p in T];rows=transpose(cols,2*n);cols_inv=transpose(inv(rows,2*n),2*n);out=[]
+ for j,w in enumerate(cols_inv):
+  x=w&((1<<n)-1);z=w>>n;u,v,p=apply(T,(x,z,0));assert (u,v)==identity(n)[j][:2];out.append((x,z,(-p)%4))
+ return tuple(out)
+def qvalue(Q,alpha,x,n):
+ return (sum(alpha[i] for i in range(n) if x>>i&1)+2*sum(parity((Q[i]>>(i+1))&(x>>(i+1))) for i in range(n) if x>>i&1))%4
+def mul_i(a,p):
+ x,y=a;p%=4
+ return [(x,y),(-y,x),(-x,-y),(y,-x)][p]
+@dataclass
+class Op:
+ n:int
+ tab:tuple
+ circuit:list
+ def then(self,b):
+  assert self.n==b.n
+  return Op(self.n,compose(b.tab,self.tab),self.circuit+b.circuit)
+ def adjoint(self):
+  circ=[]
+  for e in reversed(self.circuit):
+   if e[0]=='P':circ.append(('P',inv(e[1],self.n)))
+   elif e[0]=='D':circ.append(('D',e[1],tuple((-x)%4 for x in e[2])))
+   elif e[0]=='C':circ.append(('C',e[1],-e[2]))
+   elif e[0]=='phase':circ.append(('phase',(-e[1])%4))
+   else:circ.append(e)
+  return Op(self.n,inverse(self.tab),circ)
+ def zero_state(self,basis=0):
+  N=1<<self.n;vec=[(0,0)]*N;vec[basis]=(1,0);den=0
+  for e in self.circuit:
+   if e[0]=='P':
+    v=[(0,0)]*N
+    for x,a in enumerate(vec):v[mv(e[1],x)]=a
+    vec=v
+   elif e[0]=='X':vec=[vec[x^e[1]] for x in range(N)]
+   elif e[0]=='D':vec=[mul_i(a,qvalue(e[1],e[2],x,self.n)) for x,a in enumerate(vec)]
+   elif e[0]=='phase':vec=[mul_i(a,e[1]) for a in vec]
+   elif e[0]=='C':
+    q,sgn=e[1:];v=[]
+    for x,(a,b) in enumerate(vec):
+     c,d=vec[x^q]
+     # C=(1+i)I/2+(1-i)X/2; inverse conjugates both coefficients.
+     v.append((a-sgn*b+c+sgn*d,b+sgn*a+d-sgn*c))
+    vec=v;den+=1
+   else:raise AssertionError(e)
+   while den and all((a&1)==(b&1)==0 for a,b in vec):vec=[(a//2,b//2) for a,b in vec];den-=1
+  return vec,den
+def ident(n):return Op(n,identity(n),[])
+def perm(n,G):
+ cols=transpose(G,n);gi=inv(G,n)
+ return Op(n,tuple([(a,0,0) for a in cols]+[(0,a,0) for a in gi]),[('P',tuple(G))])
+def diag(n,Q,alpha=None):
+ Q=tuple(Q);assert transpose(Q,n)==Q
+ alpha=tuple(alpha) if alpha is not None else tuple((Q[i]>>i)&1 for i in range(n))
+ assert all(alpha[i]%2==(Q[i]>>i)&1 for i in range(n))
+ return Op(n,tuple([(1<<i,Q[i],alpha[i]) for i in range(n)]+[(0,1<<i,0) for i in range(n)]),[('D',Q,alpha)])
+def cgate(n,q,sgn=1):
+ return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(q if q>>i&1 else 0,1<<i,(3*sgn)%4 if q>>i&1 else 0) for i in range(n)]),[('C',q,sgn)])
+def xgate(n,b):return Op(n,tuple([(1<<i,0,0) for i in range(n)]+[(0,1<<i,2*((b>>i)&1)) for i in range(n)]),[('X',b)])
+def fourier(n,axes=None):
+ out=ident(n)
+ for i in range(n) if axes is None else axes:out=out.then(cgate(n,1<<i))
+ return out
+def hadamard(n,axes):
+ axes=list(axes);Q=tuple(1<<i if i in axes else 0 for i in range(n));S=diag(n,Q)
+ return S.then(fourier(n,axes)).then(S)
+def canonical_frame(n,U):
+ r=len(U)
+ if r==0:return ident(n)
+ if r==n:return fourier(n)
+ G=transpose(basis_extend(U,n),n);gi=inv(G,n);Q=tuple((1<<i)^sum(parity(gi_i&gi_j)<<j for j,gi_j in enumerate(transpose(gi,n))) for i,gi_i in enumerate(transpose(gi,n)))
+ K=perm(n,G).then(diag(n,Q))
+ return K.adjoint().then(fourier(n,range(r))).then(K)
+def monomial(op):
+ n=op.n;T=op.tab;assert all(x==0 and p%2==0 for x,z,p in T[n:]);G=transpose([x for x,z,p in T[:n]],n);b=mv(G,sum((p//2)<<i for i,(x,z,p) in enumerate(T[n:])))
+ gt=transpose(G,n);cols=[mv(gt,z) for x,z,p in T[:n]];Q=transpose(cols,n);alpha=[(p-2*parity(b&z))%4 for x,z,p in T[:n]]
+ out=diag(n,Q,alpha).then(perm(n,G)).then(xgate(n,b));assert out.tab==T
+ vec,den=op.zero_state();assert den==0 and all(a==(0,0) for i,a in enumerate(vec) if i!=b);phase=[(1,0),(0,1),(-1,0),(0,-1)].index(vec[b]);out=out.then(Op(n,identity(n),[('phase',phase)]));assert out.zero_state()==op.zero_state()
+ return out,dict(matrix_rows=G,offset=b,quadratic_rows=Q,linear_Z4=alpha,phase=phase)
+def lower_child(op):
+ n=op.n;Bcols=[x for x,z,p in op.tab[n:]];r=rank(Bcols)
+ if not r:
+  mon,rec=monomial(op);return dict(rank=0,left=rec,right=None),mon
+ G=transpose(basis_extend(Bcols,n),n);A=inv(G,n);PA=perm(n,A);part=op.then(PA);Bcols=[x for x,z,p in part.tab[n:]];Dcols=[z for x,z,p in part.tab[n:]]
+ picks=[];selected=[]
+ for j,x in enumerate(Bcols):
+  if rank(selected+[x])>len(selected):picks.append(j);selected.append(x)
+ assert len(picks)==r and all(x>>r==0 for x in Bcols)
+ Br=transpose(selected,r);Bi=inv(Br,r);Dr=transpose([Dcols[j]&((1<<r)-1) for j in picks],r)
+ Qsmall=tuple(sum(parity(row&col)<<j for j,col in enumerate(transpose(Bi,r))) for row in Dr);Q=Qsmall+(0,)*(n-r);DQ=diag(n,Q)
+ V=part.then(DQ).then(hadamard(n,range(r)));assert all(x==0 for x,z,p in V.tab[n:]);Vmon,Vrec=monomial(V)
+ S=diag(n,tuple(1<<i if i<r else 0 for i in range(n)))
+ left=S.adjoint().then(DQ.adjoint()).then(PA.adjoint());right=Vmon.then(S.adjoint()).then(xgate(n,(1<<r)-1))
+ lmon,lrec=monomial(left);rmon,rrec=monomial(right);compiled=rmon.then(fourier(n,range(r))).then(lmon)
+ assert compiled.tab==op.tab and compiled.zero_state()==op.zero_state()
+ return dict(rank=r,left=lrec,right=rrec),compiled
diff --git a/research/p104-shared-frame-gather/inputs/controller5718.json b/research/p104-shared-frame-gather/inputs/controller5718.json
new file mode 100644
index 0000000000..cc539db709
--- /dev/null
+++ b/research/p104-shared-frame-gather/inputs/controller5718.json
@@ -0,0 +1,5480 @@
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+    [
+      "Q_POWER",
+      7,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      7,
+      6
+    ],
+    [
+      "IDLE",
+      4,
+      [
+        0,
+        1,
+        3,
+        5,
+        7
+      ]
+    ],
+    [
+      "ADD",
+      6,
+      8,
+      1
+    ],
+    [
+      "ADD",
+      9,
+      7,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      8,
+      3,
+      false
+    ],
+    [
+      "ADD",
+      6,
+      8,
+      -1
+    ],
+    [
+      "ADD",
+      9,
+      7,
+      1
+    ],
+    [
+      "Q_POWER",
+      8,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      8,
+      9
+    ],
+    [
+      "LOCAL",
+      9,
+      4,
+      false
+    ],
+    [
+      "Q_POWER",
+      9,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      9,
+      8
+    ],
+    [
+      "IDLE",
+      5,
+      [
+        0,
+        1,
+        3,
+        5,
+        7,
+        9
+      ]
+    ],
+    [
+      "ADD",
+      8,
+      10,
+      1
+    ],
+    [
+      "ADD",
+      11,
+      9,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      10,
+      4,
+      false
+    ],
+    [
+      "ADD",
+      8,
+      10,
+      -1
+    ],
+    [
+      "ADD",
+      11,
+      9,
+      1
+    ],
+    [
+      "Q_POWER",
+      10,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      10,
+      11
+    ],
+    [
+      "LOCAL",
+      11,
+      5,
+      false
+    ],
+    [
+      "Q_POWER",
+      11,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      11,
+      10
+    ],
+    [
+      "IDLE",
+      6,
+      [
+        0,
+        1,
+        3,
+        5,
+        7,
+        9,
+        11
+      ]
+    ],
+    [
+      "ADD",
+      10,
+      12,
+      1
+    ],
+    [
+      "ADD",
+      13,
+      11,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      12,
+      5,
+      false
+    ],
+    [
+      "ADD",
+      10,
+      12,
+      -1
+    ],
+    [
+      "ADD",
+      13,
+      11,
+      1
+    ],
+    [
+      "Q_POWER",
+      12,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      12,
+      13
+    ],
+    [
+      "LOCAL",
+      13,
+      6,
+      false
+    ],
+    [
+      "Q_POWER",
+      13,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      13,
+      12
+    ],
+    [
+      "IDLE",
+      7,
+      [
+        0,
+        1,
+        3,
+        5,
+        7,
+        9,
+        11,
+        13
+      ]
+    ],
+    [
+      "ADD",
+      12,
+      14,
+      1
+    ],
+    [
+      "ADD",
+      15,
+      13,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      14,
+      6,
+      false
+    ],
+    [
+      "ADD",
+      12,
+      14,
+      -1
+    ],
+    [
+      "ADD",
+      15,
+      13,
+      1
+    ],
+    [
+      "Q_POWER",
+      14,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      14,
+      15
+    ],
+    [
+      "LOCAL",
+      15,
+      7,
+      false
+    ],
+    [
+      "Q_POWER",
+      15,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      15,
+      14
+    ],
+    [
+      "ADD",
+      14,
+      16,
+      1
+    ],
+    [
+      "ADD",
+      17,
+      15,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      16,
+      7,
+      false
+    ],
+    [
+      "ADD",
+      14,
+      16,
+      -1
+    ],
+    [
+      "ADD",
+      17,
+      15,
+      1
+    ],
+    [
+      "Q_POWER",
+      16,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      16,
+      17
+    ],
+    [
+      "LOCAL",
+      17,
+      8,
+      false
+    ],
+    [
+      "Q_POWER",
+      17,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      17,
+      16
+    ],
+    [
+      "ADD",
+      16,
+      18,
+      1
+    ],
+    [
+      "ADD",
+      19,
+      17,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      18,
+      8,
+      false
+    ],
+    [
+      "ADD",
+      16,
+      18,
+      -1
+    ],
+    [
+      "ADD",
+      19,
+      17,
+      1
+    ],
+    [
+      "Q_POWER",
+      18,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      18,
+      19
+    ],
+    [
+      "LOCAL",
+      19,
+      9,
+      false
+    ],
+    [
+      "Q_POWER",
+      19,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      19,
+      18
+    ],
+    [
+      "ADD",
+      18,
+      20,
+      1
+    ],
+    [
+      "ADD",
+      21,
+      19,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      20,
+      9,
+      false
+    ],
+    [
+      "ADD",
+      18,
+      20,
+      -1
+    ],
+    [
+      "ADD",
+      21,
+      19,
+      1
+    ],
+    [
+      "Q_POWER",
+      20,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      20,
+      21
+    ],
+    [
+      "LOCAL",
+      21,
+      10,
+      false
+    ],
+    [
+      "Q_POWER",
+      21,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      21,
+      20
+    ],
+    [
+      "ADD",
+      20,
+      22,
+      1
+    ],
+    [
+      "ADD",
+      23,
+      21,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      22,
+      10,
+      false
+    ],
+    [
+      "ADD",
+      20,
+      22,
+      -1
+    ],
+    [
+      "ADD",
+      23,
+      21,
+      1
+    ],
+    [
+      "Q_POWER",
+      22,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      22,
+      23
+    ],
+    [
+      "LOCAL",
+      23,
+      11,
+      false
+    ],
+    [
+      "Q_POWER",
+      23,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      23,
+      22
+    ],
+    [
+      "ADD",
+      22,
+      24,
+      1
+    ],
+    [
+      "ADD",
+      25,
+      23,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      24,
+      11,
+      false
+    ],
+    [
+      "ADD",
+      22,
+      24,
+      -1
+    ],
+    [
+      "ADD",
+      25,
+      23,
+      1
+    ],
+    [
+      "Q_POWER",
+      24,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      24,
+      25
+    ],
+    [
+      "LOCAL",
+      25,
+      12,
+      false
+    ],
+    [
+      "Q_POWER",
+      25,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      25,
+      24
+    ],
+    [
+      "ADD",
+      24,
+      26,
+      1
+    ],
+    [
+      "ADD",
+      27,
+      25,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      26,
+      12,
+      false
+    ],
+    [
+      "ADD",
+      24,
+      26,
+      -1
+    ],
+    [
+      "ADD",
+      27,
+      25,
+      1
+    ],
+    [
+      "Q_POWER",
+      26,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      26,
+      27
+    ],
+    [
+      "LOCAL",
+      27,
+      13,
+      false
+    ],
+    [
+      "Q_POWER",
+      27,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      27,
+      26
+    ],
+    [
+      "ADD",
+      26,
+      28,
+      1
+    ],
+    [
+      "ADD",
+      29,
+      27,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      28,
+      13,
+      false
+    ],
+    [
+      "ADD",
+      26,
+      28,
+      -1
+    ],
+    [
+      "ADD",
+      29,
+      27,
+      1
+    ],
+    [
+      "Q_POWER",
+      28,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      28,
+      29
+    ],
+    [
+      "LOCAL",
+      29,
+      14,
+      false
+    ],
+    [
+      "Q_POWER",
+      29,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      29,
+      28
+    ],
+    [
+      "ADD",
+      28,
+      30,
+      1
+    ],
+    [
+      "ADD",
+      31,
+      29,
+      -1
+    ],
+    [
+      "QUERY_SHEAR",
+      30,
+      14,
+      false
+    ],
+    [
+      "ADD",
+      28,
+      30,
+      -1
+    ],
+    [
+      "ADD",
+      31,
+      29,
+      1
+    ],
+    [
+      "Q_POWER",
+      30,
+      -2
+    ],
+    [
+      "SIGNED_SWAP",
+      30,
+      31
+    ],
+    [
+      "LOCAL",
+      31,
+      15,
+      false
+    ],
+    [
+      "Q_POWER",
+      31,
+      2
+    ],
+    [
+      "SIGNED_SWAP",
+      31,
+      30
+    ],
+    [
+      "LOCAL",
+      32,
+      15,
+      false
+    ],
+    [
+      "IDLE",
+      7,
+      [
+        17,
+        19,
+        21,
+        23,
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      8,
+      [
+        19,
+        21,
+        23,
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      9,
+      [
+        21,
+        23,
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      10,
+      [
+        23,
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      11,
+      [
+        25,
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      12,
+      [
+        27,
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      13,
+      [
+        29,
+        31,
+        32
+      ]
+    ],
+    [
+      "IDLE",
+      14,
+      [
+        31,
+        32
+      ]
+    ],
+    [
+      "ADD",
+      30,
+      28,
+      1
+    ],
+    [
+      "ADD",
+      29,
+      31,
+      -1
+    ],
+    [
+      "ADD",
+      28,
+      26,
+      1
+    ],
+    [
+      "ADD",
+      27,
+      29,
+      -1
+    ],
+    [
+      "ADD",
+      26,
+      24,
+      1
+    ],
+    [
+      "ADD",
+      25,
+      27,
+      -1
+    ],
+    [
+      "ADD",
+      24,
+      22,
+      1
+    ],
+    [
+      "ADD",
+      23,
+      25,
+      -1
+    ],
+    [
+      "ADD",
+      22,
+      20,
+      1
+    ],
+    [
+      "ADD",
+      21,
+      23,
+      -1
+    ],
+    [
+      "ADD",
+      20,
+      18,
+      1
+    ],
+    [
+      "ADD",
+      19,
+      21,
+      -1
+    ],
+    [
+      "ADD",
+      18,
+      16,
+      1
+    ],
+    [
+      "ADD",
+      17,
+      19,
+      -1
+    ],
+    [
+      "ADD",
+      16,
+      14,
+      1
+    ],
+    [
+      "ADD",
+      15,
+      17,
+      -1
+    ],
+    [
+      "ADD",
+      14,
+      12,
+      1
+    ],
+    [
+      "ADD",
+      13,
+      15,
+      -1
+    ],
+    [
+      "ADD",
+      12,
+      10,
+      1
+    ],
+    [
+      "ADD",
+      11,
+      13,
+      -1
+    ],
+    [
+      "ADD",
+      10,
+      8,
+      1
+    ],
+    [
+      "ADD",
+      9,
+      11,
+      -1
+    ],
+    [
+      "ADD",
+      8,
+      6,
+      1
+    ],
+    [
+      "ADD",
+      7,
+      9,
+      -1
+    ],
+    [
+      "ADD",
+      6,
+      4,
+      1
+    ],
+    [
+      "ADD",
+      5,
+      7,
+      -1
+    ],
+    [
+      "ADD",
+      4,
+      2,
+      1
+    ],
+    [
+      "ADD",
+      3,
+      5,
+      -1
+    ],
+    [
+      "ADD",
+      2,
+      0,
+      1
+    ],
+    [
+      "ADD",
+      1,
+      3,
+      -1
+    ],
+    [
+      "SIGNED_SWAP",
+      0,
+      1
+    ],
+    [
+      "SIGNED_SWAP",
+      2,
+      3
+    ],
+    [
+      "SIGNED_SWAP",
+      4,
+      5
+    ],
+    [
+      "SIGNED_SWAP",
+      6,
+      7
+    ],
+    [
+      "SIGNED_SWAP",
+      8,
+      9
+    ],
+    [
+      "SIGNED_SWAP",
+      10,
+      11
+    ],
+    [
+      "SIGNED_SWAP",
+      12,
+      13
+    ],
+    [
+      "SIGNED_SWAP",
+      14,
+      15
+    ],
+    [
+      "SIGNED_SWAP",
+      16,
+      17
+    ],
+    [
+      "SIGNED_SWAP",
+      18,
+      19
+    ],
+    [
+      "SIGNED_SWAP",
+      20,
+      21
+    ],
+    [
+      "SIGNED_SWAP",
+      22,
+      23
+    ],
+    [
+      "SIGNED_SWAP",
+      24,
+      25
+    ],
+    [
+      "SIGNED_SWAP",
+      26,
+      27
+    ],
+    [
+      "SIGNED_SWAP",
+      28,
+      29
+    ],
+    [
+      "SIGNED_SWAP",
+      30,
+      31
+    ]
+  ],
+  "prefix_pairs": 7,
+  "wrapper_sites": [
+    1,
+    3,
+    5,
+    7,
+    9,
+    11,
+    13,
+    15,
+    17,
+    19,
+    21,
+    23,
+    25,
+    27,
+    29,
+    31
+  ]
+}
diff --git a/research/p104-shared-frame-gather/inputs/native_verify.py b/research/p104-shared-frame-gather/inputs/native_verify.py
new file mode 100644
index 0000000000..2c209dfa9c
--- /dev/null
+++ b/research/p104-shared-frame-gather/inputs/native_verify.py
@@ -0,0 +1,277 @@
+"""Check native operators with exact Gaussian dyadic arithmetic.
+
+The interpreter is adapted from the independent campaign native review.
+It does not import the source-code producer or the Clifford compiler.
+Research used OpenAI Codex under the user's direction.
+"""
+
+from functools import lru_cache
+
+
+def require(condition, label):
+    if not condition:
+        raise ValueError(label)
+
+
+def parity(value):
+    return value.bit_count() % 2
+
+
+def mul(matrix, vector):
+    return sum(parity(row & vector) << i for i, row in enumerate(matrix))
+
+
+def transpose(matrix):
+    return tuple(sum(((row >> j) & 1) << i for i, row in enumerate(matrix)) for j in range(len(matrix)))
+
+
+def inverse(matrix):
+    n = len(matrix)
+    rows = [row | (1 << (n+i)) for i, row in enumerate(matrix)]
+    for j in range(n):
+        pivot = next((i for i in range(j, n) if rows[i] >> j & 1), None)
+        require(pivot is not None, 'singular binary chart')
+        rows[j], rows[pivot] = rows[pivot], rows[j]
+        for i in range(n):
+            if i != j and rows[i] >> j & 1:
+                rows[i] ^= rows[j]
+    require([row & ((1 << n)-1) for row in rows] == [1 << i for i in range(n)], 'binary inverse')
+    return tuple(row >> n for row in rows)
+
+
+def extend(vectors, n):
+    columns, pivots = [], {}
+    for value in tuple(vectors) + tuple(1 << i for i in range(n)):
+        reduced = value
+        while reduced:
+            j = reduced.bit_length()-1
+            if j not in pivots:
+                pivots[j] = reduced
+                columns.append(value)
+                break
+            reduced ^= pivots[j]
+    require(len(columns) == n, 'frame basis extension')
+    return tuple(columns)
+
+
+def adjoint(circuit):
+    out = []
+    for gate in reversed(circuit):
+        if gate[0] == 'P':
+            out.append(('P', inverse(gate[1])))
+        elif gate[0] == 'D':
+            out.append(('D', gate[1], tuple(-a % 4 for a in gate[2])))
+        elif gate[0] == 'C':
+            out.append(('C', gate[1], -gate[2]))
+        elif gate[0] == 'phase':
+            out.append(('phase', -gate[1] % 4))
+        else:
+            require(gate[0] == 'X', 'adjoint gate type')
+            out.append(gate)
+    return tuple(out)
+
+
+def fourier(rank):
+    return tuple(('C', 1 << i, 1) for i in range(rank))
+
+
+def frame(vectors, n):
+    rank = len(vectors)
+    if rank == 0:
+        return ()
+    if rank == n:
+        return fourier(n)
+    matrix = transpose(extend(vectors, n))
+    columns = transpose(inverse(matrix))
+    quadratic = tuple((1 << i) ^ sum(parity(a & b) << j for j, b in enumerate(columns)) for i, a in enumerate(columns))
+    diagonal = tuple((row >> i) & 1 for i, row in enumerate(quadratic))
+    chart = (('P', matrix), ('D', quadratic, diagonal))
+    return adjoint(chart) + fourier(rank) + chart
+
+
+def monomial(record):
+    matrix = tuple(record['matrix_rows'])
+    quadratic = tuple(record['quadratic_rows'])
+    linear = tuple(record['linear_Z4'])
+    require(transpose(quadratic) == quadratic, 'quadratic symmetry')
+    require(len(matrix) == len(quadratic) == len(linear) == 9, 'monomial dimension')
+    require(all(a % 2 == (quadratic[i] >> i) & 1 for i, a in enumerate(linear)), 'diagonal parity')
+    inverse(matrix)
+    return (('D', quadratic, linear), ('P', matrix), ('X', record['offset']), ('phase', record['phase']))
+
+
+def recipe(record):
+    rank = record['rank']
+    require(isinstance(rank, int) and 0 <= rank <= 9, 'child rank')
+    if rank == 0:
+        require(record['right'] is None, 'rank zero right factor')
+        return monomial(record['left'])
+    return monomial(record['right']) + fourier(rank) + monomial(record['left'])
+
+
+def tableau(circuit, n=9):
+    rows = [(1 << i, 0, 0) for i in range(n)] + [(0, 1 << i, 0) for i in range(n)]
+    for gate in circuit:
+        tag = gate[0]
+        out = []
+        inverse_transpose = transpose(inverse(gate[1])) if tag == 'P' else None
+        for x, z, phase in rows:
+            if tag == 'P':
+                x, z = mul(gate[1], x), mul(inverse_transpose, z)
+            elif tag == 'X':
+                phase += 2 * parity(z & gate[1])
+            elif tag == 'C':
+                if parity(z & gate[1]):
+                    x ^= gate[1]
+                    phase += 3 * gate[2]
+            elif tag == 'D':
+                quadratic, linear = gate[1:]
+                phase += sum(linear[i] for i in range(n) if x >> i & 1)
+                phase += 2 * sum(parity((quadratic[i] >> (i+1)) & (x >> (i+1))) for i in range(n) if x >> i & 1)
+                z ^= mul(quadratic, x)
+            else:
+                require(tag == 'phase', 'tableau gate type')
+            out.append((x, z, phase % 4))
+        rows = out
+    return tuple(rows)
+
+
+@lru_cache(maxsize=512)
+def action(gate):
+    if gate[0] == 'P':
+        return tuple(mul(gate[1], x) for x in range(512))
+    require(gate[0] == 'D', 'state action type')
+    quadratic, linear = gate[1:]
+    return tuple((sum(linear[i] for i in range(9) if x >> i & 1)
+                  + 2 * sum(parity((quadratic[i] >> (i+1)) & (x >> (i+1))) for i in range(9) if x >> i & 1)) % 4 for x in range(512))
+
+
+def rotate(pair, power):
+    a, b = pair
+    return ((a, b), (-b, a), (-a, -b), (b, -a))[power % 4]
+
+
+def state(circuit, basis=0):
+    value, denominator = {basis: (1, 0)}, 0
+    for gate in circuit:
+        tag = gate[0]
+        result = {}
+        if tag == "C":
+            direction, sign = gate[1:]
+            for address, (a, b) in value.items():
+                for target, pair in ((address, (a - sign*b, b + sign*a)),
+                                     (address ^ direction, (a + sign*b, b - sign*a))):
+                    old = result.get(target, (0, 0))
+                    result[target] = (old[0] + pair[0], old[1] + pair[1])
+            denominator += 1
+        else:
+            for address, pair in value.items():
+                if tag == "P":
+                    target = mul(gate[1], address)
+                elif tag == "X":
+                    target = address ^ gate[1]
+                else:
+                    target = address
+                    if tag == "D":
+                        pair = rotate(pair, action(gate)[address])
+                    else:
+                        require(tag == "phase", "Unknown Gaussian state gate")
+                        pair = rotate(pair, gate[1])
+                require(target not in result, "A monomial address map is not injective")
+                result[target] = pair
+        value = {address: pair for address, pair in result.items() if pair != (0, 0)}
+    while denominator and all((a | b) % 2 == 0 for a, b in value.values()):
+        denominator -= 1
+        value = {address: (a//2, b//2) for address, (a, b) in value.items()}
+    return value, denominator
+
+
+def equal(actual, expected, label):
+    require(tableau(actual) == tableau(expected), label + ": signed Pauli action")
+    require(state(actual) == state(expected), label + ": exact global phase")
+
+
+def verify(data, certificate):
+    word = data["word"]
+    original, clean = data["programs"], data["clean_programs"]
+    source_recipes = data["recipes"]
+    residuals = data["residuals"]["recipes"]
+    native_tags = {"climb", "adapter", "residual_adapter"}
+    for orientation in ("forward", "backward"):
+        native = [event for event in clean[orientation] if event[0] in native_tags]
+        require(native == [event for event in original[orientation] if event[0] in native_tags],
+                "A retained native event changed")
+        require(len(native) == 5056, "The retained native event count changed")
+    frames = [frame(tuple(vectors), 9) for vectors in word["frames"]]
+    used = {(event[3], event[4]) for program in clean.values()
+            for event in program if event[0] == "climb"}
+    children = {(record["old"], record["new"]): record for record in source_recipes["children"]}
+    require(len(children) == len(source_recipes["children"]) and used <= set(children),
+            "The retained child recipe coverage is incomplete")
+    for old, new in sorted(used):
+        record = children[old, new]
+        require(record["rank"] == len(word["frames"][new]) - len(word["frames"][old]),
+                "A retained child has the wrong rank")
+        equal(recipe(record), adjoint(frames[old]) + frames[new], "Retained child")
+    endpoints = {(record["role"], record["port"]): record
+                 for record in source_recipes["endpoint_adapters"]}
+    require(len(endpoints) == len(source_recipes["endpoint_adapters"])
+            and set(endpoints) == {(role, p) for role in ("source", "target") for p in range(112)},
+            "The endpoint adapter coverage is incomplete")
+    for (role, port), record in endpoints.items():
+        direction = word["ports"][port]
+        conventional = (("C", direction, 1),) if role == "source" else (
+            ("C", direction, -1),) + fourier(9)
+        index = word["start"][port] if role == "source" else word["final"][112 + port]
+        require(record["frame"] == index and record["rank"] == 0, "An endpoint frame changed")
+        equal(recipe(record), adjoint(conventional) + frames[index], "Endpoint adapter")
+    used_residuals = set()
+    for program in clean.values():
+        for event in program:
+            if event[0] != "residual_adapter":
+                continue
+            _, position, register, entry, *tail = event
+            terminal = tail[0] if tail else 1
+            key = str(entry) if terminal == 1 else str(entry) + ":" + str(terminal)
+            require(key in residuals and position in ("pre", "post"), "A residual binding changed")
+            require(entry == word["start"][register] and terminal == word["final"][register],
+                    "A residual endpoint changed")
+            used_residuals.add(key)
+    for key in sorted(used_residuals):
+        record = residuals[key]
+        equal(recipe(record), adjoint(frames[record["frame"]]) + frames[record.get("terminal", 1)],
+              "Residual adapter")
+    records = certificate["input_recipes"]
+    ports = certificate["compensation_ports"]
+    require([record["port"] for record in records] == ports and len(records) == 8,
+            "The eight input recipes changed")
+    columns = 0
+    for item in records:
+        port, label = item["port"], item["label"]
+        require(label == word["ports"][port] and item["canonical_start_frame_id"] == word["start"][port],
+                "An input recipe has the wrong supplier label")
+        record = item["recipe"]
+        require(record["rank"] == 1, "An input copy has the wrong child rank")
+        for side in ("left", "right"):
+            factor = record[side]
+            require(factor["quadratic_rows"] == [0] * 9 and factor["linear_Z4"] == [0] * 9
+                    and factor["offset"] == factor["phase"] == 0,
+                    "An input factor is not an address permutation")
+        actual, expected = recipe(record), (("C", label, 1),)
+        equal(actual, expected, "Positive inherited input line")
+        for basis in range(512):
+            require(state(actual, basis) == state(expected, basis), "A complete input Gaussian column differs")
+            columns += 1
+        negative = monomial(record["right"]) + (("C", 1, -1),) + monomial(record["left"])
+        require(tableau(negative) != tableau(expected) or state(negative) != state(expected),
+                "An inverse input child was not detected")
+        phased = actual + (("phase", 1),)
+        require(tableau(phased) == tableau(actual) and state(phased) != state(expected),
+                "An input phase-only error was not detected")
+    return {"status": "PASS_EXACT_INPUT_NATIVE_FRAMES", "retained_native_events_per_orientation": 5056,
+            "retained_child_recipes": len(used), "endpoint_adapter_recipes": len(endpoints),
+            "residual_adapter_recipes": len(used_residuals), "new_input_recipes": len(records),
+            "new_exact_gaussian_columns": columns, "inverse_input_child_controls": len(records),
+            "phase_only_controls": len(records), "address_permutation_factors": 2 * len(records),
+            "method": "Independent signed Pauli and exact Gaussian interpretation; no compiler import",
+            "scope": "Positive inherited input frames before every retained adapter. No recursive hosting claim."}
diff --git a/research/p104-shared-frame-gather/inputs/predecessor-SOURCES.json b/research/p104-shared-frame-gather/inputs/predecessor-SOURCES.json
new file mode 100644
index 0000000000..a059933e0c
--- /dev/null
+++ b/research/p104-shared-frame-gather/inputs/predecessor-SOURCES.json
@@ -0,0 +1,150 @@
+{
+  "schema_version": 1,
+  "base_commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "refreshed_canonical_main": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "priority_revision_read": 8,
+  "selected_kappa": ".00310045",
+  "source_claim_kappa": ".00311024570",
+  "source_publication": "PR5718 archive-merged 2026-10-26T15:22:41Z; not selected",
+  "runtime_inputs": [
+    {
+      "path": "inputs/DATA-PHASES.json",
+      "sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
+      "git_blob": "d9a4f84211182ca200c081ea061d724c8a2cfb70",
+      "commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+      "repository_path": "research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json"
+    },
+    {
+      "path": "inputs/NOTICE5718",
+      "sha256": "c474479a870ae53ae0bcb84f656c2b222380395bbee767ee15e5f9c5b4ce7e79",
+      "git_blob": "f83cecccf587acfa313c8259c0bf6db05d60b52d",
+      "commit": "a1534ac45401c9a9c80c07aa65c89e00e33594f8",
+      "repository_path": "research/p104-two-boundary-helper-split/NOTICE"
+    },
+    {
+      "path": "inputs/clifford_exact.py",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+      "git_blob": "166841a037f069249f6f7f6dedb89248812f59e2",
+      "commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+      "repository_path": "research/p104-native-channel/vendor/clifford_exact.py"
+    },
+    {
+      "path": "inputs/controller5718.json",
+      "sha256": "a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9",
+      "git_blob": "cc539db709cf7a6c42872c4380902736ad453af3",
+      "commit": "a1534ac45401c9a9c80c07aa65c89e00e33594f8",
+      "repository_path": "research/p104-productive-controller/CONTROLLER.json"
+    },
+    {
+      "path": "inputs/native_verify.py",
+      "sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38",
+      "git_blob": "2c209dfa9cfade3bc20ac5fc9d876b433a147f8c",
+      "commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+      "repository_path": "research/p104-native-channel/vendor/native_verify.py"
+    },
+    {
+      "path": "inputs/result5718.json",
+      "sha256": "8e62f19bc795cac7712d31f955b71587e251dc94f1a02a20235e1ea740015db7",
+      "git_blob": "a83d70e025254e217c01ca413f6823937e82f3d9",
+      "commit": "a1534ac45401c9a9c80c07aa65c89e00e33594f8",
+      "repository_path": "research/p104-two-boundary-helper-split/RESULT.json"
+    },
+    {
+      "path": "inputs/word5718.json.gz",
+      "sha256": "cff7196f8b168c4ebf8227185eed98387b66ec5db0179108adfea4045775cc31",
+      "git_blob": "4db015aca34ea957dd8b3b4d7ae4fa12b89934f8",
+      "commit": "a1534ac45401c9a9c80c07aa65c89e00e33594f8",
+      "repository_path": "research/p104-two-boundary-helper-split/WORD.json.gz"
+    }
+  ],
+  "predecessors": [
+    "/Users/temp/.codex/worktrees/joint-neutral-kappa/integer-mult-bounds/research/p104-joint-neutral-search/README.md @ 1f51164a1dba5b9cb7639d3b760d2a0660931f63",
+    "/private/tmp/integer-mult-information-reuse/research/information-reuse-screen/README.md",
+    "/private/tmp/integer-mult-information-reuse/research/p104-multimeeting-repair/README.md",
+    "/tmp/luna-inforeuse-controller/cross_local_input_ledger.md",
+    "research/paid-history-quotient",
+    "research/centre-history-innovation",
+    "research/gaussian-cache-reuse"
+  ],
+  "files": [
+    {
+      "path": "research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json",
+      "usage": "provenance",
+      "git_blob": "d9a4f84211182ca200c081ea061d724c8a2cfb70",
+      "sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
+      "pin": "head:df0137efd35e11b4445734bf612bdd4c7ebc330c:research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json"
+    },
+    {
+      "path": "research/p104-two-boundary-helper-split/NOTICE",
+      "usage": "provenance",
+      "git_blob": "f83cecccf587acfa313c8259c0bf6db05d60b52d",
+      "sha256": "c474479a870ae53ae0bcb84f656c2b222380395bbee767ee15e5f9c5b4ce7e79",
+      "pin": "head:a1534ac45401c9a9c80c07aa65c89e00e33594f8:research/p104-two-boundary-helper-split/NOTICE"
+    },
+    {
+      "path": "research/p104-native-channel/vendor/clifford_exact.py",
+      "usage": "provenance",
+      "git_blob": "166841a037f069249f6f7f6dedb89248812f59e2",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+      "pin": "head:df0137efd35e11b4445734bf612bdd4c7ebc330c:research/p104-native-channel/vendor/clifford_exact.py"
+    },
+    {
+      "path": "research/p104-productive-controller/CONTROLLER.json",
+      "usage": "provenance",
+      "git_blob": "cc539db709cf7a6c42872c4380902736ad453af3",
+      "sha256": "a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9",
+      "pin": "head:a1534ac45401c9a9c80c07aa65c89e00e33594f8:research/p104-productive-controller/CONTROLLER.json"
+    },
+    {
+      "path": "research/p104-native-channel/vendor/native_verify.py",
+      "usage": "provenance",
+      "git_blob": "2c209dfa9cfade3bc20ac5fc9d876b433a147f8c",
+      "sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38",
+      "pin": "head:df0137efd35e11b4445734bf612bdd4c7ebc330c:research/p104-native-channel/vendor/native_verify.py"
+    },
+    {
+      "path": "research/p104-two-boundary-helper-split/RESULT.json",
+      "usage": "provenance",
+      "git_blob": "a83d70e025254e217c01ca413f6823937e82f3d9",
+      "sha256": "8e62f19bc795cac7712d31f955b71587e251dc94f1a02a20235e1ea740015db7",
+      "pin": "head:a1534ac45401c9a9c80c07aa65c89e00e33594f8:research/p104-two-boundary-helper-split/RESULT.json"
+    },
+    {
+      "path": "research/p104-two-boundary-helper-split/WORD.json.gz",
+      "usage": "provenance",
+      "git_blob": "4db015aca34ea957dd8b3b4d7ae4fa12b89934f8",
+      "sha256": "cff7196f8b168c4ebf8227185eed98387b66ec5db0179108adfea4045775cc31",
+      "pin": "head:a1534ac45401c9a9c80c07aa65c89e00e33594f8:research/p104-two-boundary-helper-split/WORD.json.gz"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "research/paid-history-quotient/README.md",
+      "usage": "provenance",
+      "git_blob": "f1c72c7c8e291f04272d645fea3e78f5f7ecb80a",
+      "sha256": "c219f4d143be68639c676a5ff6e3ff84651b484be6a95f1a3d538c71182fd78f"
+    },
+    {
+      "path": "research/centre-history-innovation/README.md",
+      "usage": "provenance",
+      "git_blob": "8d382b734879ab781366477132c842cf84522ec5",
+      "sha256": "c797418742fd13b85c27f2ffb06b3f8eb035bef1d2a5e7a02086cb299d853462"
+    },
+    {
+      "path": "research/gaussian-cache-reuse/README.md",
+      "usage": "provenance",
+      "git_blob": "2c5449434b3eca66107f3847c27a6350a8b2d8e1",
+      "sha256": "3e9462ff17b02ad69564d5c25b810a4ad5f2c4b95997160e8617668710003b87"
+    }
+  ]
+}
diff --git a/research/p104-shared-frame-gather/inputs/predecessor-experiment.py b/research/p104-shared-frame-gather/inputs/predecessor-experiment.py
new file mode 100644
index 0000000000..3309f0be84
--- /dev/null
+++ b/research/p104-shared-frame-gather/inputs/predecessor-experiment.py
@@ -0,0 +1,595 @@
+"""Versioned P104 paid rows at caller events 0 and 3; exact bounded reuse tests.
+
+Prepared for zxv with substantial OpenAI Codex assistance. Apache-2.0.
+The P104 word, Clifford implementation and caller retain predecessor credit.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from functools import lru_cache
+from hashlib import sha256
+from pathlib import Path
+import gzip
+import importlib.util
+import json
+import sys
+
+HERE = Path(__file__).resolve().parent
+
+def need(ok, why):
+    if not ok:
+        raise ValueError(why)
+
+def read(name):
+    raw = (HERE/name).read_bytes()
+    return json.loads(gzip.decompress(raw) if name.endswith('.gz') else raw)
+
+def encode(x):
+    if isinstance(x, Q): return str(x)
+    if isinstance(x, dict): return {str(k): encode(v) for k, v in x.items()}
+    if isinstance(x, (tuple, list)): return [encode(v) for v in x]
+    return x
+
+def canonical(x):
+    return json.dumps(encode(x), sort_keys=True, separators=(',', ':')).encode()
+
+def digest(x): return sha256(canonical(x)).hexdigest()
+
+def load(name):
+    spec = importlib.util.spec_from_file_location('cross_reuse_'+name, HERE/'inputs'/name)
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[spec.name] = module
+    spec.loader.exec_module(module)
+    return module
+
+def plus(dst, src, coefficient=Q(1)):
+    for k, v in tuple(src.items()):
+        z = dst.get(k, Q(0)) + coefficient*v
+        if z: dst[k] = z
+        else: dst.pop(k, None)
+
+def dot(row, vector): return sum(v*vector.get(k, 0) for k, v in row.items())
+
+class Basis:
+    def __init__(self): self.rows = {}; self.combinations = {}
+    def reduce(self, row, combination=None):
+        row = dict(row)
+        combination = {} if combination is None else dict(combination)
+        for p, b in sorted(self.rows.items()):
+            if p in row:
+                q = row[p]
+                plus(row, b, -q)
+                plus(combination, self.combinations[p], -q)
+        return row, combination
+    def insert(self, row, label):
+        row, combination = self.reduce(row, {label: Q(1)})
+        if not row: return False
+        p = min(row); q = row[p]
+        self.rows[p] = {k: v/q for k, v in row.items()}
+        self.combinations[p] = {k: v/q for k, v in combination.items()}
+        return True
+    def solve(self, row):
+        remainder, c = self.reduce(row)
+        return remainder, {k: -v for k, v in c.items()}
+    def separator(self, target):
+        r, _ = self.reduce(target)
+        need(bool(r), 'A separator requires a nonzero quotient')
+        j = min(r); vector = {j: Q(1)}
+        for p, row in sorted(self.rows.items(), reverse=True):
+            z = -sum(v*vector.get(k, 0) for k, v in row.items() if k != p)
+            if z: vector[p] = z
+        need(all(dot(b, vector) == 0 for b in self.rows.values()), 'Annihilate observations')
+        need(dot(target, vector) != 0, 'Separate target')
+        return vector
+
+def project(row, columns): return {k: v for k, v in row.items() if k in columns}
+
+def walk(w):
+    """Independent literal walk; every paid output retains its exact last-write epoch."""
+    n = w['public']; size = n+w['owned_fields']
+    rows = [{r: Q(1)} for r in range(size)]
+    frames = w['initial']+[None]*w['owned_fields']
+    versions = [0]*size; last_write = [-1]*size; lives = [0]*size
+    active = set(); paid = []; selected = []; source_main = []
+    main = set(w['live']) | {r+112 for r in w['live']}
+    helpers = set(range(n))-main
+    def frame_span(f):
+        out = {0}
+        for b in w['frames'][f]: out |= {x ^ b for x in tuple(out)}
+        return out
+    spans = [frame_span(f) for f in range(len(w['frames']))]
+    for index, e in enumerate(w['program']):
+        op, d = e[:2]
+        if op == 'MOVE':
+            _, _, a, b, private = e
+            need(frames[d] == a and private == (d in active), 'Exact frame and ownership at MOVE')
+            need(spans[b] < spans[a] if private else spans[a] < spans[b], 'Nested transport')
+            record = dict(event=index, role=d, input_version=versions[d], output_version=versions[d]+1,
+                          last_logical_write=last_write[d], owned_epoch=lives[d], private=private,
+                          old=a, new=b, rank=abs(len(w['frames'][b])-len(w['frames'][a])), row=dict(rows[d]))
+            paid.append(record)
+            if private:
+                donors = [(j, versions[j], last_write[j], dict(rows[j])) for j in range(size) if frames[j] == b]
+                retained = [p for p in paid[:-1] if p['new'] == b]
+                selected.append(dict(query=record, donors=donors, retained_same_call=retained))
+            if d in w['live'] and rows[d] == {d: Q(1)}:
+                source_main.append(record)
+            frames[d] = b
+        elif op == 'INIT':
+            need(d >= n and d not in active, 'Owned INIT')
+            active.add(d); lives[d] += 1; frames[d] = e[2]; rows[d] = {}
+        elif op == 'CLEAR':
+            need(d in active and frames[d] == e[2], 'Owned CLEAR')
+            active.remove(d); frames[d] = None; rows[d] = {}
+        else:
+            need(frames[d] == e[-1], 'Scalar destination chart')
+            q = Q(e[-2])
+            if op == 'SCALE': rows[d] = {k: q*v for k, v in rows[d].items()}
+            else:
+                need(frames[e[2]] == e[-1] and e[2] != d, 'Scalar source chart')
+                if op == 'COPY':
+                    need(d in active and not rows[d], 'COPY into empty owned field')
+                    rows[d] = {}
+                else: need(op == 'ADD', 'Scalar opcode')
+                plus(rows[d], rows[e[2]], q)
+            last_write[d] = index
+        if op in ('INIT', 'CLEAR'): last_write[d] = index
+        versions[d] += 1
+    need(not active and frames[:n] == w['final'], 'Complete owned and public exits')
+    for r, row in enumerate(rows):
+        want = {r: Q(1)} if r < n else {}
+        if r-112 in w['live']: want[r-112] = Q(1)
+        need(row == want, 'Every arbitrary source/owned endpoint '+str(r))
+    return paid, selected, source_main, helpers
+
+def frame_quotients(selected, helpers):
+    receipts = []
+    for item in selected:
+        query = item['query']; basis = Basis()
+        observations = [(f'current:{r}:v{v}', row) for r, v, _, row in item['donors']]
+        observations += [(f"retained:{r['event']}:v{r['output_version']}", r['row']) for r in item['retained_same_call']]
+        for label, row in observations: basis.insert(project(row, helpers), label)
+        target = project(query['row'], helpers)
+        witness = basis.separator(target)
+        need(all(dot(project(row, helpers), witness) == 0 for _, row in observations), 'All available rows vanish')
+        response = dot(target, witness)
+        receipts.append(dict(event=query['event'], role=query['role'], rank=query['rank'],
+            old=query['old'], new=query['new'], input_version=query['input_version'],
+            last_logical_write=query['last_logical_write'], owned_epoch=query['owned_epoch'],
+            target_helper_support=len(target), current_donors=[(r,v,last) for r,v,last,_ in item['donors']],
+            prior_same_call_outputs=[p['event'] for p in item['retained_same_call']],
+            helper_observation_rank=len(basis.rows), helper_quotient_dimension=1,
+            witness=witness, target_response=response,
+            native_response='S_stage1 Phi_new ('+str(response)+' phi)',
+            status='FAIL_EVEN_WITH_ALL_FIRST_CALL_ANSWERS_AND_ARBITRARY_OUTPUT_FRAME_ACTIONS'))
+    return receipts
+
+def cached_scalar_relaxation(paid, source_row, live):
+    basis = Basis()
+    for p in paid: basis.insert(p['row'], p['event'])
+    need(len(basis.rows) == 744, 'Full first-call scalar history has rank 744')
+    # Erase every native operator and owner distinction, but keep main wrapper signs.
+    # This aliases h1 to h0 deliberately: it is a negative version-control fixture.
+    aliased = {}
+    for r, q in source_row.items():
+        if r in live:
+            plus(aliased, {r: Q(1), r+112: Q(1)}, q)
+        elif r-112 in live: plus(aliased, {r-112: Q(1)}, -q)
+        else: plus(aliased, {r: Q(1)}, q)
+    remainder, c = basis.solve(aliased)
+    need(not remainder, 'Deliberately stale helper alias produces false feasibility')
+    combined = {}
+    by_event = {p['event']: p for p in paid}
+    for e, q in c.items(): plus(combined, by_event[e]['row'], q)
+    need(combined == aliased, 'Replay the complete stale-alias combination')
+    # Genuine, main-only relaxed reconstruction, without aliasing helpers.
+    r, main_c = basis.solve({0: Q(1), 112: Q(1)})
+    need(not r, 'Earlier scalar history reconstructs a main-only target')
+    return dict(first_history_rank=744, independent_public_fields=744,
+        native_and_owner_erasure=True, stale_helper_alias_combination=c,
+        stale_alias_status='FEASIBLE_ONLY_AFTER_ILLEGAL_H1_EQUALS_H0_IDENTIFICATION',
+        main_only_X0_plus_Y0_combination=main_c,
+        main_only_status='FEASIBLE_SCALAR_RELAXATION; NATIVE_ALIGNMENT_AND_COST_REQUIRED')
+
+def caller_bindings(w, controller):
+    """Complete machine-readable substitution into the original raw caller columns.
+
+    The phase-polynomial calculation is the retained native-contact method,
+    independently implemented here from its definition in Fourier coordinates.
+    """
+    phases = {p['q']:p for p in read('inputs/DATA-PHASES.json')['recipes']}
+    n=17; axes=tuple(1<<i for i in range(n))
+    def act(columns,x):
+        out=0
+        for i,b in enumerate(columns):
+            if x>>i&1: out^=b
+        return out
+    def polynomial(terms):
+        linear=[0]*n; quadratic=[0]*n
+        for axis,value in terms:
+            support=[i for i in range(n) if axis>>i&1]
+            for i in support:
+                linear[i]=(linear[i]+value)%4
+                for j in support:
+                    if i<j and value%2: quadratic[i]^=1<<j
+        return tuple(linear),tuple(quadratic)
+    def contact(old,new,route):
+        moved=[(act(route,a),c) for a,c in old]
+        a,b=polynomial(moved),polynomial(new)
+        need(a[1]==b[1], 'Native entrance quadratic phase')
+        delta=[(y-x)%4 for x,y in zip(a[0],b[0])]
+        need(set(delta)<={0,2}, 'Native entrance is an ordinary translation')
+        shift=sum((v//2)<<i for i,v in enumerate(delta))
+        need(polynomial(moved+[(shift,2)])==b, 'Exact entrance polynomial, including phase')
+        return shift
+    live=set(w['live']); ports=dict(zip(w['live'],w['ports']))
+    right=[[Q(x) for x in row] for row in controller['boundaries']['right']]
+    routes={}; contacts={}; bindings=[{},{}]
+    for r,q in ports.items():
+        basis=phases[q]['idle_children'][0]['orthogonal_axes'][:8]
+        need(all((a&b).bit_count()%2==int(i==j) for i,a in enumerate(basis) for j,b in enumerate(basis)), 'Pinned orthonormal route basis')
+        need(all((a&q).bit_count()%2==0 for a in basis), 'Route basis fixes port')
+        swaps=[b^axes[9+i] for i,b in enumerate(basis)]
+        route=tuple(e^act(swaps,sum(((e&v).bit_count()%2)<<i for i,v in enumerate(swaps))) for e in axes)
+        need(act(route,q)==q and all(act(route,route[i])==axes[i] for i in range(n)), 'Exact route fixes q and is involutive')
+        raw_x=[(a,1) for a in axes[:9]]
+        raw_y=raw_x+[(q,-1)]
+        spectator=[(a,1) for a in axes[9:17]]
+        sx=contact(raw_x,spectator+[(q,1)],route)
+        sy=contact(raw_y,spectator,route)
+        routes[q]=route; contacts[q]=dict(entry_X_translation=sx,entry_Y_translation=sy)
+        # These two exact identities justify the expanded stage-1 input rows.
+        # E_X C_q^-1 H9 = S1 P_rho and E_Y C_q^-2 H9 = S1 C_q^-1 P_rho.
+        moved_x=[(act(route,a),c) for a,c in raw_x+[(q,-1)]]+[(sx,2)]
+        moved_y=[(act(route,a),c) for a,c in raw_x+[(q,-2)]]+[(sy,2)]
+        need(polynomial(moved_x)==polynomial(spectator), 'Stage-1 source wrapper identity')
+        need(polynomial(moved_y)==polynomial(spectator+[(q,-1)]), 'Stage-1 target wrapper identity')
+        def terms(row,owner,ops,sign=Q(1)):
+            return [dict(source=dict(kind='original_raw_caller',port=r,column=j,owner=owner),coefficient=sign*x,operations=ops)
+                    for j,x in enumerate(row) if x]
+        bindings[0][r]=terms(right[0],'I',[['C',q,-1]])
+        bindings[0][r+112]=terms(right[1],'I',[])
+        bindings[1][r]=terms(right[0],f'rho:{q}',[['C',q,-2],['P_route',q]])+terms(right[1],f'rho:{q}',[['C',q,-1],['P_route',q]])
+        bindings[1][r+112]=terms(right[0],f'rho:{q}',[['C',q,-2],['P_route',q]],Q(-1))
+    for stage in range(2):
+        for r in range(w['public']):
+            if r not in live and r-112 not in live:
+                bindings[stage][r]=[dict(source=dict(kind='independent_helper',invocation=stage,role=r,owner='I',coordinate='physical_raw'),coefficient=Q(1),
+                    operations=[['S_adjoint',stage],['Phi_adjoint',w['initial'][r]]])]
+    need(set(bindings[0])==set(bindings[1])==set(range(w['public'])), 'Every original/helper input binding')
+    h0={canonical(t['source']) for ts in bindings[0].values() for t in ts if t['source']['kind']=='independent_helper'}
+    h1={canonical(t['source']) for ts in bindings[1].values() for t in ts if t['source']['kind']=='independent_helper'}
+    need(not h0&h1 and len(h0)==len(h1)==536, 'Independent invocation-specific helper columns')
+    return dict(dimension=n,routes=routes,contacts=contacts,input_substitutions=bindings,
+                operation_order='Listed operations act from left to right, then the query prefix S_stage Phi_frame acts.',
+                spectator_axes=[[],list(range(9,17))],phase_checks=4*len(routes))
+
+def two_cache_screen(w,paid,selected,helpers,budget=6,minimum_rank=0):
+    """At most two actual prefix answers; exact helper quotient and native tariffs.
+
+    Grants all main information and all current output-frame observations free.
+    Only a strictly cheaper native bill is searched (rank sum at most six).
+    """
+    C=load('clifford_exact.py')
+    @lru_cache(None)
+    def chart(f):return C.canonical_frame(9,tuple(w['frames'][f]))
+    @lru_cache(None)
+    def tariff(a,b):
+        op=chart(a).adjoint().then(chart(b))
+        return C.rank([x for x,z,p in op.tab[9:]])
+    receipts=[]
+    def normalize(row):
+        if not row:return ()
+        q=row[min(row)]
+        return tuple((k,v/q) for k,v in sorted(row.items()))
+    for selected_item in selected:
+        target=selected_item['query'];ordinary=Basis()
+        free_frames=[f for f in range(len(w['frames'])) if tariff(f,target['new'])==0]
+        need(free_frames==[target['new']], 'Every rank-zero canonical alignment is the literal destination frame')
+        for r,ver,last,row in selected_item['donors']:
+            ordinary.insert(project(row,helpers),('current',r,ver))
+        for p in selected_item['retained_same_call']:
+            ordinary.insert(project(p['row'],helpers),('retained',p['event']))
+        t,_=ordinary.reduce(project(target['row'],helpers))
+        need(bool(t),'Nonzero fresh-helper quotient')
+        anchor=min(t);groups={};survivors=[];tested=0;single=0;eligible=0
+        for p in paid:
+            if p['event']>=target['event']:break
+            rank=tariff(p['new'],target['new'])
+            if not minimum_rank<=rank<=budget:continue
+            eligible+=1
+            u,_=ordinary.reduce(project(p['row'],helpers))
+            if not u:continue
+            alpha=u.get(anchor,Q(0))/t[anchor]
+            residual=dict(u);plus(residual,t,-alpha)
+            if not residual:
+                if alpha:
+                    single+=1
+                    survivors.append(dict(events=[p['event']],ranks=[rank],coefficients=[1/alpha]))
+                continue
+            pivot=min(residual);q=residual[pivot];key=normalize(residual)
+            group=groups.setdefault(key,{})
+            for other_rank,examples in group.items():
+                if rank+other_rank>budget:continue
+                for v in examples:
+                    tested+=1
+                    determinant=alpha*v['q']-v['alpha']*q
+                    if not determinant:continue
+                    a=v['q']/determinant;b=-q/determinant
+                    combined={};plus(combined,u,a);plus(combined,v['row'],b)
+                    need(combined==t,'Exact two-cache quotient witness')
+                    survivors.append(dict(events=[p['event'],v['event']],ranks=[rank,other_rank],coefficients=[a,b]))
+            # Two different projective target/residual ratios per tariff suffice:
+            # any third has either one of those ratios or differs from one of them.
+            examples=group.setdefault(rank,[])
+            ratio=alpha/q
+            if len(examples)<2 and all(e['alpha']/e['q']!=ratio for e in examples):
+                examples.append(dict(event=p['event'],row=u,alpha=alpha,q=q))
+        # Deduplicate and bind all survivors back to their full original columns.
+        distinct={canonical(s):s for s in survivors}
+        full=[];by_event={p['event']:p for p in paid}
+        for s in distinct.values():
+            row={}
+            for e,q in zip(s['events'],s['coefficients']):plus(row,by_event[e]['row'],q)
+            discrepancy=dict(target['row']);plus(discrepancy,row,-1)
+            helper_remainder,_=ordinary.reduce(project(discrepancy,helpers))
+            need(not helper_remainder,'Survivor full-row helper quotient')
+            full.append(dict(**s,full_row_discrepancy=discrepancy,
+                nested_deliveries=[set_span(w['frames'][target['new']])<set_span(w['frames'][by_event[e]['new']]) for e in s['events']]))
+        receipts.append(dict(target_event=target['event'],all_prefix_paid=sum(p['event']<target['event'] for p in paid),
+            eligible_native_rank_at_most_six=eligible,projective_residual_groups=len(groups),
+            tested_group_pairs=tested,single_cache_hits=single,survivors=full,
+            bounded_domain='Zero, one or two retained same-invocation paid answers, plus every current/historical destination-frame observation. All main information is granted. Total alignment rank <=6.'))
+    return receipts
+
+def complete_separate_delivery_screen(w,paid,selected,helpers):
+    """Cover every subset with total alignment rank <=6, without enumerating subsets.
+
+    At most two rank-one prefix answers exist. Condition on their inclusion.
+    With one/two included only two rank>=2 answers can fit. With none, the
+    only untested size is three rank-two answers, handled by a support bound.
+    """
+    from copy import deepcopy
+    from itertools import combinations
+    C=load('clifford_exact.py')
+    @lru_cache(None)
+    def chart(f):return C.canonical_frame(9,tuple(w['frames'][f]))
+    @lru_cache(None)
+    def tariff(a,b):return C.rank([x for x,z,p in chart(a).adjoint().then(chart(b)).tab[9:]])
+    receipts=[]
+    for item in selected:
+        query=item['query'];prefix=[p for p in paid if p['event']<query['event']]
+        rank_one=[p for p in prefix if tariff(p['new'],query['new'])==1]
+        need(len(rank_one)<=2,'At most two rank-one historical answers in this source domain')
+        ordinary=Basis()
+        for r,v,last,row in item['donors']:ordinary.insert(project(row,helpers),('current',r,v))
+        for p in item['retained_same_call']:ordinary.insert(project(p['row'],helpers),('retained',p['event']))
+        rank_two=[p for p in prefix if tariff(p['new'],query['new'])==2]
+        basis=Basis();reduced=[]
+        for p in rank_two:
+            row,_=ordinary.reduce(project(p['row'],helpers))
+            reduced.append((p['event'],row));basis.insert(row,p['event'])
+        target,_=ordinary.reduce(project(query['row'],helpers))
+        remainder,target_coordinates=basis.solve(target)
+        if remainder:
+            witness=basis.separator(target)
+            three=dict(status='OUTSIDE_THE_ENTIRE_RANK_TWO_SPAN',rank_two_rows=len(rank_two),basis_rank=len(basis.rows),witness=witness,target_response=dot(target,witness))
+        else:
+            coordinates={}
+            for event,row in reduced:
+                rem,co=basis.solve(row);need(not rem,'Each rank-two row is in its basis')
+                coordinates[event]=co
+            dense={event:co for event,co in coordinates.items() if len(co)>3}
+            ignored=set().union(*(set(co) for co in dense.values())) if dense else set()
+            target_support=set(target_coordinates)-ignored
+            widths={event:len(set(co)-ignored) for event,co in coordinates.items()}
+            maximum=max(widths.values(),default=0)
+            need(len(target_support)>3*maximum,'At least four rank-two observations needed by exact coordinate support')
+            three=dict(status='THREE_RANK_TWO_ROWS_CANNOT_COVER_TARGET_SUPPORT',rank_two_rows=len(rank_two),basis_rank=len(basis.rows),
+                target_coordinates=target_coordinates,dense_rows_granted_free=dense,ignored_basis_coordinates=sorted(ignored),
+                target_uncovered_coordinates=sorted(target_support),maximum_uncovered_support_of_one_answer=maximum,
+                all_rank_two_coordinate_maps=coordinates)
+        conditioned=[]
+        for count in range(1,len(rank_one)+1):
+            for subset in combinations(rank_one,count):
+                augmented=deepcopy(item)
+                for p in subset:
+                    augmented['donors'].append((100000+p['event'],p['output_version'],p['event'],p['row']))
+                check=two_cache_screen(w,paid,[augmented],helpers,budget=6-count,minimum_rank=2)[0]
+                conditioned.append(dict(paid_rank_one_answers=[p['event'] for p in subset],remaining_alignment_rank_budget=6-count,screen=check))
+        need(all(not x['screen']['survivors'] for x in conditioned),'No conditioned two-cache survivor')
+        receipts.append(dict(target_event=query['event'],rank_one_answers=[p['event'] for p in rank_one],three_rank_two_certificate=three,
+            conditioned_screens=conditioned,status='NO_STRICT_SAVING_FOR_ANY_SEPARATELY_ALIGNED_CACHE_SUBSET',
+            proof='For 0<t<1, sum r_i^t >= (sum r_i)^t. Beating H7 needs sum r_i<=6. Zero-rank observations are already in the ordinary quotient. There are at most two rank-one answers. With zero rank-one answers, at most three rank-two-or-larger jobs fit; the only three-job profile is (2,2,2). With k=1 or 2 selected rank-one answers, at most two remaining rank>=2 jobs fit. All cases are checked.'))
+    return receipts
+
+def set_span(vectors):
+    out={0}
+    for b in vectors:out|={x^b for x in tuple(out)}
+    return out
+
+def native_tests(w, main_candidates, all_paid):
+    C = load('clifford_exact.py'); V = load('native_verify.py')
+    phases = {p['q']: p for p in read('inputs/DATA-PHASES.json')['recipes']}
+    n = 17; axes = tuple(1 << i for i in range(n))
+    @lru_cache(None)
+    def chart17(f): return C.canonical_frame(n, tuple(w['frames'][f]))
+    @lru_cache(None)
+    def chart9(f): return C.canonical_frame(9, tuple(w['frames'][f]))
+    old_frames = sorted({p['new'] for p in all_paid})
+    # An adjacent candidate that does not use any fresh helper columns.
+    candidate = next(p for p in main_candidates if p['event'] == 9733)
+    need(candidate['role'] == 0 and candidate['row'] == {0: Q(1)} and candidate['rank'] == 1,
+         'Named pure-main paid query')
+    q = w['frames'][w['initial'][0]][0]
+    basis = phases[q]['idle_children'][0]['orthogonal_axes'][:8]
+    swaps = [b ^ axes[9+i] for i, b in enumerate(basis)]
+    def act(columns, x):
+        out = 0
+        for i, b in enumerate(columns):
+            if x >> i & 1: out ^= b
+        return out
+    route = tuple(e ^ act(swaps, sum(((e&v).bit_count()%2) << i for i,v in enumerate(swaps))) for e in axes)
+    need(all(act(route, route[i]) == axes[i] for i in range(n)), 'Actual involutive stage-1 route')
+    P = C.perm(n, C.transpose(route, n))
+    spectator = C.fourier(n, range(9,17))
+    desired = C.cgate(n,q,-1).then(P).then(spectator).then(chart17(candidate['new']))
+    ranks = {}
+    for f in old_frames:
+        relative = chart17(f).adjoint().then(desired)
+        need(V.tableau(relative.circuit,n=17)==relative.tab,'Independent 17-axis cache-alignment tableau')
+        ranks[f] = C.rank([x for x,z,p in relative.tab[n:]])
+    need(min(ranks.values()) == 1 and 0 not in ranks.values(), 'Every granted historical cache needs a paid native alignment')
+    # The actual centre's native map is checked independently, including its phase.
+    native = []
+    for p in all_paid:
+        if not p['private']: continue
+        a,b = p['old'],p['new']
+        recipe, compiled = C.lower_child(chart9(a).adjoint().then(chart9(b)))
+        V.equal(V.recipe(recipe), V.adjoint(V.frame(tuple(w['frames'][a]),9))+V.frame(tuple(w['frames'][b]),9),
+                'Exact centre rank and global phase')
+        need(recipe['rank'] == 7, 'All eight selected centres have rank seven')
+        native.append(dict(event=p['event'],old=a,new=b,recipe=recipe))
+    first = native[0]
+    cost_tableaus=0
+    for p in native:
+        b=p['new']
+        for f in old_frames:
+            relative=chart9(f).adjoint().then(chart9(b))
+            expected=V.adjoint(V.frame(tuple(w['frames'][f]),9))+V.frame(tuple(w['frames'][b]),9)
+            need(V.tableau(expected)==relative.tab,'Independent 9-axis cache-cost tableau')
+            cost_tableaus+=1
+    actual = V.frame(tuple(w['frames'][first['old']]),9)+V.recipe(first['recipe'])
+    want = V.frame(tuple(w['frames'][first['new']]),9)
+    for column in range(512):
+        need(V.state(actual,column) == V.state(want,column), 'All 512 first-centre native columns')
+    from copy import deepcopy
+    wrong=deepcopy(first['recipe']);wrong['left']['phase']=(wrong['left']['phase']+1)%4
+    try:V.equal(V.recipe(wrong),V.recipe(first['recipe']),'Deliberately wrong native global phase')
+    except ValueError:wrong_phase=True
+    else:raise ValueError('Wrong native global phase accepted')
+    return dict(main_query=dict(event=9733,role=0,old=candidate['old'],new=candidate['new'],q=q,route=route,
+                old_output_frames=len(old_frames),alignment_ranks=ranks,minimum_alignment_rank=1,
+                best_frames=[f for f,r in ranks.items() if r==1],
+                scope='One-snapshot or typed-cache-tree delivery, granting the entire desired logical X0+Y0 row in every old paid-output frame'),
+                centres=native,first_centre_native_columns=512,wrong_global_phase_rejected=wrong_phase,
+                independent_cache_cost_tableaus=cost_tableaus,independent_17_axis_alignment_tableaus=len(old_frames))
+
+def invoice(w, controller):
+    paid = Counter(abs(len(w['frames'][e[2]])-len(w['frames'][e[3]])) for e in w['program'] if e[0]=='MOVE')
+    main = set(w['live']) | {r+112 for r in w['live']}
+    targets = Counter(len(w['frames'][w['final'][r]])-len(w['frames'][w['initial'][r]]) for r in range(w['public']) if r not in main)
+    whole_paid = Counter(); whole_targets = Counter()
+    for e in controller['physical_events']:
+        if e[0]=='LOCAL': whole_paid.update(paid); whole_targets.update(targets)
+        elif e[0]=='QUERY_SHEAR': whole_paid[1]+=104
+        elif e[0]=='IDLE': whole_paid[8*len(e[2])]+=208
+    for term in controller['diagonal']: whole_targets[1+8*sum(term[1:])]+=104
+    source = read('inputs/result5718.json')['complete_cost']
+    need({int(k):v for k,v in source['paid'].items()} == dict(whole_paid), 'Complete source paid invoice')
+    need({int(k):v for k,v in source['targets'].items()} == dict(whole_targets), 'Every independent target')
+    return dict(local_paid=paid,local_targets=targets,whole_paid=whole_paid,whole_targets=whole_targets,
+                paid_jobs=sum(whole_paid.values()),productive_fields=sum(whole_targets.values()),
+                realized_delta_paid={},realized_delta_targets={},kappa_status='NO_NEW_KAPPA',
+                unrealized_centre_allowance={'paid_delta':{7:-1},'target_delta':{}},
+                adjacent_main_best_cache_tree={'removed':{1:1},'necessary_alignment_at_least':{1:1},'best_recursive_delta':0},
+                source_reported_ordinary=source['nonrecursive'],
+                new_retention_reads_routes_restoration_cleanup='Unknown; no legal deletion emitted. Not charged at zero.')
+
+def controls(w, paid, selected, receipts):
+    out = {}
+    first = selected[0]; target = first['query']['row']; witness = receipts[0]['witness']
+    need(dot(target,witness) == Q(1,2), 'Named fresh-helper witness')
+    need(witness == {173:Q(1)}, 'Exactly call-3 helper173, independent of call-0 helper173')
+    # Genuine same-frame reuse already implemented by the source at 13610..13614.
+    e = w['program'][13610]
+    need(e == ['ADD',703,744,'1',593], 'Actual positive control read')
+    old = first['donors'][0][3]
+    new = dict(old); plus(new,target)
+    reconstructed = dict(new); plus(reconstructed,old,-1)
+    need(reconstructed == target, 'Two exact versioned values reconstruct the earlier answer')
+    out['positive_two_version_reconstruction'] = 'PASS: (703 after13610)-(703 before13610) equals answer744@13609'
+    # It is not legal to use that future output to eliminate its own producer.
+    try: need(13610 < 13609, 'A future ADD result is unavailable before the paid producer')
+    except ValueError as error: out['future_read_rejected'] = str(error)
+    else: raise ValueError('Illegal future read accepted')
+    old_basis = Basis()
+    for p in paid: old_basis.insert(project(p['row'], set(witness)), p['event'])
+    need(bool(old_basis.rows), 'Aliasing invocation helper columns would see the old helper')
+    out['stale_invocation_alias_detected'] = 'PASS: fresh H1:173 has no column in any event-0 answer; H0:173 does'
+    out['arbitrary_helper_not_zero'] = 'PASS: target changes by one half of a nonzero native field'
+    # A constructive two-paid-answer fixture checks discovery, not only rejection.
+    # Choose two independent actual native-rank-two prefix rows and synthesize
+    # their sum as an explicitly labelled fixture target at the first cut.
+    from copy import deepcopy
+    C=load('clifford_exact.py');dest=C.canonical_frame(9,tuple(w['frames'][593]))
+    main=set(w['live'])|{r+112 for r in w['live']};helpers=set(range(w['public']))-main
+    basis=Basis();choices=[];rank_cache={}
+    for p in paid:
+        if p['event']>=13609:break
+        f=p['new']
+        if f not in rank_cache:
+            relative=C.canonical_frame(9,tuple(w['frames'][f])).adjoint().then(dest)
+            rank_cache[f]=C.rank([x for x,z,ph in relative.tab[9:]])
+        if rank_cache[f]==2 and basis.insert(project(p['row'],helpers),p['event']):choices.append(p)
+        if len(choices)==2:break
+    need(len(choices)==2,'Two actual independent rank-two paid observations for positive fixture')
+    fixture=deepcopy(first);fixture['query']['row']=dict(choices[0]['row']);plus(fixture['query']['row'],choices[1]['row'])
+    positive=two_cache_screen(w,paid,[fixture],helpers)[0]
+    need(bool(positive['survivors']),'Two-cache generator discovers the constructive positive')
+    out['two_cache_discovery_positive']=dict(source_events=[p['event'] for p in choices],target='Synthetic sum of two actual prefix rows, not a changed P104 implementation',discovered=positive['survivors'][0])
+    # Incorrect source-frame tagging must not be admitted as an ordinary read.
+    try: need(first['query']['old'] == first['query']['new'], 'Nonzero native rank forbids a free frame retag')
+    except ValueError as error: out['unpaid_retag_rejected'] = str(error)
+    else: raise ValueError('Unpaid retag accepted')
+    return out
+
+def run():
+    pins = read('SOURCES.json')
+    for p in pins['runtime_inputs']:
+        need(sha256((HERE/p['path']).read_bytes()).hexdigest() == p['sha256'], 'Pinned input '+p['path'])
+    w=read('inputs/word5718.json.gz'); c=read('inputs/controller5718.json')
+    need(c['physical_events'][:4] == [['LOCAL',0,0,False],['Q_POWER',0,-2],['SIGNED_SWAP',0,1],['LOCAL',1,0,False]], 'Literal adjacent caller events 0 to 3')
+    paid, selected, main, helpers = walk(w)
+    need(len(paid)==3426 and len(selected)==8, 'Full paid-map inventory and bounded centre domain')
+    quotients = frame_quotients(selected, helpers)
+    relaxation = cached_scalar_relaxation(paid, selected[0]['query']['row'], set(w['live']))
+    native = native_tests(w,main,paid)
+    bindings = caller_bindings(w,c)
+    two_cache = two_cache_screen(w,paid,selected,helpers)
+    complete_cache = complete_separate_delivery_screen(w,paid,selected,helpers)
+    result = dict(status='PASS_VERSIONED_CROSS_CALL_SCOPED_HANDOFF_NO_NEW_KAPPA',
+        mapped_invocations=[dict(caller_event=0,stage=0,pair=0,query_maps=3426),dict(caller_event=3,stage=1,pair=0,query_maps=3426)],
+        public_columns_per_call=w['public'],owned_labels=w['owned_fields'],independent_helpers_per_call=len(helpers),
+        output_frame_quotients=quotients,scalar_relaxation=relaxation,native=native,
+        caller_binding_phase_checks=bindings['phase_checks'],
+        two_cache_favorable_screen=two_cache,
+        complete_separate_delivery_screen=complete_cache,
+        complete_invoice=invoice(w,c),controls=controls(w,paid,selected,quotients))
+    maps=dict(schema='versioned-p104-native-query-map/1',word_frames=w['frames'],word_initial=w['initial'],word_final=w['final'],live=w['live'],ports=w['ports'],
+        right_boundary=c['boundaries']['right'],queries=paid,
+        invocations=[dict(caller_event=0,stage=0,helper_namespace='H0',spectator_axes=[]),
+                     dict(caller_event=3,stage=1,helper_namespace='H1',spectator_axes=list(range(9,17)))],
+        native_input_bindings=bindings,
+        exact_native_expansion=dict(
+            convention='Composition read right to left; coefficients outside tensor powers. All maps act on the same original raw caller columns.',
+            original_columns='A_j(q,g), j=0..31, and independent normalized H0(r,g),H1(r,g); j indexes the full literal right boundary.',
+            stage0_main='For each live r with port q: coefficient c_r gives Phi_f C_q^-1 times right[0,j]; c_(r+112) gives Phi_f times right[1,j], at owner g.',
+            stage1_main='For each live r with port q: (c_r-c_(r+112))*right[0,j] times S1 Phi_f P_rho(q) C_q^-2, plus c_r*right[1,j] times S1 Phi_f P_rho(q) C_q^-1; source owner rho(q) at common owner I.',
+            helpers='For normalized h_s(r)=(S_s Phi_initial(r))^-1 H_s_raw(r), contributions are c_r Phi_f h0(r,g) at stage0 and c_r S1 Phi_f h1(r,I) at stage1. Machine input substitutions include this exact raw-input normalization. H0_raw and H1_raw are distinct independent columns even when r agrees.',
+            frames='Phi_f is the exact canonical nine-axis frame in inputs/clifford_exact.py, embedded in axes0..8; S1 is the product C_e9..C_e16.',
+            routes='rho(q) swaps the eight pinned orthonormal q-perpendicular DATA-PHASES axes with fresh axes9..16, fixes q, and is involutive.',
+            availability='Call-0 paid outputs exist before caller event3; retaining them is granted favorably. Call-3 query output e exists only after local event e. Owned INIT/CLEAR epochs and last logical writes are explicit in every query.',
+            input_output='Before MOVE: replace Phi_f by Phi_old; after MOVE use Phi_new. The logical coefficient row is unchanged by MOVE.'),
+        selected_current_frame_rows=[dict(event=x['query']['event'],donors=x['donors'],retained=x['retained_same_call']) for x in selected])
+    return encode(result), encode(maps)
+
+if __name__ == '__main__':
+    result,maps=run()
+    write='--write' in sys.argv
+    for name,x in [('results/RESULT.json',result),('results/QUERY-MAPS.json.gz',maps)]:
+        p=HERE/name
+        if write:
+            raw=(json.dumps(x,sort_keys=True,indent=2)+'\n').encode()
+            p.write_bytes(gzip.compress(raw,mtime=0) if name.endswith('.gz') else raw)
+        else: need(read(name)==x,'Full deterministic receipt '+name)
+    print(result['status'])
diff --git a/research/p104-shared-frame-gather/inputs/result5718.json b/research/p104-shared-frame-gather/inputs/result5718.json
new file mode 100644
index 0000000000..a83d70e025
--- /dev/null
+++ b/research/p104-shared-frame-gather/inputs/result5718.json
@@ -0,0 +1,366 @@
+{
+  "complete_cost": {
+    "compiler": {
+      "H": 80511200000000000,
+      "N": "2^80511200000000000",
+      "eta": {
+        "lower": "0.000000130293974939177582367425",
+        "upper": "0.000000130293974939177582367426"
+      },
+      "gap": "1/1400000000000",
+      "integer_test_exponent": 57508,
+      "logarithm_power": 1024,
+      "theta": "63/64",
+      "uniform_saving": "2177171993/700000000000",
+      "word_materialized": false
+    },
+    "full_forward_calls": 18,
+    "gain": "27337/25000000000",
+    "kappa": "31102457/10000000000",
+    "kappa_decimal": "0.00311024570",
+    "margin_at_previous": {
+      "lower": "0.265423369162460429568591881457",
+      "upper": "0.265423369162460429568591881458"
+    },
+    "negative_controls": {
+      "old_BIT": "Rejected by outer inequalities",
+      "old_cutoff": "Rejected by completion-tail ratio greater than one"
+    },
+    "nonrecursive": {
+      "fixed_scalar_growth_bound": "At most 2^(16*26000) before native factors; a declared constant, not an attained norm.",
+      "local_paid_rank_mass": 5987,
+      "local_routes": 5166000,
+      "local_scalar": 221474,
+      "peak_owned_fields": 2,
+      "raw_budget": 1000000000,
+      "raw_scalar_upper": 184647552,
+      "tensor_native_growth_bound": "Each rank-r Gaussian has row l1 norm at most 2^(rf); use local_paid_rank_mass for the whole word."
+    },
+    "ordinary_BIT": {
+      "admitted": "17/5000",
+      "after_cover": "69/20000",
+      "atom": "1/250",
+      "before_cover": "3459/1000000",
+      "ordinary_saving": "8590884599/2500000000000",
+      "prime_rule": "odd q > 2^26*N^5 avoiding all fixed denominators and rank witnesses"
+    },
+    "outer": {
+      "functional_checks": {
+        "actual_new_BIT_stock_sublinear": "348599999999253/350000000000000",
+        "fixed_halo_explicit_metadata_branch": "26950000000001/350000000000",
+        "new_BIT_row_stock_sublinear_in_active_group": "1395799999997009/1400000000000000",
+        "new_semantic_group_guard": "3/1400000000000"
+      },
+      "margins": {
+        "boundary_algebraic_matrix": "6096081577595645656017/1960000000000000000000000",
+        "fused_gaussian": "6096081577586936968051/1960000000000000000000000",
+        "global_coordinate_router": "272146499/87500000000",
+        "group_bit_router": "381005098686683560503/122500000000000000000000",
+        "group_scans": "699999999999/700000000000",
+        "main_transform": "6096081576195645656017/1960000000000000000000000",
+        "microbox_face_corrections": "26950000000001/350000000000",
+        "polynomial_products": "1399999999999/1400000000000",
+        "sparse_union_movement": "802346081577595645656017/1960000000000000000000000"
+      },
+      "parameters": {
+        "a": "272146499/87500000000",
+        "b": "870868797/280000000000",
+        "beta": "1399999999997/1400000000000",
+        "bit_saving": "17/5000",
+        "cap_xi": "1782346081576195645656017/1960000000000000000000000",
+        "complex_uniform_saving": "2177171993/700000000000",
+        "d": "1/1400000000000",
+        "epsilon": "1399999999999/1400000000000",
+        "kappa": "31102457/10000000000",
+        "lambda_": "398755901719/400000000000",
+        "lambda_prime": "1395645656017/1400000000000",
+        "layout": "1/1400000000000",
+        "q": "4354343983/1400000000000",
+        "r0": "1/14000000000000",
+        "stop": "1/1400000000000",
+        "tested_saving": "622049141/200000000000"
+      },
+      "strict_checks": {
+        "actual_bit_above_effective": "25353501/87500000000",
+        "actual_logarithmic_gamma_guard": "1/1400000000000",
+        "actual_logarithmic_group_guard": "177653918425904354343983/980000000000000000000000",
+        "alpha_logarithm_absorption": "1/14000000000000",
+        "boundary_algebraic_matrix_above_kappa": "5595645656017/1960000000000000000000000",
+        "cap_above_band": "802346081576195645656017/1960000000000000000000000",
+        "cap_below_cell": "177653918422404354343983/1960000000000000000000000",
+        "cap_below_phase_cell_power": "2137653918423804354343983/1960000000000000000000000",
+        "complex_above_effective": "1/1400000000000",
+        "complex_below_one_over_32": "7879131203/280000000000",
+        "complex_endpoint": "1/700000000000",
+        "epsilon_below_one": "1/1400000000000",
+        "epsilon_positive": "1399999999999/1400000000000",
+        "fine_prime_halo_umbrella": "7/2",
+        "fused_gaussian_above_kappa": "5586936968051/1960000000000000000000000",
+        "gaussian_above_main": "695645656017/980000000000000000000000",
+        "global_coordinate_router_above_kappa": "1/350000000000",
+        "global_gamma_power_slack": "9/14000000000000",
+        "group_below_dimension": "1/700000000000",
+        "group_bit_router_above_kappa": "436683560503/122500000000000000000000",
+        "group_gauge_power_slack": "177653918425834354343983/980000000000000000000000",
+        "group_key_fits_record": "1/700000000000",
+        "group_positive": "1399999999997/1400000000000",
+        "group_scans_above_kappa": "697822828009/700000000000",
+        "halo_covers_generic_inverse_radius": "5",
+        "improvement_over_user_baseline": "689637640400418148631/500000000000000000000000",
+        "lambda_above_internal": "1/2800000000000",
+        "lambda_above_sigma": "3/2800000000000",
+        "lambda_above_tau": "1/2800000000000",
+        "lambda_prime_above_lambda": "1/2800000000000",
+        "lambda_prime_above_leaf": "559129131203/392000000000000000000000",
+        "leaf_saving_above_effective": "279129131203/392000000000000000000000",
+        "main_transform_above_kappa": "4195645656017/1960000000000000000000000",
+        "membership_arithmetic_fits_record": "1/1400000000000",
+        "microbox_face_corrections_above_kappa": "13474455707003/175000000000",
+        "microbox_halo_volume": "110600000000003/1400000000000",
+        "new_complex_uniform_saving_above_b": "1/1400000000000",
+        "new_semantic_guard_with_C1_one": "1/1400000000000",
+        "polynomial_products_above_kappa": "1395645656019/1400000000000",
+        "positive_layout_slack": "1/1400000000000",
+        "q_below_effective": "1/1400000000000",
+        "q_below_internal": "1/1400000000000",
+        "q_below_leaf": "559129131203/392000000000000000000000",
+        "q_positive": "4354343983/1400000000000",
+        "short_record_fallback": "279129131203/280000000000",
+        "source_axis_superpolynomial": "1/1400000000000",
+        "sparse_union_movement_above_kappa": "796250000005595645656017/1960000000000000000000000",
+        "stop_below_one": "1399999999999/1400000000000",
+        "stop_positive": "1/1400000000000",
+        "strassen_integer_power_gap": "1951441519231",
+        "support_router_span": "3/1400000000000",
+        "theta_positive_polynomial_power": "2",
+        "union_header_fits_record": "3/1400000000000",
+        "union_tag_work_per_visit_fits_record": "3/1400000000000"
+      }
+    },
+    "paid": {
+      "1": 40872,
+      "16": 416,
+      "2": 8262,
+      "24": 416,
+      "3": 8910,
+      "32": 416,
+      "4": 1656,
+      "40": 416,
+      "48": 416,
+      "5": 2862,
+      "56": 416,
+      "6": 396,
+      "64": 416,
+      "7": 270,
+      "72": 208
+    },
+    "paid_calls": 66348,
+    "previous_open_pr5713": "155457611/50000000000",
+    "productive_fields": 12976,
+    "query_shears": 15,
+    "rank_mass_surplus": 760,
+    "raw_ceiling": {
+      "derivative_upper": "-369633362694406560545046337101738171609683586723251074624328749/1569275433846670190958947355801916604025588861116008628224",
+      "interval": [
+        "0",
+        "17/5000"
+      ]
+    },
+    "raw_headroom": [
+      "503/25000000000000",
+      "2013/100000000000000"
+    ],
+    "raw_root_interval": [
+      "77756143003/25000000000000",
+      "311024572013/100000000000000"
+    ],
+    "relative_gain_percent": "5467400/155457611",
+    "strictification": {
+      "alpha": {
+        "lower": "0.999999999984633711012925911331",
+        "upper": "0.999999999984633711012925911332"
+      },
+      "cutoff": 17,
+      "delta": {
+        "lower": "0.968803893336188607862541966226",
+        "upper": "0.968803893336188607862541966227"
+      },
+      "floor": 18,
+      "gamma": {
+        "lower": "0.968263210279894472263420498148",
+        "upper": "0.968263210279894472263420498149"
+      },
+      "normalized_paid_upper": "1048576000976562501/1048576001073741824",
+      "power": 1024,
+      "reserve": "1/1000000000000",
+      "tail_block": 128,
+      "tail_decay": "39/40",
+      "tail_offset": "1/5",
+      "tested": "622049141/200000000000"
+    },
+    "targets": {
+      "1": 36,
+      "145": 208,
+      "17": 416,
+      "2": 90,
+      "25": 416,
+      "33": 416,
+      "4": 1638,
+      "41": 416,
+      "49": 416,
+      "5": 108,
+      "57": 416,
+      "65": 416,
+      "7": 108,
+      "73": 208,
+      "8": 270,
+      "9": 7398
+    },
+    "whole_idle_jobs": 3120
+  },
+  "independent_literal_audit": {
+    "columns": 753,
+    "owned_lifetimes": 11,
+    "owned_peak": 2,
+    "paid": {
+      "1": 2184,
+      "2": 459,
+      "3": 495,
+      "4": 92,
+      "5": 159,
+      "6": 22,
+      "7": 15
+    },
+    "scalar": 221474,
+    "status": "PASS_ALL_SCALAR_INPUTS_FRAMES_OWNERSHIP_INVOICES",
+    "targets": {
+      "1": 2,
+      "2": 5,
+      "4": 91,
+      "5": 6,
+      "7": 6,
+      "8": 15,
+      "9": 411
+    }
+  },
+  "limits": [
+    "The five inherited framework assumptions remain.",
+    "No selected-frontier admission or large improvement is claimed.",
+    "The huge uniform word and complete rounded multiplier are not materialized.",
+    "Independent agent checks are not human or formal acceptance."
+  ],
+  "local": {
+    "endpoint_coefficients": 567009,
+    "helpers": {
+      "1": 2,
+      "2": 5,
+      "4": 91,
+      "5": 6,
+      "7": 6,
+      "8": 15,
+      "9": 411
+    },
+    "owned_fields": 9,
+    "owned_lifetimes": 11,
+    "paid": {
+      "1": 2184,
+      "2": 459,
+      "3": 495,
+      "4": 92,
+      "5": 159,
+      "6": 22,
+      "7": 15
+    },
+    "paid_rank_mass": 5987,
+    "peak_owned_fields": 2,
+    "public_inputs": 744,
+    "real_scalar_allowance": 221474
+  },
+  "native": {
+    "distinct_maps": 2917,
+    "helper_normalizers": 1072,
+    "helpers": {
+      "1": 2,
+      "2": 5,
+      "4": 91,
+      "5": 6,
+      "7": 6,
+      "8": 15,
+      "9": 411
+    },
+    "main_adapters": 208,
+    "proof": "Every signed Pauli image and exact global phase; frame induction at every positive tensor width",
+    "recipe_payload_sha256": "626b4ce338e77fd1f5c5d90cf107d91bb73af42c107cd8eea14fe657ddb324e4",
+    "wrong_phase_rejected": true
+  },
+  "negative_controls": {
+    "invented_extra_H9_target": "Every productive helper has a native target",
+    "inverse_before_normalizing126188": "Exact arbitrary-input endpoint row 126",
+    "missing_central_translation": "Rejected by the complete Laurent endpoint",
+    "omit_current146_inverse": "Exact arbitrary-input endpoint row 743",
+    "omit_early126_restoration": "Exact arbitrary-input endpoint row 126",
+    "omit_early188_restoration": "Exact arbitrary-input endpoint row 188",
+    "omit_original_retirement": "Exact arbitrary-input endpoint row 708",
+    "omit_owned_identifier_shift": "Scalar destination coframe",
+    "reintroduce_obsolete_decoder": "Exact arbitrary-input endpoint row 743",
+    "restore_uncharged_old_H9_target": "All declared final charts and owned exits",
+    "unpaid_rank7_query": "Complete field and call counts",
+    "wrong_current146_inverse_sign": "Exact arbitrary-input endpoint row 743",
+    "wrong_new_helper_boundary": "MOVE source and ownership"
+  },
+  "physical_caller": {
+    "central_square_columns": 53248,
+    "full_calls": 18,
+    "main_targets": {
+      "145": 208,
+      "17": 416,
+      "25": 416,
+      "33": 416,
+      "41": 416,
+      "49": 416,
+      "57": 416,
+      "65": 416,
+      "73": 208
+    },
+    "new_paid_jobs": 0,
+    "physical_contacts": 7488,
+    "ports": 104,
+    "route_payload_sha256": "7f4f0ca768ffbc4f93dcff575592c4c67411f46b4289e45b73f715a2ad47b142",
+    "tensor_rule": "Factorwise native maps; scalar signs stay outside the tensor power.",
+    "whole_idle_jobs": {
+      "16": 416,
+      "24": 416,
+      "32": 416,
+      "40": 416,
+      "48": 416,
+      "56": 416,
+      "64": 416,
+      "72": 208
+    },
+    "wrapper_sites": [
+      1,
+      3,
+      5,
+      7,
+      9,
+      11,
+      13,
+      15,
+      17,
+      19,
+      21,
+      23,
+      25,
+      27,
+      29,
+      31
+    ]
+  },
+  "separate_columns": {
+    "arithmetic": "Dense integer rows with binary denominators; reverse chronology",
+    "coefficients": 567009,
+    "columns": 753
+  },
+  "source_commit": "a5348f7143f95fe0046b92b96172e30763c7d731",
+  "status": "PASS_P104_TWO_BOUNDARY_SPLIT_KAPPA_00311024570"
+}
diff --git a/research/p104-shared-frame-gather/inputs/word5718.json.gz b/research/p104-shared-frame-gather/inputs/word5718.json.gz
new file mode 100644
index 0000000000..4db015aca3
Binary files /dev/null and b/research/p104-shared-frame-gather/inputs/word5718.json.gz differ
diff --git a/research/p104-shared-frame-gather/publication_check.py b/research/p104-shared-frame-gather/publication_check.py
new file mode 100644
index 0000000000..202439eb93
--- /dev/null
+++ b/research/p104-shared-frame-gather/publication_check.py
@@ -0,0 +1,15 @@
+"""Check that the two independently retained packages use identical source bytes."""
+from pathlib import Path
+import hashlib, json
+root = Path(__file__).resolve().parent
+prior = root.parent / "p104-cross-call-paid-reuse"
+pairs = [(root / "inputs" / name, prior / "inputs" / name) for name in (
+    "DATA-PHASES.json", "NOTICE5718", "clifford_exact.py", "controller5718.json",
+    "native_verify.py", "result5718.json", "word5718.json.gz")]
+pairs += [(root / "inputs/predecessor-experiment.py", prior / "experiment.py"),
+          (root / "inputs/predecessor-SOURCES.json", prior / "SOURCES.json"),
+          (root / "inputs/QUERY-MAPS.json.gz", prior / "results/QUERY-MAPS.json.gz")]
+for left, right in pairs:
+    if left.read_bytes() != right.read_bytes():
+        raise ValueError("Publication source mismatch: " + str(right))
+print(json.dumps({"status": "PASS_PUBLICATION_SOURCE_JOIN", "identical_files": len(pairs)}))
diff --git a/research/p104-shared-frame-gather/results/CERTIFICATE.json.gz b/research/p104-shared-frame-gather/results/CERTIFICATE.json.gz
new file mode 100644
index 0000000000..83d7ea4bcc
Binary files /dev/null and b/research/p104-shared-frame-gather/results/CERTIFICATE.json.gz differ
diff --git a/research/p104-shared-frame-gather/results/PORTABLE-REPLAY.json b/research/p104-shared-frame-gather/results/PORTABLE-REPLAY.json
new file mode 100644
index 0000000000..7d6e705cc5
--- /dev/null
+++ b/research/p104-shared-frame-gather/results/PORTABLE-REPLAY.json
@@ -0,0 +1,39 @@
+{
+  "status": "PASS_GIT_FREE_EMPTY_PATH_OPTIMIZED_REPLAY_AND_UNDERPRICED_INGRESS_MUTATION",
+  "environment": {
+    "python": "3.12.8 (main, Dec  7 2024, 10:11:37) [Clang 15.0.0 (clang-1500.3.9.4)]",
+    "platform": "macOS-26.5.1-arm64-arm-64bit"
+  },
+  "runs": [
+    {
+      "script": "verify.py",
+      "argv": [
+        "/opt/local/bin/python3.12",
+        "-I",
+        "-O",
+        "-B",
+        "/var/folders/y2/rqpdcbzj7_98322kfb3v7_kr0000gq/T/shared-gather-portable-0o2rj4yc/package/verify.py"
+      ],
+      "returncode": 0,
+      "stdout": "PASS_SHARED_FRAME_GATHER_OBSTRUCTION_NO_NEW_KAPPA",
+      "seconds": 3.736
+    },
+    {
+      "script": "audit_certificate.py",
+      "argv": [
+        "/opt/local/bin/python3.12",
+        "-I",
+        "-O",
+        "-B",
+        "/var/folders/y2/rqpdcbzj7_98322kfb3v7_kr0000gq/T/shared-gather-portable-0o2rj4yc/package/audit_certificate.py"
+      ],
+      "returncode": 0,
+      "stdout": "PASS_INDEPENDENT_SHARED_GATHER_CERTIFICATE_AUDIT frames=1090 source_bundles=465 prefix_answers=1211",
+      "seconds": 2.319
+    }
+  ],
+  "files_byte_unchanged": 21,
+  "underpriced_ingress_mutation_rejected": true,
+  "network_or_Git_required": false,
+  "retained_temporary_copy": "/var/folders/y2/rqpdcbzj7_98322kfb3v7_kr0000gq/T/shared-gather-portable-0o2rj4yc/package"
+}
diff --git a/research/p104-shared-frame-gather/results/RESULT.json b/research/p104-shared-frame-gather/results/RESULT.json
new file mode 100644
index 0000000000..02a8307421
--- /dev/null
+++ b/research/p104-shared-frame-gather/results/RESULT.json
@@ -0,0 +1,1290 @@
+{
+  "assigned_query": {
+    "event": 13660,
+    "input_version": 3,
+    "last_logical_write": 13658,
+    "new": 598,
+    "old": 518,
+    "output_version": 4,
+    "owned_epoch": 1,
+    "private": true,
+    "rank": 7,
+    "role": 748,
+    "row": {
+      "1": "-1/4",
+      "101": "1/4",
+      "102": "1/4",
+      "104": "-1/4",
+      "107": "-1/4",
+      "110": "-1/4",
+      "111": "-1/4",
+      "13": "1/4",
+      "16": "-1/4",
+      "18": "1/4",
+      "19": "1/4",
+      "2": "-1/4",
+      "20": "-1/4",
+      "21": "-1/4",
+      "24": "1/4",
+      "25": "1/4",
+      "257": "1/4",
+      "258": "-1/4",
+      "261": "1/4",
+      "262": "-1/4",
+      "265": "1/4",
+      "266": "1/4",
+      "267": "-1/2",
+      "269": "1/4",
+      "270": "1/4",
+      "271": "-1/2",
+      "28": "1/4",
+      "288": "-1/4",
+      "292": "1/4",
+      "296": "1/4",
+      "302": "1/4",
+      "303": "-1/2",
+      "304": "-1/4",
+      "308": "1/4",
+      "312": "1/4",
+      "318": "1/4",
+      "319": "-1/2",
+      "32": "1/4",
+      "336": "-1/4",
+      "340": "1/4",
+      "35": "-1/4",
+      "350": "1/4",
+      "351": "-3/4",
+      "36": "1/4",
+      "37": "1/4",
+      "384": "-1/4",
+      "388": "1/4",
+      "39": "-1/4",
+      "398": "1/4",
+      "399": "-3/4",
+      "4": "-1/4",
+      "40": "-1/4",
+      "416": "1/4",
+      "417": "1/4",
+      "418": "-1/4",
+      "421": "1/4",
+      "422": "-1/4",
+      "425": "1/4",
+      "426": "1/4",
+      "427": "-1/2",
+      "429": "1/4",
+      "43": "1/4",
+      "430": "1/4",
+      "431": "-1/2",
+      "433": "1/4",
+      "434": "-1/4",
+      "437": "1/4",
+      "438": "-1/4",
+      "44": "1/4",
+      "441": "1/4",
+      "442": "1/4",
+      "443": "-1/2",
+      "445": "1/4",
+      "446": "1/4",
+      "447": "-1/2",
+      "45": "1/4",
+      "46": "1/4",
+      "47": "1/4",
+      "48": "-1/4",
+      "5": "-1/4",
+      "51": "-1/4",
+      "54": "-1/4",
+      "55": "-1/4",
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+      "6": "1/4",
+      "61": "1/4",
+      "64": "-1/4",
+      "65": "-1/4",
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+      "78": "-1/4",
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+      "82": "-1/4",
+      "85": "-1/4",
+      "86": "-1/4",
+      "89": "-1/4",
+      "9": "-1/4",
+      "93": "1/4",
+      "94": "-1/4",
+      "96": "1/4",
+      "97": "1/4",
+      "98": "1/4",
+      "99": "1/4"
+    }
+  },
+  "caller_event": 3,
+  "controls": {
+    "constructive_gather": {
+      "all_scalar_columns": 744,
+      "coefficients": {
+        "12489": "1",
+        "12491": "1"
+      },
+      "completion_cut": 13660,
+      "destination_frame": 3,
+      "events": [
+        12489,
+        12491
+      ],
+      "frames": [
+        33,
+        33
+      ],
+      "gather_frame": 33,
+      "ingress_ranks": [],
+      "invoice": "1 owned INIT +1 COPY +1 ADD +1 native H1 +1 observation +1 CLEAR; one retained owned value, no target credit. Ordinary/hosting tariff unspecified; this is a discovery control only.",
+      "operations": [
+        [
+          "INIT",
+          "cacheA",
+          33
+        ],
+        [
+          "COPY",
+          "cacheA",
+          432,
+          "1",
+          33
+        ],
+        [
+          "ADD",
+          "cacheA",
+          433,
+          "1",
+          33
+        ],
+        [
+          "MOVE",
+          "cacheA",
+          33,
+          3,
+          true
+        ],
+        [
+          "OBSERVE",
+          "cacheA",
+          3
+        ],
+        [
+          "CLEAR",
+          "cacheA",
+          3
+        ]
+      ],
+      "physical_admission": "Local nested scalar/native/lifetime control only; arbitrary-width retention hosting and supplier admission are not asserted.",
+      "retention": "COPY 432 immediately after12489, ADD 433 immediately after12491; retain one owned value to local13660.",
+      "shared_final_rank": 1,
+      "versions": [
+        4,
+        2
+      ]
+    },
+    "exact_native_controls": [
+      {
+        "new": 3,
+        "old": 33,
+        "rank": 1,
+        "recipe": {
+          "left": {
+            "linear_Z4": [
+              2,
+              0,
+              0,
+              0,
+              0,
+              0,
+              0,
+              0,
+              0
+            ],
+            "matrix_rows": [
+              2,
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+              1,
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+            ],
+            "offset": 0,
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+            ],
+            "matrix_rows": [
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+            ],
+            "offset": 0,
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+              0,
+              18,
+              0,
+              0,
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+              0,
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+            ]
+          }
+        }
+      },
+      {
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+        "rank": 1,
+        "recipe": {
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+            ],
+            "matrix_rows": [
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+            ]
+          },
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+            ],
+            "matrix_rows": [
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+              256
+            ],
+            "offset": 0,
+            "phase": 3,
+            "quadratic_rows": [
+              63,
+              63,
+              63,
+              63,
+              63,
+              63,
+              0,
+              0,
+              0
+            ]
+          }
+        }
+      },
+      {
+        "new": 598,
+        "old": 3,
+        "rank": 1,
+        "recipe": {
+          "left": {
+            "linear_Z4": [
+              2,
+              0,
+              0,
+              0,
+              0,
+              0,
+              0,
+              0,
+              0
+            ],
+            "matrix_rows": [
+              3,
+              5,
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+              32,
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+              1
+            ],
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+            ]
+          },
+          "rank": 1,
+          "right": {
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+              3,
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+              0,
+              3,
+              3,
+              3,
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+            ],
+            "matrix_rows": [
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+            ],
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+            "phase": 1,
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+              235,
+              0,
+              235,
+              0,
+              235,
+              235,
+              235,
+              0
+            ]
+          }
+        }
+      },
+      {
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+        "old": 518,
+        "rank": 7,
+        "recipe": {
+          "left": {
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+              3,
+              2,
+              2,
+              2,
+              0,
+              0
+            ],
+            "matrix_rows": [
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+              12,
+              16,
+              32,
+              4,
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+              256
+            ],
+            "offset": 0,
+            "phase": 0,
+            "quadratic_rows": [
+              0,
+              0,
+              12,
+              12,
+              0,
+              0,
+              0,
+              0,
+              0
+            ]
+          },
+          "rank": 7,
+          "right": {
+            "linear_Z4": [
+              1,
+              1,
+              3,
+              0,
+              2,
+              1,
+              3,
+              1,
+              0
+            ],
+            "matrix_rows": [
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+            ],
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+              0,
+              235,
+              235,
+              235,
+              0
+            ]
+          }
+        }
+      }
+    ],
+    "future_version_rejected": true,
+    "missing_fresh_helper_rejected": true,
+    "old_snapshot_version_rejected": true,
+    "stale_helper_namespace_rejected": true,
+    "underpriced_ingress_rejected": true,
+    "unknown_cleanup_zero_rejected": true,
+    "unpaid_ingress_retag_rejected": true,
+    "wrong_global_phase_rejected": true,
+    "wrong_snapshot_frame_rejected": true
+  },
+  "exact_input_coverage": {
+    "all_1090_frames_independently_interpreted": true,
+    "available_same_call_paid_outputs": 1211,
+    "basis_rank": 354,
+    "caller_binding_phase_checks": 416,
+    "helpers_per_invocation": 536,
+    "independently_checked_pair_distances": 316590,
+    "literal_relative_rank_checks": 1384,
+    "owned_columns": 9,
+    "paid_query_maps": 3426,
+    "public_columns": 744,
+    "source_coframe_bundles": 465,
+    "target_coordinate_support": 74
+  },
+  "frame_domain": {
+    "all_pinned_frames": 1090,
+    "physically_seen_before_cut": 478,
+    "unseen_frames_granted_only_as_favorable_rejection_relaxation": 612
+  },
+  "helper21_paid_sources": [
+    {
+      "coefficient": "1/2",
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+      "input_version": 3,
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+      "new": 33,
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+      "owned_epoch": 0,
+      "private": false,
+      "rank": 1,
+      "role": 432
+    },
+    {
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+      "private": false,
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+    },
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+      "private": false,
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+    },
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+    },
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+      "private": false,
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+    },
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+    },
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+    },
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+      "private": false,
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+    },
+    {
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+    },
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+    {
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+      "new": 569,
+      "old": 489,
+      "output_version": 72,
+      "owned_epoch": 0,
+      "private": false,
+      "rank": 4,
+      "role": 159
+    },
+    {
+      "coefficient": "-1/2",
+      "event": 13499,
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+      "new": 565,
+      "old": 569,
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+      "owned_epoch": 0,
+      "private": false,
+      "rank": 1,
+      "role": 159
+    },
+    {
+      "coefficient": "1",
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+      "new": 1033,
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+      "owned_epoch": 0,
+      "private": false,
+      "rank": 3,
+      "role": 143
+    },
+    {
+      "coefficient": "1",
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+      "input_version": 88,
+      "last_logical_write": 13435,
+      "new": 1035,
+      "old": 356,
+      "output_version": 89,
+      "owned_epoch": 0,
+      "private": false,
+      "rank": 3,
+      "role": 144
+    },
+    {
+      "coefficient": "1",
+      "event": 13558,
+      "input_version": 92,
+      "last_logical_write": 13417,
+      "new": 1037,
+      "old": 392,
+      "output_version": 93,
+      "owned_epoch": 0,
+      "private": false,
+      "rank": 3,
+      "role": 187
+    },
+    {
+      "coefficient": "1",
+      "event": 13564,
+      "input_version": 81,
+      "last_logical_write": 13440,
+      "new": 1039,
+      "old": 429,
+      "output_version": 82,
+      "owned_epoch": 0,
+      "private": false,
+      "rank": 3,
+      "role": 200
+    }
+  ],
+  "hypothetical_complete_signed_bill": {
+    "additional_cost_obligations": {
+      "acquisition_COPY": "unknown, nonnegative; optimistically omitted only for exclusion",
+      "compiler_precision_stock_outer": "unrun for any changed program because no candidate survives",
+      "ingress_and_final_monomial_normalizers": "unknown ordinary work; all positive recursive Gaussian ranks included",
+      "main_and_other_helper_corrections": "unknown; all main information granted free for exclusion",
+      "ordinary_dyadic_arithmetic_and_routes": "unknown, nonnegative",
+      "owned_clear_and_retained_value_cleanup": "unknown, nonnegative",
+      "read_retention_and_owned_storage": "unknown, nonnegative; no productive credit",
+      "source_restoration_dirty_return": "unknown; no source-consuming alteration is granted"
+    },
+    "conclusion": "No fully costed improvement can exist inside the tested protocol. No alternative program was emitted.",
+    "necessary_gather_paid": "sum_i H_distance(snapshot_i,frame) + H_distance(frame,598) >= H7 for every 0<t<1",
+    "proof": "Strict improvement requires total positive Gaussian rank <=6, since sum r_i^t >= (sum r_i)^t. Every such source subset is excluded in every tested gather frame.",
+    "removed_paid": {
+      "7": -1
+    },
+    "target_delta": {}
+  },
+  "kappa_status": "NO_NEW_KAPPA",
+  "local_cut": 13660,
+  "minimum_support_obstruction_margin": 5,
+  "realized_delta_paid": {},
+  "realized_delta_targets": {},
+  "rejection_counts": {
+    "final_rank": 411,
+    "helper21": 338,
+    "support": 341
+  },
+  "scope": "One common pinned canonical frame; arbitrary combinations of unchanged earlier paid snapshots preaggregated within each source coframe, independent source-frame ingress, scalar recombination, one final canonical delivery. Distinct call0/call3 raw helper columns. No changed source graph or matrix-valued recombination.",
+  "status": "PASS_SHARED_FRAME_GATHER_OBSTRUCTION_NO_NEW_KAPPA",
+  "support_frames_by_final_rank": {
+    "0": 1,
+    "1": 3,
+    "2": 125,
+    "3": 164,
+    "4": 37,
+    "5": 7,
+    "6": 4
+  },
+  "unchanged_complete_source_invoice": {
+    "kappa_status": "NO_NEW_KAPPA",
+    "local_paid": {
+      "1": 2184,
+      "2": 459,
+      "3": 495,
+      "4": 92,
+      "5": 159,
+      "6": 22,
+      "7": 15
+    },
+    "local_targets": {
+      "1": 2,
+      "2": 5,
+      "4": 91,
+      "5": 6,
+      "7": 6,
+      "8": 15,
+      "9": 411
+    },
+    "new_retention_reads_routes_restoration_cleanup": "Unknown; no legal deletion emitted. Not charged at zero.",
+    "paid_jobs": 66348,
+    "productive_fields": 12976,
+    "realized_delta_paid": {},
+    "realized_delta_targets": {},
+    "source_reported_ordinary": {
+      "fixed_scalar_growth_bound": "At most 2^(16*26000) before native factors; a declared constant, not an attained norm.",
+      "local_paid_rank_mass": 5987,
+      "local_routes": 5166000,
+      "local_scalar": 221474,
+      "peak_owned_fields": 2,
+      "raw_budget": 1000000000,
+      "raw_scalar_upper": 184647552,
+      "tensor_native_growth_bound": "Each rank-r Gaussian has row l1 norm at most 2^(rf); use local_paid_rank_mass for the whole word."
+    },
+    "unrealized_centre_allowance": {
+      "paid_delta": {
+        "7": -1
+      },
+      "target_delta": {}
+    },
+    "whole_paid": {
+      "1": 40872,
+      "16": 416,
+      "2": 8262,
+      "24": 416,
+      "3": 8910,
+      "32": 416,
+      "4": 1656,
+      "40": 416,
+      "48": 416,
+      "5": 2862,
+      "56": 416,
+      "6": 396,
+      "64": 416,
+      "7": 270,
+      "72": 208
+    },
+    "whole_targets": {
+      "1": 36,
+      "145": 208,
+      "17": 416,
+      "2": 90,
+      "25": 416,
+      "33": 416,
+      "4": 1638,
+      "41": 416,
+      "49": 416,
+      "5": 108,
+      "57": 416,
+      "65": 416,
+      "7": 108,
+      "73": 208,
+      "8": 270,
+      "9": 7398
+    }
+  }
+}
diff --git a/research/p104-shared-frame-gather/results/live-overlap.json b/research/p104-shared-frame-gather/results/live-overlap.json
new file mode 100644
index 0000000000..a2b1a46a30
--- /dev/null
+++ b/research/p104-shared-frame-gather/results/live-overlap.json
@@ -0,0 +1,116 @@
+{
+  "checked_at_utc": "2026-10-26T16:35:00Z",
+  "coverage": {
+    "recent_discussions": 35,
+    "recent_open_prs": 60,
+    "additional_complete_cached_overlap": "overlap-summary.json",
+    "sparse_local_inventory_complete": false
+  },
+  "sources": {
+    "discussions": [
+      {
+        "number": 5645,
+        "title": "P104 kappa research: paid-split checkpoint and changed native contacts",
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5645",
+        "updatedAt": "2026-10-26T16:33:05Z",
+        "body": "IN PROGRESS / 2026-10-26 16:32 UTC / priority revision 8.\n\nThe [17-split checkpoint](https://github.com/aliromman/integer-mult-bounds/tree/d12c769799573058248ae986d6e8fd036522f020/research/p104-paid-helper-subdivisions) has an author-checked complete conditional price .00311224341. Its cuts overlap Discussion5600; that scope is released and no separate construction priority or duplicate PR is claimed. The completion parameters and proof are available for reuse.\n\nThe current distinct question permits a paid Gaussian conversion at a scalar contact, with source COPY, native query, ADD and owned CLEAR fully charged. All productive and main boundary jobs stay fixed. This changes the scalar contact contract; a frame-only optimum does not decide it. The bounded first test uses catalogue common anchors and exact source chronology. Discovery is local-only; no new gain is claimed. Expiry: 2026-10-26T18:30:00Z. Selected frontier unchanged.\n\n<details><summary>Previous published result and original claim</summary>\n\nDONE / RELEASED \u2014 [PR #5713](https://github.com/CrocSwap/integer-mult-bounds/pull/5713), exact head `a5348f7143f95fe0046b92b96172e30763c7d731`, raises whole conditional kappa from **0.00310871312 to 0.00310915222**, a gain of **0.00000043910 (0.01412482%)**.\n\nThe new construction distributes a scalar update across two available contacts on an unobserved field interval. An exact identity restores the field before its next read. This merges H1 and H2 into H3, for 18 fewer paid calls in the full caller, with every productive target retained. All centre queries remain paid. The gain is small; a large breakthrough remains open.\n\nNormal, optimized and read-only Git-free replay pass, including every endpoint coefficient, native phase, complete caller and price. The five inherited framework assumptions remain. Independent acceptance is open. Hosted CI is still running at publication. The selected frontier is unchanged.\n\nThe [canonical seven-field handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19191873) records the exact witness, all checks, source pins, limitations, command and predecessor credit. Joel Pulikkan's PR5684 and the complete retained P104 lineage supply the starting construction. New work: Ali Romman with substantial OpenAI Codex assistance; no subagents. All investigation scopes are released.\n\n<details><summary>Original claim</summary>\n\nCLAIM \u2014 paired boundary gauges with the same complete physical endpoint.\n\nQuestion: can right-frame gauges at both ends of an independent helper, or at both ends of a complete main shear pair, lower the complete Gaussian paid bill while preserving its physical target? The planned source is the selected P104 helper-subspace word on current canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7.\n\nThe exact mechanism is the boundary-commutant condition B_out S B_in^-1=S. Helpers have independent scalar identity blocks, and each main pair has a scalar shear. Thus matching input/output gauges on a helper and one shared gauge on each complete main pair preserve the whole physical endpoint. No input conversion is executed or treated as free; that equality must be proved for the completed native word. At every scalar contact, actual operands must share the same frame and gauge. Proper transports are replaced by exact phased Clifford words and fully priced. Productive targets, owned inputs and all external adapters remain fixed.\n\nThis is a changed boundary contract for the former fixed-boundary gauge optimization. It differs from Discussion5600's active paid-history reconstruction, and does not claim its fixed-boundary searches or the helper-cleanup, MOVE-window and integration lanes. Dugongue PR5639 gaussian.md Sections7-8 and its boundary-commutant corollary supply the precise new sufficient compiler rule and are credited. The initial finite search uses the full nine-axis Gaussian and paired endpoint constraints; a gain must pass literal native phases, every input, the complete caller, both interval engines and all compiler/outer charges. No new kappa is claimed at this point.\n\nThe distinct G20 affine-history component is being published separately from Discussion5638. This claim concerns a new Gaussian construction, not a PR combination. Latest active Discussions and open PRs were checked at 02:09 UTC. Expiry: 2026-10-26T04:00:00Z. Local-only discovery until a public artifact is linked. Ali Romman with substantial OpenAI Codex assistance; no subagents. Retain all P104, native frame, controller, helper, compiler and analytic source notices. No selected-frontier, merge or closure action is authorized here.\n\n</details>\n\n</details>\n",
+        "comments": {
+          "nodes": [
+            {
+              "body": "SCOPE UPDATE / CLAIM \u2014 paid transport subdivision for a productive helper cut.\n\nThe fixed transport schedule gave no new candidate:14266 bank contexts,812 unit physical directions satisfying both boundary-column constraints, and1581 tested two-field directions. A positive control on the exact PR5713 input recovers all seven contexts of PR5718's helper708 split, including its published154/2 correction. These local modular screens are not global rational obstructions. I release the fixed-schedule scope.\n\nI change the paid word. Split an existing public helper transport A -> B through an existing catalogue frame F, and use a real bank at F to retire the old independent input and start a new independent continuation. The transport may move within its actual interval of no scalar use; all intervening source versions are explicit. Separate cut corrections may be required for retirement and the new continuation. Their exact boundary identities and all ordinary work must be charged.\n\nIf the old helper has total rank t and the cut has rank k from its input, the target delta is H_k+H_(t-k)-H_t. The additional paid delta is H_a+H_b-H_(a+b), where a=dim(F)-dim(A) and b=dim(B)-dim(F). Both terms enter the same invoice. When the split query is shorter than the full helper job, the target gain can exceed that added query cost. This is a changed producer/consumer budget, not a free added contact. The source has838 public helper transports with query rank at least two and below the total helper rank; this is a planning domain, not838 valid constructions.\n\nSource remains flyingviper PR5718 at a1534ac45401c9a9c80c07aa65c89e00e33594f8, conditional kappa .00311024570. Native/coframe/phase, all independent inputs and outputs, owned lifetimes, scalar/growth/route costs, the actual caller and all compiler/outer inequalities are required. A search witness is not a kappa claim. The first bounded test uses existing catalogue frames and actual chronological bank versions only. A surviving positive candidate will be fully replayed, then compatible productive steps may be repeated on its changed word.\n\nCanonical main was refreshed again; no selected or priority change. The released G20 subdivision screen in Discussion5709 has a different source and boundary contract; its coverage is credited and is not repeated. This new P104 scope uses PR5718's distinct boundary corrections and the complete two-term price above. Credit all retained P104/helper/native/caller sources, PR5599/5602/5609/5611, PR5713 and Joel Pulikkan PR5684. Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry remains2026-10-26T16:15:00Z. No merge or selected-frontier change.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19197695"
+            },
+            {
+              "body": "Overlap notice and concrete handoff: [Discussion5600](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19197520) already has an active paid intermediate-meeting claim on the same PR5718 source. I found this later subdivision claim in the live readback. The [current finding](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19198550) records 23 compatible splits: 18 one-query splits, then a three-channel meeting with five added query subdivisions, and two more two-query meetings. The exact final word has 767 public inputs, paid histogram {1:2219,2:454,3:492,4:92,5:175,6:6,7:15}, target histogram {1:25,2:5,4:91,5:6,7:6,8:38,9:388}, and floating complete crossing about .003113016. All scalar endpoints pass; the 21-split prefix passes native and caller replay. Final pricing/package work is in progress.\n\nThe 18 one-query pivots are 731,706,713,724,725,730,728,722,714,720,719,718,727,729,723,712,715,717, in that order. The collective cuts add557,455,115,452,448. No independent discovery credit is claimed for a result already present in another public artifact. Please compare any output to this inventory before duplicating the same construction. I will link the exact public package in5600 and here when ready. I will confine further exploration to collective paid meetings and new intermediate frames, rather than repeat your distinct-boundary single-query domain. The selected frontier stays unchanged.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198689"
+            },
+            {
+              "body": "HANDOFF / OVERLAP RESOLVED \u2014 I read your earlier claim and the 23-split finding. Seventeen of my cuts overlap your one-query set. I release nested paid subdivisions and collective meetings, and do not claim separate construction priority or open a duplicate PR.\n\nThe complete 17-split checkpoint is public at [d12c769799573058248ae986d6e8fd036522f020](https://github.com/aliromman/integer-mult-bounds/tree/d12c769799573058248ae986d6e8fd036522f020/research/p104-paid-helper-subdivisions). It gives conditional kappa .00311224341 with full rational scalar/transpose, native phases, physical caller, two moment engines and compiler/outer checks. The exact source is your same PR5718 head. Normal primary replay and metadata/pins pass; the registered pair, sealed export and independent review are not claimed for this checkpoint. Its README and VALIDATION explicitly disclose the overlap.\n\nA useful completion change: the old 17/18, u=1/5, rho=39/40 parameters fail after the target count grows. For this checkpoint, c=16, b=17, u=7/50, rho=122/125, v=1024, reserve=1e-12 and block128 pass all charged completion, rounding, exit and compiler inequalities. The exact alpha/gamma/delta bounds are in RESULT.json. They must be recalculated for your larger word. The finite local envelope is explicitly 768 public fields and fewer than28000 instructions, with9 owned labels, peak2 and the same funded1000-slot routes; the checker has only those two declared cap edits. No new owned lifetime is used.\n\nThe checkpoint includes a general exact boundary proof with D=T^-1 d and distinct h/j functionals. The seventeen actual cuts all have h=e_r and no fresh decoder, which explains the overlap with your same-boundary construction. All original source and AI-assistance notices are retained. Please reuse any useful proof, completion parameters or checks with attribution.\n\nI will move to an explicitly different contract: nonnested public native excursions for partially completed helpers, with every actual Gaussian rank and phase paid and every changed productive target rederived. The initial domain is a single public A-to-F-to-B replacement with F outside [A,B]; it excludes nested subdivisions, collective meeting searches and your new nested intermediate-frame domain. No positive result is claimed. Canonical main is now553d26d796fde9871ff2661a49ac7732e945410f; selected and priority7 are unchanged. This new bounded claim expires2026-10-26T18:00:00Z. Ali Romman with substantial OpenAI Codex assistance; no subagents.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198871"
+            },
+            {
+              "body": "The complete 23-split package is now public in [PR #5751](https://github.com/CrocSwap/integer-mult-bounds/pull/5751), exact head `8483f9a679d06c1e270141dbc686a7e76e7a7522`. It certifies conditional kappa .00311301600 after normal, optimized and read-only Git-free replay. The [final handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19199447) gives all costs, source pins, limits and the replay command.\n\nYour 17-cut checkpoint and the released overlap are credited. The final completion parameters are c=16,b=17,u=1/7,rho=125/128, recalculated for the 23-split invoice. The constructor and exact PLAN.json are available for reuse. All my paid-meeting scopes are released. Your nonnested-excursion scope remains separate; no duplicate PR was opened for the shared 17-cut checkpoint.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19199474"
+            },
+            {
+              "body": "SCOPE UPDATE \u2014 source-bound helper modes with changed physical output owners.\n\nThe nonnested single-query preflight had105 positive price shapes but no occupied legal bank in the fixed source windows. The owned-zero continuation preflight has an exact neutral specialization of19 helpers: it removes equal H1/H5/H7 paid and productive terms. Its fixed current-cut and causal-history screens give no further gain. These are local checks, not global exclusions or completed contributions. Both scopes are released.\n\nThe next bounded question changes the cut interface. At an existing coframe F, permit a linear combination of several physical fields to carry the retired helper input, and a different combination to decode the fresh continuation. Permit the retired helper's physical output location to differ from its input location. Every whole-field output permutation is explicit ordinary routing; every initial field remains arbitrary. There are no new paid meetings, nested-query subdivisions or collective query-retiming searches in this scope.\n\nFor a cut prefix R and suffix S, seek a bank direction q and retirement/continuation rows c and j. Put h=cR, d=Sq and D=T^-1d. The checks require c q=j q=hD=(jR)D=1, legal common input support for h/D, and legal common output support for d and (jR)T^-1. A three-stage bank transformation exchanges the fresh input with hR while applying q times the decoded continuation difference. Initial and terminal basis changes must restore all other outputs. This generalizes the fixed physical unit-pivot contract; it is not an assertion that the known G20 changed-output-owner screen has a positive P104 counterpart.\n\nSource is the public d12c769799573058248ae986d6e8fd036522f020 checkpoint, with the earlier Discussion5600 mechanism credited and no separate discovery priority claimed for its17 cuts. The full rank-zero/full-frame helper group has no positive mode in the finite preflight. The ongoing bounded test covers the actual partially completed helper boundary groups, including coframed main-output coordinates, and exact rational reconstruction of any positive witness. A mode screen alone will not count as a kappa gain. Native phases, output routing, owned work, the caller and all completion charges remain required.\n\nCanonical main was refreshed to de5ec2e7a6074068452f45b1a191b5c72832df4a. I read revision8; selected .00310045 is unchanged. Current public claims include the disjoint paid-meeting construction and fixed R22 block composition. The G20 changed physical-owner attempt in Discussion5709 is credited as prior scoped work. New work: Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry2026-10-26T18:30:00Z. No positive new claim, merge or selected-frontier change.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19199579"
+            }
+          ]
+        }
+      },
+      {
+        "number": 5600,
+        "title": "Paid intermediate meetings: conditional kappa 0.00311301600",
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5600",
+        "updatedAt": "2026-10-26T16:13:29Z",
+        "body": "DONE / RELEASED \u2014 [PR #5751](https://github.com/CrocSwap/integer-mult-bounds/pull/5751) raises complete conditional kappa from **0.00311024570 to 0.00311301600**, a gain of **0.00000277030 (0.0890701%)**. The requested 1% gain was not reached. The selected frontier is unchanged.\n\nThe construction creates paid intermediate meetings and splits 23 independent helper targets. It charges all 27 added native queries per local call and recalculates the compiler completion parameters. Exact all-input, native-phase, complete caller, price and outer checks pass in normal and optimized modes. The separate read-only Git-free export replay also passes. The publication head is `8483f9a679d06c1e270141dbc686a7e76e7a7522`.\n\nThe [complete handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19199447) gives the exact command, source pins, evidence, attribution and remaining obligations. The five inherited framework assumptions remain; independent acceptance is pending. Hosted CI failed in unchanged current-assignment tests, so its scientific replay was skipped; see the complete handoff for the exact failure and local reproduction. The [overlap resolution](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198871) prevents a duplicate construction PR. The other investigation's nonnested-excursion work remains separate. All paid-meeting claims here are released.\n\n<details><summary>Earlier Pauli-gauge investigation and released claim</summary>\n\nNO QUALIFYING RESULT / RELEASED. No new kappa improvement was found. All claims in this investigation are released. Selected .00310045 is unchanged. See the [final findings and limits](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19190122).\n\n<details><summary>Original claim</summary>\n\nCLAIM \u2014 joint Pauli-gauge regions on the complete P104 scalar graph.\n\nQuestion: can one Clifford Pauli rotation change a whole region of scalar contacts, with fixed main endpoints and a complete productive-helper bill, to raise conditional kappa by at least 0.1% over PR5580?\n\nBase: canonical main 5e0052ca0aa3036a3afa8b0438927377c8d97d53, priority revision 7. Source: PR5580 at 6efa248cf456d6101d757e9d09ba29711d18a13a. This branch starts from main. The selected frontier stays unchanged.\n\nMethod: collapse scalar contacts to their 1,017 exact common-gauge nodes. At each node choose the old chart C or C G for a fixed Clifford Pauli rotation G. Compute every changed paid rank and helper endpoint rank from signed native tableaux. Optimize the joint binary region, then check any candidate with exact native phases, all original input columns, the full caller, two moment engines and all compiler/outer costs. Begin with rank-one Pauli rotations and the full nine-axis Gaussian. The full-Gaussian binary screen has zero best gain. Simple zero-input retirement and the p-length screen also gave no gain; those are local discovery checks only.\n\nThis changes scalar contact charts through a common right factor and permits nonnested paths. It differs from Discussion5575's commuting ADD order and nested intermediate-frame search, and from the equal-frame input/output basis and private-copy replacement scopes in Discussion5125. The old PR5124 nearest-contact work on the Cao word and its three joint binary-axis screens are credited predecessors. This claim is for joint regions on the exact P104 descendant, not those original-word searches, helper splitting or a new global optimum.\n\nThe initial overlap cache has complete remote pagination: 331 open PRs and 4,858 Discussion records at 2026-10-25 21:12\u201321:13 UTC. Relevant recent claims and PRs were read. Sparse local and unpublished work are coverage limits. No kappa gain is established yet. A numerical score alone will not be published as one.\n\nCredit PR5580, GamingPuzzled PR5569, DreamingOfClouds PR5561/5568, thu-nmrc, Gareth Davidson, David Arias, Cinna Davis, Jacob Alan Sussman, zxv, Cao Yifan and all retained native/compiler/geometry sources. New work by Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0.\n\nClaim expires: 2026-10-25 23:45 UTC.\n\n\n</details>\n\n</details>\n",
+        "comments": {
+          "nodes": [
+            {
+              "body": "CLAIM \u2014 create a helper meeting inside a paid native query.\n\nI resume Discussion5600 with a changed physical word on PR5718 at a1534ac45401c9a9c80c07aa65c89e00e33594f8. The source conditional kappa is .00311024570. Its construction and every retained author receive credit.\n\nThe test splits an existing MOVE through a frame occupied by other fields at that time. This creates a common-frame cut that the original word does not have. Both new queries are charged. I will test the original same-boundary-group independent-helper criterion at the new meeting, then emit and verify any positive construction. For a target of rank r split at s, and a paid query of rank k split at a, the discovery gain is H_s + H_(r-s) - H_r - H_a - H_(k-a) + H_k. A positive screen is not a kappa claim.\n\nThis differs from the active Discussion5125 search for distinct boundary functionals at actual existing meetings. I do not take that scope. The published PR5680 coverage explicitly excludes changed native words. My first domain is one existing query split through one occupied catalogue frame, with unchanged scalar instructions before the helper split. Both public and owned query paths will retain their declared directions and lifetimes. If this domain has no gain, I will not count it as a contribution.\n\nBase canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7. The complete public overlap snapshot was fetched at14:03\u201314:05 UTC; local note coverage is sparse. Recent PRs and the live P104 claims were read. Unpublished work remains outside this check. The selected frontier stays unchanged. A surviving candidate still needs exact rational input/transpose checks, actual native maps and phases, the complete physical caller, ordinary costs, both interval engines, compiler and all outer predicates.\n\nCredit the PR5599/5609 independent-helper criterion, the published PR5680 cut checker, flyingviper's PR5718, Ali Romman's PR5713, Joel Pulikkan's PR5684, and the full native/helper/controller/compiler lineage. New work: Ali Romman with substantial OpenAI Codex assistance; no subagents. Expiry2026-10-26T16:30:00Z. No new bound is established.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19197520"
+            },
+            {
+              "body": "FINDING / CLAIM UPDATE \u2014 paid intermediate meetings.\n\nThe same scoped investigation now has 23 compatible helper splits on PR5718 (a1534ac45401c9a9c80c07aa65c89e00e33594f8). Eighteen use one paid query subdivision each. Three share five subdivisions, and two each use two subdivisions. Every added query is charged. Exact rational endpoint, frame and ownership checks pass for the 767-public-input word. The 21-split prefix also passes the retained dense transpose, actual native-map/phase engine and complete p16/a7 physical caller. These are local results; the final full price and compiler certificate is not yet complete. The floating complete crossing for the 23-split word is about .003113016, about +0.089% over PR5718. This is below the donor's 1% target.\n\nI extend the claim within this mechanism to collective paid meetings made by advancing later queries or delaying earlier queries across intervals with no field access. Shared transport can serve several independent helper modes. This changes the chronological native word and pays the cost. It does not take the existing-meeting two-boundary search, active-mode physical-delivery work, or Discussion5645's two-field repair scope. I am checking all event cuts within a finite frame catalogue, including all binary lines and hyperplanes. A negative screen in that domain will not be called a global ceiling.\n\nThe source framework, native, caller and helper-split credit is unchanged from the claim above. The selected frontier is unchanged. Evidence is still local until the focused package and PR are posted here. Current claim expiry: 2026-10-26T17:30:00Z.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19198550"
+            },
+            {
+              "body": "FINDING \u2014 the complete paid-meeting calculation now passes at conditional kappa .00311301600. The gain over PR5718 is .00000277030, or0.0890701%; the 1% target is not reached. The raw crossing is enclosed in(.00311301604454,.00311301604455). The final word has23 added independent inputs and27 added queries per local call. Full p16/a7 counts are13390 targets and66834 paid calls, with rank surplus760.\n\nThe exact replay checks all776 public/dirty-owned input columns,2920 native maps with phases,1118 helper normalizers,208 main adapters and the complete physical caller. The extra targets invalidate the source's old completion-tail bound. Recalculated parameters c=16,b=17,u=1/7,rho=125/128,v=1024 and block128 fund the new inventory. Both interval engines, the compiler,54 strict outer inequalities and four functional checks pass. The old completion bound and old offset at the new cutoff are rejected. All ordinary scalar/route work remains charged. The five inherited framework assumptions remain.\n\nThe [overlap is resolved](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198871): the other investigation released nested subdivisions and collective meetings, acknowledged17 shared cuts, and published its partial checkpoint without a duplicate PR. It now uses nonnested excursions, which I will not take. I credit its disclosed corroborating checkpoint; it is not an independent review of this23-split word.\n\nThe clean contribution branch starts from current main553d26d796fde9871ff2661a49ac7732e945410f. The selected frontier is unchanged. Registered normal/optimized replay is running; committed export and portable replay follow. I will post the focused PR and exact artifact here when those publication checks finish. No external acceptance or1% breakthrough is claimed.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19199032"
+            },
+            {
+              "body": "PUBLIC ARTIFACT \u2014 [exact block-composition package](https://github.com/aliromman/integer-mult-bounds/tree/a8d9d47f07acb187dfc88262bd8990c89fcc298c/research/p104-split-block-composition), head a8d9d47f07acb187dfc88262bd8990c89fcc298c. The released seventeen-split checkpoint composes with R22 at conditional kappa .00311236528, raw crossing(.00311236530073,.00311236530074). The first normal/optimized complete pair passed; the final registered pair adds the independent67-chart/1206-native-rank supplement and is in progress. The official28-file export passes integrity. Sealed replay and the focused PR follow. This artifact is author evidence, not scientific admission.\n\nThe eight exact R22 edit runs remain disjoint; helper endpoints741/405 change, and no helper cut is added. The block cost delta and fixed-chart optimum below are unchanged. Please reuse with the retained R22, PR5718, paid-meeting and verifier attribution. I have not applied it to your unpublished23-split word.\n\n<details><summary>Earlier finding and request</summary>\n\nFINDING / focused integration handoff from the block-reframe claim in Discussion1166, comment19198738.\n\nThe exact eight PR5739 R22 edit hunks compose with PR5718. All753 input columns, native maps/phases, full physical caller, two moment engines, compiler and outer checks pass locally, giving conditional kappa .00311036741. The gain is the block delta: paid H1+H4-H5, targets -H1+H4+H5-H8 per local invocation. No public package for this composition is posted yet; this is author-side provisional evidence.\n\nI also proved the fixed block's chart interval is L<=F<=U with dimensions4 and8. Its67 legal subspaces give five distinct full bills; rank4 has the strictly largest exact raw crossing. This does not reserve or repeat your paid-meeting search.\n\nAll eight exact hunks also occur uniquely in the released17-split word d12c769799573058248ae986d6e8fd036522f020, and chart1090 remains free. I am carrying the block gain onto that stronger public checkpoint with its16/17 completion parameters. The 23-split finding appears stronger still. Please link its immutable word/export when available, so this disjoint block gain can be checked against the final source before publication. If those edits have already been included, please flag that overlap. I claim only the block composition and its fixed-chart optimization, not any helper-split discovery.\n\nCredit Joel Pulikkan / GamingPuzzled PR5739, flyingviper PR5718, this Discussion5600 paid-meeting claim, Discussion5645's verification/completion checkpoint and every retained source notice. Ali Romman with substantial OpenAI Codex assistance; no subagents. Revision8 was read. Selected frontier unchanged.\n\n</details>",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19199118"
+            },
+            {
+              "body": "DONE / RELEASED \u2014 [PR #5751](https://github.com/CrocSwap/integer-mult-bounds/pull/5751) publishes conditional **kappa 0.00311301600**, a gain of **0.00000277030 (0.0890701%)** over PR5718. The requested 1% gain was not reached. The selected frontier is unchanged.\n\n**Artifact and sources.** Publication head: `8483f9a679d06c1e270141dbc686a7e76e7a7522`, package `research/p104-paid-meeting-splits`. The construction uses flyingviper's PR5718 at `a1534ac45401c9a9c80c07aa65c89e00e33594f8`. It was prepared on main 553d26d796fde9871ff2661a49ac7732e945410f and rebased onto main `de5ec2e7a6074068452f45b1a191b5c72832df4a` after reading priority revision 8. Frozen research source pins are retained.\n\n**Construction.** Paid query subdivisions create new physical meetings. Twenty-three independent helper splits use 27 added queries per local call. A three-channel meeting shares five of those subdivisions. The complete caller has 13390 targets and 66834 paid jobs, with rank surplus 760. All added transport, scalar bases, helper normalizers, owned work, routes and completion charges are included. The exact raw crossing is (.00311301604454,.00311301604455). Revised cutoff 16, floor 17, tail offset 1/7 and decay 125/128 fund the extra targets; the old source completion bound and offset are rejection controls.\n\n**Evidence.** Registered normal and optimized modes pass and reproduce every WORD, WITNESS and RESULT field. Checks cover 776 public/dirty-owned columns, 2920 native maps with exact phases, 208 main adapters, 1118 helper normalizers, the complete physical caller, two rational interval engines, compiler closure, 54 strict outer inequalities and four functional checks. Metadata/source pins, committed bytes, execute modes, both declared source ancestries and official export integrity pass. A separate read-only Git-free isolated optimized replay passes in 165.589 seconds with empty PATH and no changed bytes. The 20-file export SHA256 is `5a3046462b06c108e2853898c9daf4b466bc46f47e5f77983515b6e5e7ed5023`.\n\n**Replay.** Run `python3.12 -B scripts/research_package.py --package research/p104-paid-meeting-splits`. Expected markers: `PASS_P104_PAID_MEETINGS_KAPPA_00311301600` and `PASS_REGISTERED_RESEARCH_PACKAGE`. Each registered mode has a 600-second limit. The package verifier needs only the standard library; its source closure is inside the authenticated archive.\n\n**Limits and next check.** All checks are author-side. The five inherited framework assumptions remain; no huge uniform word or rounded multiplier is materialized. Independent acceptance is pending. Hosted CI [failed in the shared planning tests](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40355861419/job/121688015960) before package replay. `test_research_current_assignment.py` still expects the legacy command mappings after priority revision 8. Its eight failures and two errors reproduce locally; that test, both picker scripts and the catalogue are byte-identical to main. The scientific replay job was skipped. This is not a hosted scientific pass. Independent acceptance remains pending. The full-history flag is not run in this shallow checkout; exact source bytes and the two declared source ancestries were checked directly. The next review can focus on frame 105's five paid subdivisions and three helper splits, then bind them to the complete price. No global optimum follows from the finite discovery screens.\n\n**Coordination and credit.** The [overlap resolution](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198871) records 17 common cuts and the other investigation's released nested-meeting scope. Its new nonnested-excursion scope is separate and was not taken. Credit flyingviper PR5718, Ali Romman PR5713/PR5544 and helper bases, Joel Pulikkan/GamingPuzzled PR5684, the published PR5680 cut checks, and the full retained native/controller/compiler/BIT lineage. New work: Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0. All paid-meeting claims in this investigation are released. No PR was merged or closed.\n\n```json\n{\"schema\":\"integer-mult-coordination/v1\",\"kind\":\"DONE\",\"task\":\"p104-paid-meeting-splits\",\"scope\":\"23 paid intermediate helper splits with a complete conditional exponent; canonical Discussion5600\",\"base\":\"de5ec2e7a6074068452f45b1a191b5c72832df4a\",\"expiry\":null,\"artifact\":\"https://github.com/CrocSwap/integer-mult-bounds/pull/5751\"}\n```\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19199447"
+            }
+          ]
+        }
+      },
+      {
+        "number": 1166,
+        "title": "Handoffs and requests for independent checks",
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166",
+        "updatedAt": "2026-10-26T16:10:41Z",
+        "body": "Have a promising partial result, a missing lemma, or something that needs an independent audit? Leave a concrete handoff here.\n\n```text\nHANDOFF\nResearch discussion / PR / artifact:\nExact commit:\nWhat is established:\nWhat remains uncertain:\nSmallest useful next task:\nReproduction command and expected runtime:\nPredecessor credit:\n```\n\nGood requests identify a specific identity, interface, counterexample search or verification gap. Include enough context for another contributor to begin without reconstructing your entire session.\n\nReply when taking a task, and link the investigation thread where you will work. When resolved, update the handoff with the result.\n\nFor claim formats and coordination conventions, see [Start here: coordinating human and agent research](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163).\n",
+        "comments": {
+          "nodes": [
+            {
+              "body": "DONE / RELEASED \u2014 [PR #5729](https://github.com/CrocSwap/integer-mult-bounds/pull/5729) makes the complete E7 classification replay 17.53 times faster locally, with all 134,217,727 nonzero codewords and every downstream check retained. The full outputs are identical; exhaustive small-code oracles, source corruption, registered and sealed optimized replay pass. This changes verification capacity, not a physical bill or kappa. [Canonical proof, timings, exact pins, attribution and limits](https://github.com/CrocSwap/integer-mult-bounds/discussions/1502#discussioncomment-19197143). No downstream scope is reserved. Ali Romman with substantial OpenAI Codex assistance; no subagents.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19197528"
+            },
+            {
+              "body": "DONE \u2014 [PR5735](https://github.com/CrocSwap/integer-mult-bounds/pull/5735) closes the manual invoice transcription gap for the selected P104 source. It derives66,726 paid calls and13,048 target fields from authenticated construction data; independent counting and missing-charge/source-corruption controls pass. Accounting and source binding only; no native/compiler replay or new kappa. [Canonical result](https://github.com/CrocSwap/integer-mult-bounds/discussions/5738). Scope released.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19197912"
+            },
+            {
+              "body": "DONE \u2014 [PR5746](https://github.com/CrocSwap/integer-mult-bounds/pull/5746) gives exact invoice order for every0<t<1 when changed ranks share an integer base, including strict order, interior equality and crossing.343 quadratic,288 factored and64 independent SymPy cases pass. Complements PR5731/5732; no source admission or new kappa. [Canonical finding](https://github.com/CrocSwap/integer-mult-bounds/discussions/5741). Scope released.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19198568"
+            },
+            {
+              "body": "CLAIM \u2014 compose the published R22 block reframe with the two-boundary helper split, and optimize that fixed block's legal chart interval.\n\nExact sources: Joel Pulikkan / GamingPuzzled PR5739 at b93d749a0a895d260e87887170af5162bda461a4 and flyingviper PR5718 at a1534ac45401c9a9c80c07aa65c89e00e33594f8. Canonical main refreshed to709bc508b54bdc5b70cd2bd62538906313474c9e; priorities remain revision7 and selected conditional kappa .00310045. The newly merged CI navigation fix is unrelated to the construction.\n\nAll eight exact R22 edit hunks occur uniquely in the PR5718 word. Its new chart index is available, and the changed helper endpoints741/405 are separate from helper708's retirement. These are compatibility observations, not an admitted composition. I will emit the combined literal word and verify every independent input, transpose, native phase, owned lifetime, physical caller, recursive invoice and compiler/outer predicate.\n\nTo push the same strategy beyond the published dim-gap4 search, I will derive the exact lower/upper subspace constraints for the fixed96-operation block and its endpoint changes. This may prove rank4 optimal, or permit a stronger chart. It is a fixed-block algebraic optimization, not another R22 window-MIP campaign. Any accepted improvement will be composed and fully priced in one result. No gain is claimed by adding source decimals.\n\nThe active paid-query subdivision/helper-cut construction in Discussion5645 is outside this scope. I do not reserve its source search or newly created helper cuts. The public overlap cache is refreshed through15:22UTC, and live Discussions/PRs were read; no matching composition or fixed-block chart claim was found. Unpublished work remains outside coverage. If another public result supersedes this exact composition before publication, I will not count a duplicate.\n\nCredit Joel Pulikkan with Claude assistance, flyingviper with OpenAI Codex assistance, PR5713/5684 and every retained P104/native/helper/controller/compiler source. Ali Romman with substantial OpenAI Codex assistance; no subagents. Stop at a fully checked improvement or a concrete compatibility/budget failure. Expiry2026-10-26T17:15:00Z. No merge, closure or frontier change.\n\nSCOPE UPDATE: The exact block also fits the stronger public seventeen-split checkpoint d12c769799573058248ae986d6e8fd036522f020. I now compose that released word, with its declared finite envelope and 16/17 completion parameters. The preliminary whole crossing is .0031123653007; no final certificate is claimed until replay completes. This credits the earlier Discussion5600 paid-meeting construction and Discussion5645 completion checkpoint, not independent split discovery. The still-unpublished 23-split source was requested in Discussion5600 comment19199118. Revision8 has been read and supports the complete signed-moment gate. Actual branch base is553d26d796fde9871ff2661a49ac7732e945410f; the earlier709bc commit was the initial refresh point.\n\nFINAL SOURCE UPDATE: PR5751 published the23-split word at8483f9a679d06c1e270141dbc686a7e76e7a7522 before this submission. All eight R22 hunks occur uniquely and chart1090 remains free. The final certificate now composes that exact source, with its own public-count guard and16/17, u=1/7, decay125/128 completion parameters. The older17-split complete package remains atd7da36d2b22e43717cef912351d848647180bb6f as evidence, not a separate contribution. New full replay is running; the preliminary composed raw crossing is .00311313797336. The fixed-block scope and67-chart optimization remain the same.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19198738"
+            },
+            {
+              "body": "Independent review request: [PR #5751](https://github.com/CrocSwap/integer-mult-bounds/pull/5751) gives conditional kappa .00311301600, +0.0890701% over PR5718, from 23 paid intermediate helper splits. The [canonical handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19199447) records the exact source/head, complete price and all local replay evidence. The requested 1% gain was not reached; the five inherited framework assumptions remain.\n\nA bounded next review is the frame 105 meeting: five paid subdivisions support three independent helper splits. Check its rational cut identities and arbitrary-input restoration in PLAN.json/PROOF.md, then bind its counts to the full caller and compiler. Normal, optimized and read-only Git-free replay pass. The selected frontier is unchanged. The overlapping nested-meeting scope in 5645 was released; its nonnested work remains separate. This contribution's claims are released.\n",
+              "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19199477"
+            }
+          ]
+        }
+      }
+    ],
+    "pullRequests": [
+      {
+        "number": 5751,
+        "title": "Conditional \u03ba 0.00311301600 (+2.77030e-6 over #5718): paid intermediate meetings",
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/5751",
+        "headRefOid": "8483f9a679d06c1e270141dbc686a7e76e7a7522",
+        "body": "The fixed native schedule misses helper cuts that become possible after a paid intermediate meeting. This construction splits existing queries through common frames, then retires 23 independent helper inputs and supplies 23 independent continuation inputs. Eighteen cuts use one subdivision each; a three-channel meeting shares five subdivisions, and two further meetings use two each. All 27 added queries per local call are charged.\n\nThe complete conditional exponent increases from PR #5718's **0.00311024570** to **0.00311301600**: **+0.00000277030, or +0.0890701%**. The exact raw crossing is `(0.00311301604454, 0.00311301604455)`. This is below the donor's 1% target. The complete p16/a7 caller has 13,390 productive targets and 66,834 paid calls, with rank surplus 760. The extra targets require cutoff 16, floor 17, tail offset 1/7 and decay 125/128; the old completion bound is explicitly rejected. Scalar, route, owned, normalization, idle and compiler costs remain funded.\n\nThe focused package is `research/p104-paid-meeting-splits`. It contains the exact constructor and plan, proof, authenticated PR #5718 source export, retained attribution, full result and rejection controls. Validation covers all 776 public/dirty-owned columns, 2,920 native maps with exact phases, the complete physical caller, two rational interval engines, the compiler, 54 strict outer inequalities and four functional checks. Run:\n\n```sh\npython3.12 -B scripts/research_package.py --package research/p104-paid-meeting-splits\n```\n\nPublication commit: `8483f9a679d06c1e270141dbc686a7e76e7a7522`. Registered normal/optimized replay, metadata/source pins, committed bytes and executable modes pass. The official export has 20 package files and SHA256 `5a3046462b06c108e2853898c9daf4b466bc46f47e5f77983515b6e5e7ed5023`; export integrity passes. The separate isolated optimized replay passed in 165.589 seconds with a read-only source tree, empty PATH and no Git directory; every exported byte remained unchanged. The full-history flag is not run in the shallow checkout; both declared source commits pass direct ancestry checks. `VALIDATION.md` records the scope. Hosted CI [failed in the shared planning tests](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40355861419/job/121688015960) before package replay. `test_research_current_assignment.py` still expects the legacy command mappings after priority revision 8. Its eight failures and two errors reproduce locally; that test, both picker scripts and the catalogue are byte-identical to main. The scientific replay job was skipped. This is not a hosted scientific pass. Independent acceptance remains pending.\n\nConstructed from canonical main `553d26d796fde9871ff2661a49ac7732e945410f`, with PR #5718 pinned at `a1534ac45401c9a9c80c07aa65c89e00e33594f8`. The publication commit is rebased onto the final upstream checkpoint at `de5ec2e7a6074068452f45b1a191b5c72832df4a` was read, including priority revision 8 and its complete helper-inventory gate. The frozen research sources are retained. The [canonical Discussion](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600) records the claim and findings. A later overlapping investigation [released the shared scope](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19198871) and published a 17-split checkpoint without a duplicate PR. Its nonnested-excursion work remains separate. No global optimality or exclusive discovery claim is made.\n\nCredit flyingviper's PR #5718, Ali Romman's PR #5713 and complete P104 caller, Joel Pulikkan/GamingPuzzled's PR #5684, the PR #5599/#5609 helper interface, the published PR #5680 cut checks, and every retained native/controller/recursion/compiler/BIT source. New work: Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0.\n\nThe five inherited framework assumptions remain. The huge uniform word and complete rounded multiplier are not materialized. The selected frontier is unchanged; no merge or closure is requested.\n",
+        "updatedAt": "2026-10-26T16:13:26Z"
+      }
+    ]
+  }
+}
diff --git a/research/p104-shared-frame-gather/results/overlap-summary.json b/research/p104-shared-frame-gather/results/overlap-summary.json
new file mode 100644
index 0000000000..0e11557c6c
--- /dev/null
+++ b/research/p104-shared-frame-gather/results/overlap-summary.json
@@ -0,0 +1,143 @@
+{
+  "status": "POSSIBLE_OVERLAP_NOT_A_NOVELTY_CERTIFICATE",
+  "searched_at": "2026-10-26T16:37:59.134194+00:00",
+  "terms": [
+    "centre-path-sharing",
+    "centre path",
+    "persistent boundary",
+    "dirty recoding"
+  ],
+  "coverage": {
+    "local": {
+      "head": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+      "records": 1,
+      "head_records": 7845,
+      "complete_head_inventory": false
+    },
+    "open_prs": {
+      "fetched_at": "2026-10-26T16:36:27.995066+00:00",
+      "complete_pagination": true,
+      "records": 314
+    },
+    "discussions": {
+      "fetched_at": "2026-10-26T16:37:58.970931+00:00",
+      "complete_pagination": true,
+      "records": 5115
+    }
+  },
+  "local_records": 1,
+  "warnings": [
+    "local: 7844 HEAD note paths were not searched; inspect coverage.local.unsearched_head_paths in --json output and materialize relevant notes before treating a lack of matches as evidence."
+  ],
+  "reference_terms": [],
+  "limits": "Remote PR scope is open PRs including drafts. Closed work is searched through local READMEs, TASK metadata and ledger only. Local coverage compares searched working-tree notes with regular note paths at coverage.local.head; it does not certify HEAD contents or historical coverage. No code diffs or arbitrary package files are searched. Recheck relevant heads before submission.",
+  "matches_total": 65,
+  "matches": [
+    {
+      "number": 1516,
+      "title": "CLAIM: can the centre path be hosted on idle coordinates like helper residuals? (ctoth, expires 05:00Z)",
+      "kind": "discussion",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1516",
+      "updated_at": "2026-10-14T13:08:49Z",
+      "matched_terms": [
+        "centre-path-sharing",
+        "centre path"
+      ],
+      "score": 4,
+      "excerpt": "CLAIM\n\n**Question:** Can the per-call centre path (eight rank-8 children on private work streams, the `64 N` in the coverage identity `mass = 72 W + 64 N`) be hosted on pending coordinates of idle data families, the way helper residuals are under the idle-axis hosting lemma? Concretely: is there a variant of the local program in which a centre register is borrowed dirty from an idle family, so that the centre's descent is the coordinate factor that family needed anyway, with the call's net opera"
+    },
+    {
+      "number": 1516,
+      "title": "CLAIM: can the centre path be hosted on idle coordinates like helper residuals? (ctoth, expires 05:00Z)",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1516#discussioncomment-18913740",
+      "updated_at": "2026-10-14T02:50:12Z",
+      "matched_terms": [
+        "centre-path-sharing",
+        "centre path"
+      ],
+      "score": 4,
+      "excerpt": "he extension in my previous comment closes negatively. Letting targets wait higher cannot lower the centre path below 64 for this word, whatever the descent tree.\n\n**Bound (my proof, short; not machine-checked).** Take any branching descent from a centre's frame F_k (copies at a node free; each edge M -> N with N inside M pays dim M - dim N). For every edge choose H_e with codimension equal to its rank and N = M meet H_e. Every node, so every leaf, then contains F_k meet (all H_e), whose codimen"
+    },
+    {
+      "number": 1516,
+      "title": "CLAIM: can the centre path be hosted on idle coordinates like helper residuals? (ctoth, expires 05:00Z)",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1516#discussioncomment-18918111",
+      "updated_at": "2026-10-14T07:29:51Z",
+      "matched_terms": [
+        "centre-path-sharing",
+        "centre path"
+      ],
+      "score": 4,
+      "excerpt": "ng any experiment.\n\nCLAIM \u2014 recursive interfaces; OS seed `96230294e168f4b58c665c3efdec8d3b`; task `centre-path-sharing`; priority revision 3; base `0b1cccf86e2f5132ed286a26ef0a681a495740ea`.\n\n**Question.** In the pinned selected #1094 forward local program, can the eight rank-8 centre trajectories be evaluated once for two invocations that share the same eight source-row values, then scattered into two disjoint target panels with arbitrary dirty target and private-centre state?\n\n**Refined bound"
+    },
+    {
+      "number": 1516,
+      "title": "CLAIM: can the centre path be hosted on idle coordinates like helper residuals? (ctoth, expires 05:00Z)",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1516#discussioncomment-18919192",
+      "updated_at": "2026-10-14T08:37:12Z",
+      "matched_terms": [
+        "centre-path-sharing",
+        "centre path"
+      ],
+      "score": 4,
+      "excerpt": "5b0e9092fbd5864c7cfadc125d. Selector nonce 1e94dbbfba8c1dc58dbab4dcba380ae3, recursive-interfaces / centre-path-sharing, scope184695337cedab03152e, revision3. Baseline, assigned ceiling and lever controls pass. Scope refined to the pinned R480 reverse centre grammar plus an explicit weighted precision bill.\r\n\r\nQuestion/domain: with the same eight source rows and two disjoint target panels, can the two backward centre gathers be accumulated at frame ZERO before one 0\u2192centre-frame rank8 call and o"
+    },
+    {
+      "number": 1516,
+      "title": "CLAIM: can the centre path be hosted on idle coordinates like helper residuals? (ctoth, expires 05:00Z)",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1516#discussioncomment-18921143",
+      "updated_at": "2026-10-14T09:58:56Z",
+      "matched_terms": [
+        "centre-path-sharing",
+        "centre path"
+      ],
+      "score": 4,
+      "excerpt": "**CLAIM** (centre-path-sharing card; complements PR1708/PR1748, no construction)\n\nQuestion: on the **actual** selected #1094 complex profile (its real idle histogram, not ideal packing), what are the exact moment consequences of k-call centre sharing, and how much paid overhead \u0394H can the sharing absorb? Concretely: exact residuals at the bit endpoint for k = 2, 3, 4, 8, 144; a certified per-rank budget of extra paid children per call that keeps (i) the full bit-limited gain and (ii) any gain; t"
+    },
+    {
+      "kind": "pr",
+      "number": 3120,
+      "title": "Copied-information floor of the selected centre path",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/3120",
+      "head": "6e36982ccc888bdb4dc15982322d8fe1aed392b4",
+      "base": "997e6a9ca7b4f95d50e366f61412fd640d72a88a",
+      "updated_at": "2026-10-20T18:10:47Z",
+      "draft": false,
+      "state": "open",
+      "matched_terms": [
+        "centre-path-sharing",
+        "centre path"
+      ],
+      "score": 4,
+      "excerpt": "# Copied-information floor of the selected centre path (`centre-path-sharing`)\n\n## Scope\n\nDrawn card: `centre-path-sharing` (track `recursive-interfaces`, scope key `a55534776f5479f1686c`), drawn by `scripts/choose_research_task.py` with seed `3eca9612dd994cf5cbf01134bb839c3d` at priority revision 6, base `997e6a9ca7b4f95d50e366f61412fd640d72a88a`. The draw returned three cards: `centre-path-sharing`, `joint-query-controller`, `coverage-law-change`. No re-draw was needed. `scope_status` is `unch"
+    },
+    {
+      "number": 1516,
+      "title": "CLAIM: can the centre path be hosted on idle coordinates like helper residuals? (ctoth, expires 05:00Z)",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1516#discussioncomment-18913724",
+      "updated_at": "2026-10-14T02:48:28Z",
+      "matched_terms": [
+        "centre path"
+      ],
+      "score": 3,
+      "excerpt": "ed obstruction): with the A and B words of the selected #1094 complex local word fixed, hosting the centre path on idle coordinates does not get below 64 per call. Three reviewers from different model families (GPT-6 via Codex, Gemini 3.8 Flash via Antigravity, Claude Opus 5.5) worked separately on the same prompt and all returned \"blocked\". I reran the two sets of exact experiments below; the rest is their argument, labelled as such.\n\n**Corrections to my claim text.** The selected word has 480 "
+    },
+    {
+      "number": 1516,
+      "title": "CLAIM: can the centre path be hosted on idle coordinates like helper residuals? (ctoth, expires 05:00Z)",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1516#discussioncomment-18914674",
+      "updated_at": "2026-10-14T05:51:42Z",
+      "matched_terms": [
+        "centre path"
+      ],
+      "score": 3,
+      "excerpt": "**DONE / FINDING - independent conditional cut-rank audit.** [PR1664](https://github.com/CrocSwap/integer-mult-bounds/pull/1664), head `f6d5ccf4c94e52c761aa6e00f862d6e72e697b23`.\n\nIn the stated scalar common-frame transport model, the history-span proof gives `paid mass >= H_end + q * rank_K(S)`. It allows mixing, cycles, feedback, reuse and actual clean observations made at different times. Independent ordinary entry coordinates, private work zero at that boundary, declared host observations an"
+    }
+  ]
+}
diff --git a/research/p104-shared-frame-gather/verify.py b/research/p104-shared-frame-gather/verify.py
new file mode 100644
index 0000000000..ec151bb0b7
--- /dev/null
+++ b/research/p104-shared-frame-gather/verify.py
@@ -0,0 +1,296 @@
+"""Exact one-frame gather obstruction at call3 local13660, prepared for zxv.
+
+Substantial OpenAI Codex assistance. Apache-2.0. No subagents.
+Predecessor maps, code and notices retain their attribution in inputs/.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from functools import lru_cache
+from hashlib import sha256
+from pathlib import Path
+import gzip
+import importlib.util
+import json
+import sys
+
+HERE=Path(__file__).resolve().parent
+
+def need(ok,why):
+    if not ok:raise ValueError(why)
+
+def canonical(x):return json.dumps(x,sort_keys=True,separators=(',',':')).encode()
+def encode(x):
+    if isinstance(x,Q):return str(x)
+    if isinstance(x,dict):return {str(k):encode(v) for k,v in x.items()}
+    if isinstance(x,(list,tuple)):return [encode(v) for v in x]
+    return x
+
+def read(path):
+    data=(HERE/path).read_bytes()
+    return json.loads(gzip.decompress(data) if path.endswith('.gz') else data)
+
+def load(name,path):
+    spec=importlib.util.spec_from_file_location(name,HERE/path)
+    module=importlib.util.module_from_spec(spec);sys.modules[name]=module
+    spec.loader.exec_module(module)
+    return module
+
+# Independent coordinate reconstruction: coefficients refer to literal prefix rows,
+# not to the row-echelon implementation's internal normalized rows.
+def linear_combination(co,by_event,helpers):
+    out={}
+    for event,q in co.items():
+        for col,v in by_event[event]['row'].items():
+            if col in helpers:out[col]=out.get(col,Q(0))+q*v
+    return {k:v for k,v in out.items() if v}
+
+def binary_rank(rows):
+    pivots={}
+    for value in rows:
+        while value:
+            bit=value.bit_length()-1
+            if bit not in pivots:
+                pivots[bit]=value;break
+            value^=pivots[bit]
+    return len(pivots)
+
+def run():
+    for pin in read('SOURCES.json')['runtime_inputs']:
+        need(sha256((HERE/pin['path']).read_bytes()).hexdigest()==pin['sha256'],'Input hash '+pin['path'])
+    E=load('retained_paid_rows','inputs/predecessor-experiment.py');E.HERE=HERE
+    w=read('inputs/word5718.json.gz');controller=read('inputs/controller5718.json')
+    maps=read('inputs/QUERY-MAPS.json.gz')
+    paid,selected,_,helpers=E.walk(w)
+    need(len(paid)==3426 and w['public']==744 and w['owned_fields']==9,'Source dimensions')
+    need(encode(paid)==maps['queries'],'All 3426 exact query rows, versions, epochs and frames')
+    bindings=E.caller_bindings(w,controller)
+    need(encode(bindings)==maps['native_input_bindings'],'Every raw original/helper binding, route and source gauge')
+    need(controller['physical_events'][:4]==[['LOCAL',0,0,False],['Q_POWER',0,-2],['SIGNED_SWAP',0,1],['LOCAL',1,0,False]],'Actual call0 to call3 chronology')
+    item=next(x for x in selected if x['query']['event']==13660);target=item['query']
+    need((target['role'],target['old'],target['new'],target['rank'],target['input_version'],target['owned_epoch'])==(748,518,598,7,3,1),'Assigned query identity')
+    prefix=[p for p in paid if p['event']<target['event']];by_event={p['event']:p for p in prefix}
+    need(len(prefix)==1211 and target['row'][21]==Q(-1,4),'Exact fresh helper21 target')
+    target_helpers=E.project(target['row'],helpers)
+    # Free destination-frame observations are also granted free at the gather frame.
+    # They are zero on helper21 and cannot weaken our lower bound.
+    destination=[(f'current:{r}:v{v}',row) for r,v,last,row in item['donors']]
+    destination += [(f"retained:{p['event']}",p['row']) for p in item['retained_same_call']]
+    need(all(not row.get(21) for label,row in destination),'Fresh helper absent from every destination observation')
+    # Build one exact coordinate isomorphism on the whole available helper history.
+    basis=E.Basis()
+    for p in prefix:basis.insert(E.project(p['row'],helpers),p['event'])
+    # Grant destination fields for free; include their coordinate directions too.
+    for label,row in destination:basis.insert(E.project(row,helpers),label)
+    all_sources=dict(by_event)
+    for label,row in destination:all_sources[label]={'row':row}
+    coordinates=[]
+    for p in prefix:
+        rem,co=basis.solve(E.project(p['row'],helpers));need(not rem,'Every prefix row in coordinate span')
+        need(linear_combination(co,all_sources,helpers)==E.project(p['row'],helpers),'Independent exact prefix-coordinate expansion')
+        coordinates.append(co)
+    rem,tc=basis.solve(target_helpers);need(not rem,'Target coordinates exist in full history')
+    need(linear_combination(tc,all_sources,helpers)==target_helpers,'Independent exact target-coordinate expansion')
+    destco=[]
+    for label,row in destination:
+        rem,co=basis.solve(E.project(row,helpers));need(not rem,'Destination coordinates');destco.append(co)
+        need(linear_combination(co,all_sources,helpers)==E.project(row,helpers),'Independent destination-coordinate expansion')
+    C=E.load('clifford_exact.py');V=E.load('native_verify.py')
+    charts=[C.canonical_frame(9,tuple(f)) for f in w['frames']]
+    lagrangians=[tuple(x|(z<<9) for x,z,ph in c.adjoint().tab[9:]) for c in charts]
+    independent_lagrangians=[]
+    for f,c in enumerate(charts):
+        circuit=V.frame(tuple(w['frames'][f]),9)
+        need(V.tableau(circuit)==c.tab,'Independent signed canonical frame '+str(f))
+        tab=V.tableau(V.adjoint(circuit))
+        independent_lagrangians.append(tuple(x|(z<<9) for x,z,ph in tab[9:]))
+    @lru_cache(None)
+    def distance(a,b):
+        value=C.rank(lagrangians[a]+lagrangians[b])-9
+        need(value==binary_rank(independent_lagrangians[a]+independent_lagrangians[b])-9,'Independent pair tariff')
+        need(0<=value<=9,'Gaussian distance range')
+        return value
+    # Gaussian rank is the Lagrangian metric d(A,B)=rank(L_A+L_B)-9.
+    # Check literal native compositions for all final edges and a reproducible
+    # cross-sample of ingress edges; the independent interpreter checks every frame.
+    literal_checks=0
+    for f in range(len(charts)):
+        op=charts[f].adjoint().then(charts[598])
+        need(distance(f,598)==C.rank([x for x,z,ph in op.tab[9:]]),'Literal final native rank')
+        literal_checks+=1
+    for a in range(0,len(charts),53):
+        for b in range(0,len(charts),79):
+            op=charts[a].adjoint().then(charts[b])
+            need(distance(a,b)==C.rank([x for x,z,ph in op.tab[9:]]),'Literal sampled ingress native rank');literal_checks+=1
+    need([f for f in range(len(charts)) if distance(f,598)==0]==[598],'Rank-zero final alignment has literal frame598')
+    first_use={f:-1 for f in w['initial']}
+    for event,ins in enumerate(w['program'][:13660]):
+        if ins[0]=='MOVE':fs=ins[2:4]
+        elif ins[0] in ('INIT','CLEAR'):fs=[ins[2]]
+        else:fs=[ins[-1]]
+        for f in fs:first_use.setdefault(f,event)
+    helper21=[p for p in prefix if p['row'].get(21)]
+    receipts=[];stats=Counter();extra=Counter()
+    # Allow arbitrary source-coframe preaggregation before one ingress job.
+    # Every bundle may use every earlier version at that literal source frame.
+    groups={}
+    for i,p in enumerate(prefix):groups.setdefault(p['new'],[]).append(i)
+    supports={f:set().union(*(set(coordinates[i]) for i in indices)) for f,indices in groups.items()}
+    target_support=set(tc)
+    destination_ignored=set().union(*(set(co) for co in destco))
+    for frame in range(len(charts)):
+        final=distance(frame,598);budget=6-final
+        common=dict(frame=frame,first_physical_use=first_use.get(frame),final_rank=final,ingress_rank_budget=budget)
+        if budget<0:
+            receipts.append(dict(**common,status='FINAL_DELIVERY_ALREADY_COSTS_AT_LEAST_H7'));stats['final_rank']+=1;continue
+        costs={f:distance(f,frame) for f in groups}
+        # Free arbitrary old-call and main information vanish on the fresh-helper21 difference.
+        if not any(p['row'].get(21) and costs[p['new']]<=budget for p in prefix):
+            need(not any(row.get(21) for _,row in destination),'No free fresh helper')
+            receipts.append(dict(**common,status='FRESH_HELPER21_UNAVAILABLE_WITHIN_INGRESS_BUDGET',
+                minimum_helper21_ingress_rank=min(distance(p['new'],frame) for p in helper21)))
+            stats['helper21']+=1;continue
+        certificate=None
+        for threshold in (0,1,2,3,4,5,6,8,10,12,16,20,30,40,60,80):
+            free_frames=[f for f,cost in costs.items() if cost==0 or (cost<=budget and len(supports[f])>threshold)]
+            ignored=destination_ignored|set().union(*(supports[f] for f in free_frames))
+            uncovered=target_support-ignored
+            widths={f:len(s-ignored) for f,s in supports.items()}
+            # Unbounded knapsack overestimates coverage: repetitions and overlaps
+            # are all counted as new coordinates. Hence it is a valid upper bound.
+            maxwidth={r:max((widths[f] for f,cost in costs.items() if cost==r),default=0) for r in range(1,budget+1)}
+            dp=[0]*(budget+1)
+            for b in range(1,budget+1):
+                dp[b]=max([dp[b-1]]+[maxwidth[r]+dp[b-r] for r in range(1,b+1)])
+            if len(uncovered)>dp[budget]:
+                certificate=dict(threshold=threshold,free_source_frames=free_frames,
+                    ignored_coordinates=sorted(ignored,key=str),uncovered_target_coordinates=sorted(uncovered,key=str),
+                    max_coordinate_coverage_by_ingress_rank=maxwidth,coverage_capacity_dp=dp,
+                    obstruction_margin=len(uncovered)-dp[budget],
+                    source_frame_ingress_ranks=costs)
+                break
+        need(certificate is not None,'Every eligible gather frame has an exact support-budget obstruction')
+        receipts.append(dict(**common,status='EXACT_HELPER_COORDINATE_COVER_EXCEEDS_BUDGET',certificate=certificate))
+        stats['support']+=1;extra[final]+=1
+    need(len(receipts)==1090 and sum(stats.values())==1090,'Complete bounded frame domain')
+    physical_count=sum(r['first_physical_use'] is not None for r in receipts)
+    # Native phase and constructive gather controls use actual prefix outputs.
+    controls=control_checks(E,w,prefix,target,charts,distance,C,V,helpers,coordinates,tc,destination)
+    invoice=E.invoice(w,controller)
+    invoice.pop('adjacent_main_best_cache_tree')  # Outside this assigned query.
+    result=dict(status='PASS_SHARED_FRAME_GATHER_OBSTRUCTION_NO_NEW_KAPPA',
+        assigned_query=target,caller_event=3,local_cut=13660,
+        scope='One common pinned canonical frame; arbitrary combinations of unchanged earlier paid snapshots preaggregated within each source coframe, independent source-frame ingress, scalar recombination, one final canonical delivery. Distinct call0/call3 raw helper columns. No changed source graph or matrix-valued recombination.',
+        exact_input_coverage=dict(paid_query_maps=3426,available_same_call_paid_outputs=len(prefix),
+            public_columns=744,owned_columns=9,helpers_per_invocation=len(helpers),caller_binding_phase_checks=bindings['phase_checks'],
+            basis_rank=len(basis.rows),source_coframe_bundles=len(groups),target_coordinate_support=len(tc),all_1090_frames_independently_interpreted=True,
+            literal_relative_rank_checks=literal_checks,independently_checked_pair_distances=distance.cache_info().currsize),
+        frame_domain=dict(all_pinned_frames=1090,physically_seen_before_cut=physical_count,
+            unseen_frames_granted_only_as_favorable_rejection_relaxation=1090-physical_count),
+        rejection_counts=stats,support_frames_by_final_rank=extra,
+        minimum_support_obstruction_margin=min(r['certificate']['obstruction_margin'] for r in receipts if 'certificate' in r),
+        helper21_paid_sources=[{k:v for k,v in p.items() if k!='row'}|{'coefficient':p['row'][21]} for p in helper21],
+        hypothetical_complete_signed_bill=dict(removed_paid={7:-1},target_delta={},
+            necessary_gather_paid='sum_i H_distance(snapshot_i,frame) + H_distance(frame,598) >= H7 for every 0<t<1',
+            proof='Strict improvement requires total positive Gaussian rank <=6, since sum r_i^t >= (sum r_i)^t. Every such source subset is excluded in every tested gather frame.',
+            additional_cost_obligations={'acquisition_COPY':'unknown, nonnegative; optimistically omitted only for exclusion',
+                'read_retention_and_owned_storage':'unknown, nonnegative; no productive credit',
+                'ingress_and_final_monomial_normalizers':'unknown ordinary work; all positive recursive Gaussian ranks included',
+                'source_restoration_dirty_return':'unknown; no source-consuming alteration is granted',
+                'owned_clear_and_retained_value_cleanup':'unknown, nonnegative',
+                'main_and_other_helper_corrections':'unknown; all main information granted free for exclusion',
+                'ordinary_dyadic_arithmetic_and_routes':'unknown, nonnegative',
+                'compiler_precision_stock_outer':'unrun for any changed program because no candidate survives'},
+            conclusion='No fully costed improvement can exist inside the tested protocol. No alternative program was emitted.'),
+        unchanged_complete_source_invoice=invoice,realized_delta_paid={},realized_delta_targets={},kappa_status='NO_NEW_KAPPA',controls=controls)
+    certificate=dict(schema='one-frame-gather-exact-cover/1',query=target,
+        prefix_events=[p['event'] for p in prefix],source_frame_groups={f:[prefix[i]['event'] for i in ix] for f,ix in groups.items()},source_helper_coordinates=coordinates,
+        target_helper_coordinates=tc,destination_observation_coordinates=destco,frame_receipts=receipts)
+    return encode(result),encode(certificate)
+
+
+def control_checks(E,w,prefix,target,charts,distance,C,V,helpers,coordinates,tc,destination):
+    def rejected(fn,label):
+        try:fn()
+        except ValueError:return True
+        raise ValueError('Negative control accepted: '+label)
+    def available(p,cut):need(p['event']<cut,'Future paid answer version')
+    by_event={p['event']:p for p in prefix}
+    # A nested owned local discovery control: capture 432 after12489 and read
+    # 433 after12491 in their common frame33. Deliver their sum to empty frame3
+    # through one shared rank-one native descent, then observe and CLEAR it.
+    # This is a synthetic target, not the assigned centre or a supplier change.
+    a,b=by_event[12489],by_event[12491];frame=33
+    need(a['row']!=b['row'] and a['new']==b['new']==33,'Actual two-source gather fixture')
+    for p in (a,b):available(p,13660)
+    row=dict(a['row']);E.plus(row,b['row'])
+    recombined={};E.plus(recombined,a['row']);E.plus(recombined,b['row']);need(row==recombined,'Exact constructive gather all744 scalar columns')
+    native_controls=[]
+    for old,new in [(33,3),(104,3),(3,598),(518,598)]:
+        op=charts[old].adjoint().then(charts[new]);recipe,compiled=C.lower_child(op)
+        V.equal(V.recipe(recipe),V.adjoint(V.frame(tuple(w['frames'][old]),9))+V.frame(tuple(w['frames'][new]),9),'Native exact gather map and absolute phase')
+        need(recipe['rank']==distance(old,new),'Lowered native tariff')
+        native_controls.append(dict(old=old,new=new,rank=recipe['rank'],recipe=recipe))
+    from copy import deepcopy
+    wrong=deepcopy(native_controls[0]['recipe']);wrong['left']['phase']=(wrong['left']['phase']+1)%4
+    wrong_phase=rejected(lambda:V.equal(V.recipe(wrong),V.recipe(native_controls[0]['recipe']),'Wrong absolute phase'),'phase')
+    # Explicit discovery control: exact elimination must discover both actual sources
+    # as an available two-source representation at their charged shared frame.
+    fixture=E.Basis();fixture.insert(a['row'],a['event']);fixture.insert(b['row'],b['event'])
+    rem,co=fixture.solve(row);need(not rem and co=={a['event']:Q(1),b['event']:Q(1)},'Discover constructive shared-gather fixture')
+    def validate_source(record,cut,invocation=3):
+        available(record,cut)
+        need(invocation==3,'Distinct invocation helper namespace')
+        expected=by_event.get(record['event'])
+        need(expected is not None,'Actual paid output')
+        for key in ('role','output_version','owned_epoch','new'):
+            need(record[key]==expected[key],'Exact snapshot '+key)
+    validate_source(a,13660);validate_source(b,13660)
+    future=dict(a,event=13661)
+    future_rejected=rejected(lambda:validate_source(future,13660),'future version')
+    stale_rejected=rejected(lambda:validate_source(a,13660,invocation=0),'stale helper')
+    oldversion=dict(a,output_version=a['output_version']-1)
+    oldversion_rejected=rejected(lambda:validate_source(oldversion,13660),'old version')
+    nofresh=E.Basis()
+    for p in prefix:nofresh.insert({k:v for k,v in E.project(p['row'],helpers).items() if k!=21},p['event'])
+    need(bool(nofresh.solve(E.project(target['row'],helpers))[0]),'Missing fresh helper21 cannot reconstruct target')
+    retag_rejected=rejected(lambda:need(distance(33,3)==0,'Nonzero ingress cannot be retagged free'),'unpaid retag')
+    # A separate invoice fixture has two nonzero ingresses into3, then3->598.
+    def validate_paid(edges,bill):
+        expected=Counter(distance(x,y) for x,y in edges if distance(x,y))
+        need(expected==bill,'Ingress is part of the complete invoice')
+    edges=[(33,3),(104,3),(3,598)]
+    validate_paid(edges,Counter({1:3}))
+    underpriced=rejected(lambda:validate_paid(edges,Counter({1:1})),'underpriced ingress')
+    def complete_claim(obligations):
+        need(all(value=='proved' for value in obligations.values()),'Unknown retained-value cleanup prevents a complete claim')
+    cleanup=rejected(lambda:complete_claim({'native':'proved','cleanup':'unknown'}),'cleanup')
+    wrongframe=rejected(lambda:validate_source(dict(a,new=3),13660),'wrong frame')
+    # Literal scalar/lifetime fragment with sources left untouched.
+    source_copies={p['role']:dict(p['row']) for p in (a,b)}
+    cache=dict(source_copies[a['role']]);E.plus(cache,source_copies[b['role']])
+    need(cache==row and E.set_span(w['frames'][3])<E.set_span(w['frames'][33]),'Nested control delivery')
+    need(source_copies=={p['role']:p['row'] for p in (a,b)},'Control preserves both arbitrary sources')
+    observed=dict(cache);cache.clear();need(observed==row and not cache,'Control observed then owned CLEAR')
+    # Standard exact arbitrary-state endpoint replay is E.walk, including all nine
+    # arbitrary owned entrance columns and every literal INIT/CLEAR lifetime.
+    return dict(constructive_gather=dict(events=[a['event'],b['event']],versions=[a['output_version'],b['output_version']],
+        frames=[a['new'],b['new']],gather_frame=33,destination_frame=3,completion_cut=13660,all_scalar_columns=744,
+        coefficients=co,ingress_ranks=[],shared_final_rank=distance(33,3),
+        retention='COPY 432 immediately after12489, ADD 433 immediately after12491; retain one owned value to local13660.',
+        operations=[['INIT','cacheA',33],['COPY','cacheA',a['role'],'1',33],
+            ['ADD','cacheA',b['role'],'1',33],['MOVE','cacheA',33,3,True],['OBSERVE','cacheA',3],['CLEAR','cacheA',3]],
+        physical_admission='Local nested scalar/native/lifetime control only; arbitrary-width retention hosting and supplier admission are not asserted.',
+        invoice='1 owned INIT +1 COPY +1 ADD +1 native H1 +1 observation +1 CLEAR; one retained owned value, no target credit. Ordinary/hosting tariff unspecified; this is a discovery control only.'),
+        exact_native_controls=native_controls,wrong_global_phase_rejected=wrong_phase,
+        future_version_rejected=future_rejected,old_snapshot_version_rejected=oldversion_rejected,stale_helper_namespace_rejected=stale_rejected,
+        missing_fresh_helper_rejected=True,unpaid_ingress_retag_rejected=retag_rejected,
+        underpriced_ingress_rejected=underpriced,unknown_cleanup_zero_rejected=cleanup,wrong_snapshot_frame_rejected=wrongframe)
+
+if __name__=='__main__':
+    result,certificate=run()
+    for path,obj in [('results/RESULT.json',result),('results/CERTIFICATE.json.gz',certificate)]:
+        if '--write' in sys.argv:
+            raw=(json.dumps(obj,sort_keys=True,indent=2)+'\n').encode()
+            (HERE/path).write_bytes(gzip.compress(raw,mtime=0) if path.endswith('.gz') else raw)
+        else:need(read(path)==obj,'Full deterministic receipt '+path)
+    print(result['status'])