diff --git a/research/p104-a2-paid-rank-exchange/CLAIM.json b/research/p104-a2-paid-rank-exchange/CLAIM.json
new file mode 100644
index 0000000000..d18c1e25d9
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/CLAIM.json
@@ -0,0 +1,35 @@
+{
+  "status": "LOCAL-ONLY author-checked conditional improvement; ready for independent scientific review",
+  "kappa": "155229/50000000",
+  "decimal": "0.00310458",
+  "previous_pr": 5627,
+  "previous_kappa": "155220273/50000000000",
+  "gain": "8727/50000000000",
+  "gain_decimal": "0.00000017454",
+  "raw_crossing": [
+    "77614664639/25000000000000",
+    "310458658557/100000000000000"
+  ],
+  "raw_headroom": [
+    "164639/25000000000000",
+    "658557/100000000000000"
+  ],
+  "local_paid_delta": {
+    "2": 1,
+    "3": -1,
+    "5": -1,
+    "6": 1
+  },
+  "target_delta": {},
+  "full_local_invocations": 18,
+  "complete_paid_calls": 66546,
+  "complete_productive_targets": 12958,
+  "rank_mass_surplus": 760,
+  "scope": "One literal A2 rewrite on the exact PR5627 word under the unchanged five framework assumptions and helper-endpoint contract. Complete uniform word and rounded multiplier are not materialized. No formal-kernel or independent human verification.",
+  "publication": {
+    "published": false,
+    "selected": false,
+    "hosted_ci": "not run"
+  },
+  "reproduce": "python3.12 -B scripts/research_package.py --package research/p104-a2-paid-rank-exchange"
+}
diff --git a/research/p104-a2-paid-rank-exchange/INDEPENDENT-AUDIT.json b/research/p104-a2-paid-rank-exchange/INDEPENDENT-AUDIT.json
new file mode 100644
index 0000000000..226ea3e649
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/INDEPENDENT-AUDIT.json
@@ -0,0 +1,66 @@
+{
+  "status": "PASS_INDEPENDENT_A2_IDENTITY_AND_PRICE_AUDIT",
+  "source_sha256": "162e6f13a6e3805517946309bd6247eaa45b337ab9c39621806b254b415075ec",
+  "candidate_sha256": "b2ab778bdd5d68c62df0b51c12dc33fde93ad89213639a8e7da23e4b9cf6c3dc",
+  "old_gates": [
+    [
+      "ADD",
+      388,
+      715,
+      "-4",
+      825
+    ],
+    [
+      "ADD",
+      340,
+      388,
+      "1",
+      825
+    ]
+  ],
+  "new_gates": [
+    [
+      "ADD",
+      340,
+      715,
+      "-4",
+      396
+    ],
+    [
+      "ADD",
+      340,
+      388,
+      "1",
+      825
+    ],
+    [
+      "ADD",
+      388,
+      715,
+      "-4",
+      127
+    ]
+  ],
+  "old_itinerary_dimensions": [
+    1,
+    6,
+    9
+  ],
+  "new_itinerary_dimensions": [
+    1,
+    3,
+    9
+  ],
+  "paid_delta": {
+    "2": 1,
+    "5": -1,
+    "3": -1,
+    "6": 1
+  },
+  "controls": {
+    "omitted_direct_shear": true,
+    "omitted_last_shear": true,
+    "wrong_direct_sign": true
+  },
+  "scope": "Exact three-column scalar identity; only scalar motif replacement; unchanged physical boundaries/owned instructions; strict source frame chain; complete paid-profile delta. Native, physical caller and outer replays supplied separately by root."
+}
diff --git a/research/p104-a2-paid-rank-exchange/INDEPENDENT-AUDIT.md b/research/p104-a2-paid-rank-exchange/INDEPENDENT-AUDIT.md
new file mode 100644
index 0000000000..f690208abf
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/INDEPENDENT-AUDIT.md
@@ -0,0 +1,74 @@
+# Independent scoped audit of the P104 A2 exchange
+
+Prepared for zxv with substantial OpenAI Codex assistance; Apache-2.0.
+The transport sidequest supplied the construction. Ali Romman's PR5627 and the
+credited P104 scalar/native/helper/controller/compiler lineage remain the source.
+The elementary three-shear identity is not claimed as a new general theorem.
+
+The exact source is PR5627 commit
+`ed4dc88a09a7d201971f6603b8c2074e95dd9c32`. The candidate byte hash is recorded
+in `INDEPENDENT-AUDIT.json` and was read independently from the proposed word.
+
+## Exact scalar identity
+
+Write A=register715, B=register388, C=register340. The old adjacent scalar gates
+are B←B−4A, C←C+B, so the final triple is (A,B−4A,C+B−4A). The proposed triple
+C←C−4A, C←C+B, B←B−4A has the same final values for every input. The audit checks
+all three basis columns over exact rationals. Omitting the direct or final shear,
+or flipping the direct coefficient's sign, is rejected.
+
+After deleting MOVE instructions, the candidate's entire scalar program is exactly
+the source program with this two-to-three replacement. All other scalar gates
+and their order are unchanged. In particular, all COPY, INIT and CLEAR instructions
+and their complete sequence are unchanged. Every initial/final frame, main port,
+public/owned inventory and frame table is unchanged.
+
+## Why different scalar charts do not create a hidden operator assumption
+
+For every legal chart f, let Φ_f be the actual native chart operator, with its
+full phase. Use z_r for a register's value in the common reference coordinates.
+The physical field in chart f is Φ_f z_r. At a coframed scalar gate,
+
+    Φ_f z_r + q Φ_f z_s = Φ_f (z_r+q z_s).
+
+At a paid change a→b, the supplied native map is Φ_b Φ_a⁻¹, so it sends
+Φ_a z_r to Φ_b z_r. These two exact identities prove the frame invariant by
+induction through the chronological program. The proof never commutes physical
+Gaussians through one another, discards a phase, or identifies tensors of sums
+with sums of tensors. Owned operations retain their existing checked contracts.
+
+The root's fresh native replay checks the actual new maps and full phases;
+its physical caller and exact price replay are separate evidence. Under those
+passed checks and the retained all-column native hypothesis, the invariant
+applies at every lifted width. Since both programs have the same scalar endpoint
+map and the same endpoint charts, they have the same physical endpoint map.
+There is no additional native alphabet, input source, owned-zero rule, address
+projection, or normalization assumption in this edit.
+
+## Exact paid change
+
+The altered source715 itinerary goes through the strict nested chart chain
+598⊂396⊂825⊂127, whose dimensions are1,3,6,9. Previously its contacts required
+rank5 then rank3; now they require rank2 then rank6. The independent complete-word
+histogram check gives exactly
+
+    Δpaid = H2 + H6 − H3 − H5.
+
+Every productive target is retained. The rank mass is unchanged:2+6=3+5=8.
+For 0<t<1, strict concavity gives
+
+    3^t−2^t > 6^t−5^t,
+
+hence H2+H6<H3+H5. This is a genuine positive moment change despite adding one
+ordinary ADD. The root's full invoice repeats it at18 local invocations and
+charges the additional ordinary scalar work. It changes the local bill to which
+the fixed-provider price-frontier theorem applies.
+
+## Verdict and limits
+
+No objection was found to retaining the same five framework hypotheses. This is
+an independent agent audit of the changed local algebra, frame implication and
+full paid-profile delta; it is not a new full theorem audit, formal-kernel result
+or independent human review. The root's all-native, caller, compiler and outer
+replays are not repeated here. The proposed κ and its strict rational root
+interval remain the responsibility of that separately checked full calculation.
diff --git a/research/p104-a2-paid-rank-exchange/LICENSE b/research/p104-a2-paid-rank-exchange/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
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+   5. Submission of Contributions. Unless You explicitly state otherwise,
+      any Contribution intentionally submitted for inclusion in the Work
+      by You to the Licensor shall be under the terms and conditions of
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+      Notwithstanding the above, nothing herein shall supersede or modify
+      the terms of any separate license agreement you may have executed
+      with Licensor regarding such Contributions.
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+   6. Trademarks. This License does not grant permission to use the trade
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+      unless required by applicable law (such as deliberate and grossly
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diff --git a/research/p104-a2-paid-rank-exchange/NOTICE b/research/p104-a2-paid-rank-exchange/NOTICE
new file mode 100644
index 0000000000..59d8e0cf31
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/NOTICE
@@ -0,0 +1,415 @@
+A sharper exponent for integer multiplication
+Copyright 2026 Douglas Colkitt
+
+This project includes unmodified source files from OpenAI's manuscript
+"Integer multiplication below n log n" and its accompanying repository:
+https://github.com/openai/math
+Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a
+
+The original manuscript identifies OpenAI as its author. Its Apache-2.0
+license is retained at upstream/LICENSE. The exact source URLs and hashes
+are recorded in upstream/manifest.json. Proposed modifications are supplied
+as separate files under patches/; upstream source files remain unchanged.
+
+The original local research note, scripts, certificates, patches, and
+documentation were prepared by Douglas Colkitt with assistance from OpenAI
+Codex. The community release incorporates work by the contributors identified
+in the notices below; it is not solely Douglas Colkitt's work.
+This project is not an official OpenAI release or an OpenAI endorsement.
+
+Related contributions (not imported in this checkpoint):
+Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset
+motif, finite-alphabet transfer and a different compressed producer:
+https://github.com/CrocSwap/integer-mult-bounds/pull/7
+The present checkpoint claims no priority for that motif. Related compressed
+complex-network work was submitted by eumemic in PR #3:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+The pending contributions and their stronger bounds require their own review;
+this notice does not represent adoption or verification of their claims.
+
+Additional community work, including parallel and superseded submissions, is
+acknowledged in CONTRIBUTORS.md, with pinned submission metadata in
+docs/research/contribution-snapshot.json. These acknowledgements do not imply
+joint authorship of the checkpoint, incorporation of pending code, or verification
+of submitted mathematical claims. Original source-specific notices remain in force.
+
+Community release (accepted by conditional maintainer audit):
+The following original contributor notices apply to the imported PR #39 chain.
+The preceding non-import statements describe the preserved 2^-30 checkpoint.
+
+Ternary five-subset construction and common-four-point resynthesis
+Copyright 2026 Zhihao Chen (jacklightChen)
+
+This is Zhihao Chen's first submission of this specific construction to this
+repository. Research, proof development, code and verification were prepared
+with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by
+the contributor. This is not independent review or an OpenAI endorsement.
+Future research building on this contribution should explicitly acknowledge
+Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and
+retain the preceding contributors' attribution. This notice records research
+provenance; the repository's Apache-2.0 license remains unchanged.
+
+Dependencies retained from eumemic's Claude-assisted PRs #3 and #5:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+https://github.com/CrocSwap/integer-mult-bounds/pull/5
+PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893
+These include complex_circuit.py, complex_network.py, fast_gaussian.py,
+their generators, construction notes and certificates. The Gaussian input
+scaling proof is corrected in this derivative; see the modification notice below.
+Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank
+sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6.
+Their results and ideas are credited separately from this new construction.
+
+Bulk recursion and controlled tensor basis, 8 October 2026
+Copyright 2026 icekylinx
+
+The new projector batching, controlled simultaneous basis, mixed-width tape
+recursions, whole-residual complex recursion, and dependency-path precision
+guard in the batched-23 proof package are contributed by icekylinx, developed
+with substantial OpenAI GPT-6 Astra assistance. Review corrections and
+repository integration were prepared with OpenAI Codex assistance. They build on the retained
+PR #7 finite producers and the preceding contributions credited above.
+The new package retains the repository's Apache-2.0 license and the original
+authors' attribution. It is not an official OpenAI release or endorsement.
+
+Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian
+proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2).
+The P=34p precision budget and asymptotic exponent are unchanged. New batching
+checks are integrated into make verify. Existing attribution and licenses
+are retained.
+
+Partial-swap extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The partial-swap framing, binary retained-total triple construction,
+dependency-preserving positive labels and carrier reuse, unequal (25,23,57)
+common basis, improved complex batching, and exact assembly are contributed
+by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance.
+This extension directly inherits PR #10, commit
+62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters
+modify the retained paired-exclusion and shared-point implementations credited
+above. The original licenses, attribution, and Gaussian correction are retained.
+
+Endpoint-gauge extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The complementary auxiliary endpoint gauges, (32,30,40) controlled basis,
+general nondegenerate binary phase residuals, retained-center complex producer,
+updated path guard and exact assembly are contributed by icekylinx with
+substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits
+f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable
+adapters retain the paired-exclusion and shared-point contributions credited
+above. Original licenses, notices and the Gaussian correction are retained.
+
+Structured-basis and semantic-bulk extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The selected (32,30,36) bit construction, A1/A5 block refinements, fixed
+middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic
+complex centers, complete data-front batching and circuit-specific assembly
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505.
+
+The semantic/bulk compatibility argument, literal scalar charge, product row
+stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23,
+commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints
+are adopted from the same contributor's PR #21,
+commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame
+work and the earlier nested-basis contributions.
+
+Completed-child semantic precision, arbitrary-coordinate routing, phase-cell
+Gaussian inversion and bulk tape transfer are credited to the RaD project
+(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through
+PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof
+sources retain their separate CC0 license and original NOTICE under
+references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI
+Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra
+assistance. These are source attributions, not claims about this session's
+runtime model. Related contributions by Rohan Arun and Aurel Prosz remain
+acknowledged in the imported notices. No exclusive priority or additional
+license condition is asserted.
+
+Copied retained-center extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The copied retained-center schedule in both orientations, its complex-phase
+transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. This extension inherits PR #32 at
+0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work.
+
+The two-stage topology and paid copied-stream endpoint correction are due
+to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab,
+through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition,
+9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil
+Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a.
+
+The contiguous corner method, incidence factorization and exact partition
+certificates are due to Rohan Arun (PR #31), method commit
+02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate
+specializes Dominik Scholz's parameterization (PR #33),
+a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof
+snapshots are preserved in references/copied-centers. Historical OpenAI
+Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to
+the respective imported sources. Existing PR #21/#23 and RaD attribution,
+licenses and the Gaussian correction remain in force.
+
+Copied centers with reversed (23,25) data corners, 2026-10-08
+Rohan Arun with substantial OpenAI Codex assistance supplies the composition,
+exact dimension23 controls and cost/assembly integration. James Chang's PR34
+supplies the reversed boundary family; icekylinx's PR36 supplies copied retained
+centers and the selected complex graph. The balanced assembly remains attributed
+to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan
+Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx,
+eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses.
+This is not a claim of formal verification, new foundational rank identities,
+global optimality or measured runtime speedup.
+
+Fixed middle basis with copied reversed corners (2026-10-08):
+Rohan Arun, with substantial OpenAI Codex assistance. Composition and
+simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34,
+Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures,
+eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed.
+
+Changed-graph composition with copied centers and fixed bases, 8 October 2026
+Prepared with OpenAI Codex assistance.
+The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J
+projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument,
+Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and
+Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods,
+Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc)
+PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40
+reversed integration, modular data tests and both-fixed baseline;
+RaD (hipotures) PR41 changed scalar DAGs, alternating point order and
+independent role compiler; RaD balanced routing/bulk interfaces, and all
+retained predecessors.
+Complete attribution and the conditional proof boundary are recorded in
+research/copied-fixed/PROOF.md; imported originals and source pins are retained.
+Original Apache-2.0 and CC0 notices remain applicable.
+
+Community follow-up integration (PRs #49, #45 and #26), 8 October 2026:
+Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out
+orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction
+composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and
+reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD /
+hipotures's alternating pair order and independent physical role compiler (#41).
+All predecessor attribution and conditional proof interfaces remain in force.
+
+Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic
+proofs, finite tensor execution, profile precision accounting and mixed-center
+reference checks (#45), with substantial OpenAI Codex assistance. See
+research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution.
+Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic
+frame/movement/guard lemmas and independent paired-circuit checks (#26). Original
+source notices and historical snapshots are preserved. Neither formal package
+claims a full formal proof of the multiplication algorithm or a new kappa.
+
+Joint frame compilation of skip-prefix strips
+
+The local compiler and exact finite witnesses were prepared by eumemic with
+substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic
+Claude assistance supplied the unmodified PR #53 skip-prefix producer.
+The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating
+Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang,
+Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt,
+OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors
+credited in the retained sources. See references/frame-compiler/pr48/NOTICE
+and references/frame-compiler/pr53/NOTICE for their original attribution.
+No authorship or review by those predecessors is claimed for this compiler.
+
+Joint dual-skip/frame-compiler composition (2026-10-08)
+---------------------------------------------------
+Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip
+scalar graph; original LICENSE and NOTICE are retained under
+references/frame-compiler/pr55. eumemic with substantial OpenAI Codex
+assistance: PR57 joint reversible frame compiler, serialized physical-word
+replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex
+assistance: this composition, independent integration and exact certificate.
+Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all
+predecessor authors retain the notices above and in pinned source files.
+No original copyright or Apache-2.0 notice is removed.
+
+Ranked reclamation after the PR58 joint-dual composition (2026-10-08)
+----------------------------------------------------------------
+Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical
+slots by descending current frame rank, retaining slot ID as the tie-breaker;
+complete replay, exact profiles, comparison and adversarial controls.
+The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57
+compiler. Its original shared compiler remains byte-identical. The PR55
+producer and all preceding attribution remain unchanged. The complete PR58
+comparison certificate and its original source/proof provenance are retained
+under references/frame-compiler/pr58.
+
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
+Local split-pair skip-prefix research continuation (2026-10-08)
+Prepared with substantial OpenAI Codex assistance at the user's request.
+New contribution: singleton/pair recursive partition refinement, finite summand
+order and moment search, and verified composition with inherited paid clones.
+Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips.
+Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone
+specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51
+whole-chain/source-partition cloning and earlier producer/interfaces.
+Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all
+stated historical AI assistance. All predecessor notices remain applicable.
+
+Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones):
+Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain
+source-partition clone operation is adapted from RaD / hipotures, PR #51,
+with original envelopes retained here. The scalar base is Avi Eisenberg's
+PR #53 skip-prefix layout, prepared with Anthropic Claude assistance.
+All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain
+in force. This extension is provided under the repository's Apache-2.0 terms.
+
+Community review checkpoint #50–#61, 8 October 2026
+Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex.
+Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude.
+Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance)
+and dual-suffix strips (#55, Anthropic Claude assistance).
+Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex.
+Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked
+reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame
+composition (#56), with Anthropic Claude and stated Codex validation assistance.
+RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones
+(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone
+composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter
+refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with
+his disclosed multi-tool and Codex assistance.
+Parallel #50/#52 records are reviewed and credited without replacing the earlier
+climbed-48 implementation; their exact commit history is retained. The separate
+#51 witness is retained as research; its source-graph/negative-basis data proof
+has not received the same full replay as #56's consumed compiler machinery.
+See docs/research/community-round2-review.md for precise validation boundaries.
+All earlier licenses, contributions and notices remain applicable.
+
+Interval strips and core-aware pair assembly
+
+The interval leave-one-out strips and the core-aware group-pair assembly in
+research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic
+Claude assistance. They build on PR #53's skip-prefix pipeline and its
+inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
+uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
+are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
+verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.
+
+Partial-gauge extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The pair-first cube assembly, compatible backward frame enlargement,
+partial source-gauge schedule and stopped whole-projector composition are
+contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834.
+
+The physical deferred bit word, reflected event accounting, fan-order
+clarification and signed two-stage endpoint are adopted from Zhihao Chen
+(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with
+Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen
+witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38,
+with the original Claude disclosure retained. Cyclic interval and core-aware
+pair primitives build on Avi Eisenberg (ikeboy), PR #62,
+ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance.
+Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and
+precision credits remain in force. Imported source notices and licenses
+are preserved under references/partial-gauge/; see SOURCES.json.
+
+Three-stage cover extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry,
+weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring
+fallback, uniform batching, internal row borrowing and exact composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This
+repository integration uses Codex assistance and extends
+PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86.
+
+The selected local complex DAG and support/label replay are adopted from
+Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf,
+with its original Apache-2.0 terms and Claude assistance disclosure.
+The unchanged bit physical word retains Zhihao Chen (jacklightChen),
+PR #97, and Swapnil Jain's underlying witness and assistance notices.
+The prior opposite-bank, stopped ordinary-leaf, copied-center, topology,
+semantic/bulk, precision and analytic credits remain applicable.
+Original source notices and license bytes are preserved in
+references/three-stage-cover/pr117/ and references/partial-gauge/.
+
+Paired-cube and shared-core extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The paired-coordinate triple-cube identity, signed overlap channels,
+original-source involution, coordinate-star centers, concrete frame/carrier
+compiler, chronological source gauges, PR97 gauge-subset selection,
+orthogonal cross-stage routing and exact supplier/assembly integration
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance.
+Repository integration uses Codex assistance and extends PR #130 at
+6a9970a530119174507904e23592fd59ede19a5d.
+
+The substantial completed-core workspace-sharing principle is credited to
+an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported
+OpenAI Codex assistance. No source code, partition or serialized witness
+from that PR is imported. The new routing and paired-motif integration
+are described separately and claim no exclusive priority for broad sharing,
+paired-coordinate, Fourier or carrier-matching ideas.
+
+Selected coarse queries restrict the retained eumemic PR #117 positive DAG;
+its original Apache-2.0 and Claude assistance notices remain unchanged.
+The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil
+Jain's frozen witness, notices and historical assistance disclosures.
+Earlier generalized Clifford, three-stage, uniform weighted recursion,
+ordinary-leaf, semantic/bulk, precision and analytic credits remain in force.
+Source-specific license bytes are preserved under references/.
+
+October 9 recycled-bit integration:
+DaysSky (#150), William Porter/hpst3r (#147), Thomas Marchand/Th0rgal (#146),
+Rohan Gupta/gupt1156 (#148), James Chang/jamesyc (#124), SovereignSteak (#122).
+The original notices and assistance disclosures are retained under research/,
+including research/recycled-bit-integration/NOTICE. Andrew Barnes/Bortlesboat
+(#101) repairs shared verifiers using rfu08's #64 counterexample. rfu08's
+scoped finite and formal-transfer packages are included separately; they do
+not establish the full multiplication theorem. Earlier and parallel credits
+remain applicable. Douglas Colkitt integrated these with OpenAI Codex assistance.
+
+SovereignSteak PR151 independently supplies the matching gauge/recycling
+composition and the selected 473/10^6 atom refinement; its original package
+and assistance disclosure are retained under research/gauge-recycling/.
+
+October 9 coordinated-frame and completed-entrance-bank integration:
+Dugongue PR186, 166a34d75e853e933855f0f7e940fe77c4bb6b8d, builds on
+Chafik Boukhalfa/chafreaky's PR181, 7fb2194801e3a10f54772c7f0d5505035a4fc0ad,
+and eumemic's PR168 v4, 4a3c769e5c5430e7114c4d3e099ff34664677f17.
+The frozen package and prerequisite archive preserve all author, source-license
+and assistance notices, including PR175's compiler optimization. DaysSky's
+carrier closure, James Chang's terminal substitution and balanced assembly,
+geckods' stopping refinements and all preceding framework credits remain.
+Rohan Arun's PR185 bootstrap and chafreaky's PR182 bit refinement are reviewed
+parallel contributions; their gains are not added to the selected witness.
+Douglas Colkitt integrated the package and supplied the bank scheduling
+supplement with OpenAI Codex assistance, under Apache-2.0. The all-size
+interfaces remain conditional; finite verification is not full formalization.
diff --git a/research/p104-a2-paid-rank-exchange/OPTIMIZED-REPLAY.log b/research/p104-a2-paid-rank-exchange/OPTIMIZED-REPLAY.log
new file mode 100644
index 0000000000..18694175a1
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/OPTIMIZED-REPLAY.log
@@ -0,0 +1,6 @@
+PASS exact source foundation bindings 12 and all arbitrary-input/frame rows
+PASS all dense transpose columns 752
+PASS all signed native maps and exact phases 2916
+PASS complete physical caller
+PASS two exact moment engines, compiler, BIT stock and58 outer predicates ['77614664639/25000000000000', '310458658557/100000000000000']
+PASS_P104_A2_KAPPA_00310458
diff --git a/research/p104-a2-paid-rank-exchange/OVERLAP.json b/research/p104-a2-paid-rank-exchange/OVERLAP.json
new file mode 100644
index 0000000000..81172a413b
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/OVERLAP.json
@@ -0,0 +1,316 @@
+{
+  "status": "POSSIBLE_OVERLAP_NOT_A_NOVELTY_CERTIFICATE",
+  "searched_at": "2026-10-26T01:05:26.259933+00:00",
+  "terms": [
+    "715",
+    "388",
+    "340",
+    "A2"
+  ],
+  "coverage": {
+    "local": {
+      "head": "f626db5c3a3d781d8f7695038776c621651f75e1",
+      "records": 5861,
+      "head_records": 5860,
+      "complete_head_inventory": true,
+      "unsearched_head_paths": []
+    },
+    "open_prs": {
+      "fetched_at": "2026-10-24T14:54:17.174399+00:00",
+      "complete_pagination": true,
+      "records": 314
+    },
+    "discussions": {
+      "fetched_at": "2026-10-24T14:55:18.516259+00:00",
+      "complete_pagination": true,
+      "records": 4711
+    }
+  },
+  "local_records": 5861,
+  "warnings": [
+    "open_prs: offline; cached data may miss current changes",
+    "discussions: offline; cached data may miss current changes"
+  ],
+  "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": 4950,
+  "matches": [
+    {
+      "number": 2023,
+      "title": "CLAIM: four-role heterogeneous S3 target through six calls (seed 849388c4)",
+      "kind": "discussion",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/2023",
+      "updated_at": "2026-10-14T17:34:43Z",
+      "matched_terms": [
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 5,
+      "excerpt": "eterogeneous S3 target with two long and two short transpositions\n\nAssignment: accepted OS seed `849388c44ed5461801c96c3efa2d7497`; track `bit-mechanisms`; task `bit-new-register-model`; priority revision 3; catalogue SHA-256 `6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075`; selector scope key `7f5f4426ef796683c485`; exact base `b5d138e63a10da21cc625088d1d953f962f05a8f`; selected-record SHA-256 `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.\n\nObject and ques"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/r715-cancellation-reconciliation/README.md",
+      "path": "research/r715-cancellation-reconciliation/README.md",
+      "sha256": "e19a3ad9cdca8735061310bb2638a30c87b4c152ebf925048b2e0d19c4ddbb09",
+      "matched_terms": [
+        "715",
+        "388",
+        "a2"
+      ],
+      "score": 5,
+      "excerpt": "# Portable reconciliation of the archived R715 cancellations\n\n**Research consolidation and independent finite verification; no new kappa.**\nThe historical 40-addition reduction is reproduced from retained, byte-identical\nparent sources with a standalone standard-library checker. The new evidence\nalso accounts for **425 empty-source frame visits** removed by the historical\nflattening step. All 955 register frame itineraries, the nine copied values,\nthe arbitrary-dirty binary maps and the local pa"
+    },
+    {
+      "number": 1740,
+      "title": "CLAIM: four-register S4 sharing for distinct noncommuting 3-cycles (seed a27baaa7)",
+      "kind": "discussion",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1740",
+      "updated_at": "2026-10-14T16:08:01Z",
+      "matched_terms": [
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "CLAIM \u2014 bit-mechanisms, seed `a27baaa734817e395db88b840bf4a089`, task `bit-new-register-model`, priority revision 3. Exact base `0b1cccf86e2f5132ed286a26ef0a681a495740ea`; selected-result SHA-256 `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.\n\nQuestion: with four restored dirty scalar roles over formal `F2[S4]`, can the complete paid alphabet realize `diag([s1*s2],[s2*s3],[e],[e])` in at most three calls? Here `s1=(12), s2=(23), s3=(34)` and the target 3-cycles are distinct "
+    },
+    {
+      "number": 1728,
+      "title": "DONE: bounded term-basis shear screen on the P104 decoder (seed 8be7a2ee)",
+      "kind": "discussion",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1728",
+      "updated_at": "2026-10-14T08:03:26Z",
+      "matched_terms": [
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": " the completed handoff now that posting works.\n\n**Track / seed / task:** `complex-mechanisms`, `8be7a2eedea3e46ac14a43b092df843e`, `complex-new-helper-inventory`, priority revision 3. **Base:** `0b1cccf86e2f5132ed286a26ef0a681a495740ea`. **Selected-record SHA-256:** `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.\n\n**Scope:** On PR717's exact P104 eight-term decoder (`P104_HIT_flip_3a.txt`, input SHA-256 `8e0178db02c14e1b29058cdbdbd50aac22f2b55a1ba21dc82f417610c9932526`), appl"
+    },
+    {
+      "number": 3664,
+      "title": "Physical normalized polynomial products for Fourier packets",
+      "kind": "discussion",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/3664",
+      "updated_at": "2026-10-23T17:48:43Z",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "12 UTC. I verified test merge `c8c304f4cfd181007bfc0bd9ff83a35fda960b9e` against head tree `9977d3f8a26c57e6ca05f1c7876435ff6bc454b9`. Export SHA256: `6b977956b2794094de4aa50545baceff9e43ccc68a473ce5569e6a710b97bac0`; receipt SHA256: `115580f6c53a84c7e09401095e8ea7d043406adcd50cc5cf34d0b3e9d0a162aa`. I retain the earlier timeouts and interrupted attempt below.\n\nI implement the full physical expansion/chirp/compression join, with coefficient production, full-support routing, short-field moves, pe"
+    },
+    {
+      "number": 3156,
+      "title": "C(P112) >= 64: what is left (for the swarm)",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/3156#discussioncomment-19030442",
+      "updated_at": "2026-10-19T11:23:36Z",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "act committed bytes/source history and official read-only Git-free exports pass:4012 has16 files, SHA256 f9fd5a7423d8db18dee7512ec59feef1658053da6edba82081cf19e759843ace;4009 has11 including manifest, SHA2569ae5a9d96e1441ea9350ddc339c52ebd57851f5d494c8dd03440322f9fb289cb. Original31170e0 evidence remains preserved. Separate same-team agents reviewed both proofs and the capacity corollary; no independent human or formal-kernel acceptance claimed. Feedback empty at10:46UTC.\r\n\r\nMain20ecf289/revisio"
+    },
+    {
+      "number": 1740,
+      "title": "CLAIM: four-register S4 sharing for distinct noncommuting 3-cycles (seed a27baaa7)",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1740#discussioncomment-18928637",
+      "updated_at": "2026-10-14T16:23:21Z",
+      "matched_terms": [
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": " heads were read back unchanged. Base `94e50883325b387add6891258a986f797b09cacc`; selected-record SHA256 `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`; recorded selector seed `4a495b25dc2389c1fc0b8e95157a0276`, priority revision 3.\n\nRegistered CPython3.12.12 replay passes, including 5,912 permutations, all 576 S4 target pairs and 3,456 omitted-call rejections. The separate no-import checker uses complete Cayley BFS and full formal dirty operators; independent AI-assisted pro"
+    },
+    {
+      "number": 2023,
+      "title": "CLAIM: four-role heterogeneous S3 target through six calls (seed 849388c4)",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/2023#discussioncomment-18930516",
+      "updated_at": "2026-10-14T17:34:43Z",
+      "matched_terms": [
+        "388",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "are equivalent by idempotent-alphabet conjugation.\n\nPackage `research/bit-s3-four-role-mixed-seed849388/`, exact base `b5d138e63a10da21cc625088d1d953f962f05a8f`, local commit `284019ef`. The final verifier passed on CPython 3.14.7 with 23,040 independent generic factor-basis checks, 240 involution checks, the three-call positive control, 12 direct target replays, eight omission controls and a wrong-target control. Source and package metadata pins pass. Python 3.12 is unavailable locally, so the "
+    },
+    {
+      "kind": "local-ledger",
+      "title": "docs/research/pr-ledger.md",
+      "path": "docs/research/pr-ledger.md",
+      "sha256": "5c70e3daa28cfc67411ab795e31aa0487cf98ff72a85bbfe1f399e8911c21aac",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "-audit/DISPOSITIONS.json) |\n| [#274](https://github.com/CrocSwap/integer-mult-bounds/pull/274) | shea256 | incorporated | [record 1](../../research/community-reconciliation-next100-audit/DISPOSITIONS.json) |\n| [#275](https://github.com/CrocSwap/integer-mult-bounds/pull/275) | lydakis | partially reviewed | [record 1](../../docs/research/pr-intake-decisions.json) \u00b7 [record 2](../../research/community-efficient1800-audit/DISPOSITIONS.json) \u00b7 [record 3](../../research/community-efficient1900-audit/"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/community-efficient2400-audit/README.md",
+      "path": "research/community-efficient2400-audit/README.md",
+      "sha256": "0569ca993ed72af85bc20904fe4b1a2e13632fc70dbfedfb34502133a471abf8",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "| Default-tree archive | [Exact author commit](https://github.com/CrocSwap/integer-mult-bounds/tree/a2d91edcf402f0fa95512c8228ab1259dd65ad45) |\n| [#2951](https://github.com/CrocSwap/integer-mult-bounds/pull/2951) / petalas | Default-tree archive | [Exact author commit](https://github.com/CrocSwap/integer-mult-bounds/tree/d7a9287d6564bae5c7b19e2e3e655c2b578f8b72) |\n| [#2852](https://github.com/CrocSwap/integer-mult-bounds/pull/2852) / dataisfire | held_intake_errors | [Exact author commit](https:"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/community-efficient2500-audit/README.md",
+      "path": "research/community-efficient2500-audit/README.md",
+      "sha256": "a0752cfbd9836821e515aa3584673e06db95102e1dba8a478085a2019eb2e1bb",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "w | [Exact author commit](https://github.com/CrocSwap/integer-mult-bounds/tree/5816f94741709e8082e06a2cc9a114ff87411f3d) |\n| [#2295](https://github.com/CrocSwap/integer-mult-bounds/pull/2295) / bitplane | Source archive + reviewed manual workflow | [Exact author commit](https://github.com/CrocSwap/integer-mult-bounds/tree/1bd441f45179a6d042d01a8237ad7faaf4e23a18) |\n| [#2289](https://github.com/CrocSwap/integer-mult-bounds/pull/2289) / bitplane | Source archive + reviewed manual workflow | [Exact"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/community-efficient2600-audit/README.md",
+      "path": "research/community-efficient2600-audit/README.md",
+      "sha256": "3e244cb39516d4d60f7c18ba6c78397ecd23dddf48b96c94e2c54c5ebcbad2a4",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "| [Exact author commit](https://github.com/CrocSwap/integer-mult-bounds/tree/4e392e90e3fca8f395e35d6a23644920160a9f0f) |\n| [#1048](https://github.com/CrocSwap/integer-mult-bounds/pull/1048) / flemhhh | held_shared_or_existing_obligation | [Exact author commit](https://github.com/CrocSwap/integer-mult-bounds/tree/07730cd8302c378607259a9ad0efb73e0b57a2f2) |\n| [#897](https://github.com/CrocSwap/integer-mult-bounds/pull/897) / DaysSky | held_shared_or_existing_obligation | [Exact author commit](http"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/fine-endpoint300-t74-k554/README.md",
+      "path": "research/fine-endpoint300-t74-k554/README.md",
+      "sha256": "ce5b1991ece5711aad916dd84d693e78a07bb7a4bff707a903c3a33e184f25b8",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "nditional **\u03ba = 0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263056561328446322744513169085862209720704481201654121427713006366929567157349625273906229277148454180995537563371124755588139767841405530386745578819565618798006892541712197674874836946756930569827889227573192555649999999997996640490086748910957328658012158594**\n\nThis is **6.5613284483e-91** above PR1839 (jeremykothe2, `836ef6a1`), the current leader. It uses the identical construction: the pi"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/gauge44-barch-t16/README.md",
+      "path": "research/gauge44-barch-t16/README.md",
+      "sha256": "42a87c4b8ba75d360412acb1330c3c5d72e37369b1999e4bc4d5dcab9f0cfb14",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "d +3.1404\u00b710\u207b\u2077 over #858 (0.001661642255531641).\n- **Complex lower endpoint** b = 1664722996771957673401/10\u00b2\u2074 (binding). The bit endpoint is unchanged at 1733427039589844234749/10\u00b2\u2074.\n- **Finite reserves:** prefix 72 + 37 + 144 + 144 + 4 = **401 < 512** bits; physical row coefficient **16469 < 20161**; 73\u00b2 > 2\u00b748\u00b2.\n- **Evidence.** `verify_t16.py` passes on the U8 locals (`PASS_GAUGE44_T16_STRICT_GLOBAL_REPLAY`). `RESULT.json` is that replay's `REPLAY.json`. It runs restore-parent; bind8 (this pac"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/gauge44-barch/README.md",
+      "path": "research/gauge44-barch/README.md",
+      "sha256": "7a6f17e83af95089f5a83ae8d4cbfed12677b6df54ee45a3203d2f8181f5633a",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "mo (0.00165861362787227) and +0.1585% over research/gauge44-promo-p6 (0.001659319032684569) and +0.1388% over research/gauge44-promo-p6sw (0.001659645685315104, the same exchange on three promotion groups) and +0.0189% over research/gauge44-frames-x (0.001661634653669567, the X4 word before the rewrite) and +0.0449% over [#741](https://github.com/CrocSwap/integer-mult-bounds/pull/741) (chafreaky, \u03ba 0.001661202643272239; the same word without the joint exchange) and +0.0969% over [#685](https://g"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/gauge44-ends7-t16/README.md",
+      "path": "research/gauge44-ends7-t16/README.md",
+      "sha256": "ea821ea8ba8933cecb9b70ff5c579e31dac52a72fec0e191e192321a3a22876e",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "ormalises only that field (canonical `d1bf3c6f\u2026`, programs `683f8323\u2026`/`2d184e4d\u2026`, Clifford `e3bfdfa2\u2026`, pinned in `CHAIN-PINS-k8.json`). For this package, `data/SPEC.json` and a fresh `verify_global.py` replay on locals rebuilt by the two drivers are the binding record.\n- **Independent price.** `independent/GAUGE44-INDEPENDENT-PRICE.json` is regenerated on this package's `data/PLAN.json` with the ported `independent/price_profile.py`: `PASS_FROZEN_PROFILE_TWO_EXACT_ENGINE_AND_OUTER47_CROSSCHEC"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/gauge44-frames-x/README.md",
+      "path": "research/gauge44-frames-x/README.md",
+      "sha256": "a086b31dcbfe273006a9a90b56a9703f3679114c9f988f8868f3c5f1ff783f3d",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "running it reproduces the receipt except `elapsed_seconds` and, across zlib versions, `compressed_sha256`.\n- **Historical #645 files (PR543-based, 492 helpers).** These are #645's files, retained unchanged and **not** updated for this package: `LOCAL-VALIDATION.json`, `CONSTRUCTION.md`, `hosting/`, `evidence/` (except `evidence/layers/`), the rest of `independent/`, `construction/reproduce.py` and its inputs, and `code/prepare_global.py`. They describe #645's word, owner map, PR543 supplier and "
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/gauge44-leaver/README.md",
+      "path": "research/gauge44-leaver/README.md",
+      "sha256": "28acc0bdb972bdc8c2c48cbf8e1ef663d79b7ab134c23d542fec31680a6af6dc",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "rch/gauge44-layered and +0.4854% over research/gauge44-notri-layered (0.001656193765416678) and +0.3388% over research/gauge44-promo (0.00165861362787227) and +0.2962% over research/gauge44-promo-p6 (0.001659319032684569) and +0.2764% over research/gauge44-promo-p6sw (0.001659645685315104, the same exchange on three promotion groups) and +0.1564% over research/gauge44-frames-x (0.001661634653669567) and +0.1375% over research/gauge44-barch (0.001661948695165034) and +0.1825% over [#741](https://"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/gauge44-leaver8/README.md",
+      "path": "research/gauge44-leaver8/README.md",
+      "sha256": "0c09f0685e5db77177187c12afea9c45465787c44d6ae0d7d28c0c7de8f1b2a0",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "ormalises only that field (canonical `d1bf3c6f\u2026`, programs `683f8323\u2026`/`2d184e4d\u2026`, Clifford `e3bfdfa2\u2026`, pinned in `CHAIN-PINS-k8.json`). For this package, `data/SPEC.json` and a fresh `verify_global.py` replay on locals rebuilt by the two drivers are the binding record.\n- **Independent price.** `independent/GAUGE44-INDEPENDENT-PRICE.json` is regenerated on this package's `data/PLAN.json` with the ported `independent/price_profile.py`: `PASS_FROZEN_PROFILE_TWO_EXACT_ENGINE_AND_OUTER47_CROSSCHEC"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/gauge44-slots-arcflow/README.md",
+      "path": "research/gauge44-slots-arcflow/README.md",
+      "sha256": "b16061522d8cd37cf470613965cf123a629cb0cc72fb4dcdf23a87d95a235227",
+      "matched_terms": [
+        "715",
+        "388",
+        "340",
+        "a2"
+      ],
+      "score": 4,
+      "excerpt": "ormalises only that field (canonical `d1bf3c6f\u2026`, programs `683f8323\u2026`/`2d184e4d\u2026`, Clifford `e3bfdfa2\u2026`, pinned in `CHAIN-PINS-k8.json`). For this package, `data/SPEC.json` and a fresh `verify_global.py` replay on locals rebuilt by the two drivers are the binding record.\n- **Independent price.** `independent/GAUGE44-INDEPENDENT-PRICE.json` is regenerated on this package's `data/PLAN.json` with the ported `independent/price_profile.py`: `PASS_FROZEN_PROFILE_TWO_EXACT_ENGINE_AND_OUTER47_CROSSCHEC"
+    }
+  ]
+}
diff --git a/research/p104-a2-paid-rank-exchange/PR-DRAFT.md b/research/p104-a2-paid-rank-exchange/PR-DRAFT.md
new file mode 100644
index 0000000000..f3261cc407
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/PR-DRAFT.md
@@ -0,0 +1,31 @@
+Conditional κ = 0.00310458 (+1.7454e-7, +0.0056223% over #5627): paid A2 rank exchange
+
+The exact [PR #5627](https://github.com/CrocSwap/integer-mult-bounds/pull/5627) word admits a paid A2 rank exchange that raises **conditional κ from 0.00310440546 to 0.00310458**: **+1.7454e-7, +0.0056223%**. A two-to-three scalar rewrite changes source 715's legal frame itinerary from paid ranks 5 and 3 to ranks 2 and 6. The extra ordinary ADD is charged; all 743 arbitrary inputs, 12,958 complete productive targets, endpoint frames and owned COPY/INIT/CLEAR instructions are retained.
+
+The local paid change is `H2 + H6 − H3 − H5`, repeated across 18 local invocations. Strict concavity makes this a strict moment gain for every exponent in `(0,1)`. Two exact rational moment engines enclose the unique raw crossing in **(0.00310458658556, 0.00310458658557)**, leaving about **6.58556e-9** above the claim. Complete paid calls and rank-mass surplus remain 66,546 and 760.
+
+## Significance for researchers
+
+This answers whether an A2 scalar rewrite can improve the recursive bill when its frame itinerary changes. The earlier [#3998](https://github.com/CrocSwap/integer-mult-bounds/pull/3998), [#4019](https://github.com/CrocSwap/integer-mult-bounds/pull/4019), [#4112](https://github.com/CrocSwap/integer-mult-bounds/pull/4112) and [#4152](https://github.com/CrocSwap/integer-mult-bounds/pull/4152) results concern frozen paid profiles. Here the complete paid profile changes while productive credits stay fixed. The useful search target is a scalar identity together with a legal, fully charged frame path. The elementary A2 identity retains its predecessor attribution.
+
+## Validation and scope
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-a2-paid-rank-exchange
+python3.12 -B scripts/check_research_commit.py --package research/p104-a2-paid-rank-exchange
+```
+
+The registered quick, isolated optimized full and separate scoped algebra audit pass. Coverage includes 752 dense transpose columns, all 2,916 signed native maps with exact global phases, the complete physical caller, two rational moment engines, compiler completion and ordinary allowances, BIT stock, all 54 strict and four functional outer predicates, and rejection controls. Publication evidence and the refreshed overlap check are recorded in `research/p104-a2-paid-rank-exchange/PUBLICATION.md`.
+
+The 41-file committed-byte check passes. The optional full source-history preflight declines this shallow checkout; the declared provenance base passes a direct ancestor check.
+
+The exact predecessor remains head `ed4dc88a09a7d201971f6603b8c2074e95dd9c32`; its scientific inputs and authenticated framework export are retained unchanged. The same five framework assumptions and helper-endpoint contract remain. The huge uniform word and complete rounded multiplier are not materialized. This is an author-checked conditional contribution; selected-frontier promotion, formal verification, independent human acceptance and hosted CI success are separate obligations.
+
+## Credit
+
+Credit [Ali Romman](https://github.com/aliromman) for [#5627](https://github.com/CrocSwap/integer-mult-bounds/pull/5627), [flyingviper](https://github.com/flyingviper) for the relays, [Scratch-net](https://github.com/Scratch-net) for the prepared prefix, [Pierre-Loic Doulcet](https://github.com/hexapode) for the commuting cascade, and [Joel Pulikkan](https://github.com/GamingPuzzled), [DreamingOfClouds](https://github.com/DreamingOfClouds), [THU New Media Research Center](https://github.com/thu-nmrc), [David Arias](https://github.com/DavidAriasC), [Gareth Davidson](https://github.com/bitplane), [Cinna Davis](https://github.com/1337isnot1337), [Jacob Alan Sussman](https://github.com/jacobalansussman), [Cao Yifan](https://github.com/intqwq) and [zxv](https://github.com/zxv) for the retained helper, native, geometry, BIT, compiler and analytic lineage. The prior A2 mechanism and boundary discipline retain [Chris R](https://github.com/Clear1nsight) and [ToikY](https://github.com/ToikY)'s credit. New construction and checks used substantial OpenAI Codex assistance and complementary subagent research during the original investigation. Apache-2.0; source notices are retained.
+
+## Open next questions
+
+- Independently inspect the three literal gates, nested frame chain and new paid native maps, then reproduce the full verifier before scientific admission. The separate agent audit covers local algebra and frame induction; it does not replace independent human review.
+- Test a different compatible noncommuting scalar motif on the emitted word. Any combined gain requires one reconstructed word and complete invoice; local headline gains cannot be assumed additive. No further positive candidate is claimed.
diff --git a/research/p104-a2-paid-rank-exchange/PRICING-EDITS.json b/research/p104-a2-paid-rank-exchange/PRICING-EDITS.json
new file mode 100644
index 0000000000..7ff9332f19
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/PRICING-EDITS.json
@@ -0,0 +1,14 @@
+[
+  [
+    "KAPPA, TESTED = Q('.00310440546'), Q('.003104405465')",
+    "KAPPA, TESTED = Q('.00310458'), Q('.0031045805')"
+  ],
+  [
+    "PREVIOUS = Q('.00308595')",
+    "PREVIOUS = Q('.00310440546')"
+  ],
+  [
+    "'kappa_decimal':'.00310440546'",
+    "'kappa_decimal':'.00310458'"
+  ]
+]
diff --git a/research/p104-a2-paid-rank-exchange/PUBLICATION-OVERLAP.json b/research/p104-a2-paid-rank-exchange/PUBLICATION-OVERLAP.json
new file mode 100644
index 0000000000..8dc71a5607
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/PUBLICATION-OVERLAP.json
@@ -0,0 +1,355 @@
+{
+  "status": "POSSIBLE_OVERLAP_NOT_A_NOVELTY_CERTIFICATE",
+  "searched_at": "2026-10-26T01:58:45.535820+00:00",
+  "terms": [
+    "scalar-to-moment",
+    "scalar simplification",
+    "paid profile",
+    "R439",
+    "3998",
+    "4019",
+    "4112",
+    "4152",
+    "staged544-a2-graph-rewrite",
+    "staged544-separated-a2",
+    "staged544-a2-composition-audit",
+    "staged544-a1858-contraction-audit",
+    "staged544-a2-graph-rewrite",
+    "staged544-separated-a2",
+    "staged544-a2-composition-audit",
+    "staged544-a1858-contraction-audit"
+  ],
+  "coverage": {
+    "local": {
+      "head": "f626db5c3a3d781d8f7695038776c621651f75e1",
+      "records": 5863,
+      "head_records": 5860,
+      "complete_head_inventory": true,
+      "unsearched_head_paths": []
+    },
+    "open_prs": {
+      "fetched_at": "2026-10-26T01:57:26.195145+00:00",
+      "complete_pagination": true,
+      "records": 335
+    },
+    "discussions": {
+      "fetched_at": "2026-10-26T01:58:40.509737+00:00",
+      "complete_pagination": true,
+      "records": 4924
+    }
+  },
+  "local_records": 5863,
+  "warnings": [],
+  "reference_terms": [
+    "3998",
+    "4019",
+    "4112",
+    "4152"
+  ],
+  "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": 1135,
+  "matches": [
+    {
+      "kind": "local-task",
+      "title": "research/p104-a2-paid-rank-exchange/TASK.json",
+      "path": "research/p104-a2-paid-rank-exchange/TASK.json",
+      "sha256": "c6255665e9fa7ef59daffa43bd3dfa12476f25efa6e871a6f89610004a121cb6",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439",
+        "3998",
+        "4019",
+        "4112",
+        "4152",
+        "staged544-a2-graph-rewrite",
+        "staged544-separated-a2",
+        "staged544-a2-composition-audit",
+        "staged544-a1858-contraction-audit"
+      ],
+      "score": 12,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"priority_revision\": 7,\n  \"base_commit\": \"f626db5c3a3d781d8f7695038776c621651f75e1\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can a changed frame itinerary remove a paid transition beyond the already covered scalar A2 contractions?\",\n    \"starting_points\": [\n      \"docs/research/research-family-status.md\",\n      \"certificates/selected-result.json\",\n      \"research/staged544-a2-graph-rewrite/README.md\",\n      \""
+    },
+    {
+      "kind": "local-task",
+      "title": "research/staged-compensation-forests/TASK.json",
+      "path": "research/staged-compensation-forests/TASK.json",
+      "sha256": "e8f051415f5005051ef7b959f1fb4212e34f4d149a048cbd40c77ee8779435d2",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439",
+        "3998",
+        "4019",
+        "4112",
+        "4152",
+        "staged544-a2-graph-rewrite",
+        "staged544-separated-a2",
+        "staged544-a2-composition-audit",
+        "staged544-a1858-contraction-audit"
+      ],
+      "score": 12,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"priority_revision\": 7,\n  \"base_commit\": \"12fa8c5c844be21bfefb38a19503fbd867a11244\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can a changed frame itinerary remove a paid transition beyond the already covered scalar A2 contractions?\",\n    \"starting_points\": [\n      \"docs/research/research-family-status.md\",\n      \"certificates/selected-result.json\",\n      \"research/staged544-a2-graph-rewrite/README.md\",\n      \""
+    },
+    {
+      "kind": "local-task",
+      "title": "research/staged544-separated-a2/TASK.json",
+      "path": "research/staged544-separated-a2/TASK.json",
+      "sha256": "2cf3bd01da44d7fb3732e3072ab55e4323ce0e95fc0031c5d88358705ed8ae93",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439",
+        "3998",
+        "staged544-separated-a2"
+      ],
+      "score": 8,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"priority_revision\": 6,\n  \"base_commit\": \"20ecf2891d80b5acf18d980b2164fb44ea7faa90\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can one known scalar simplification be converted into an actual paid-profile improvement?\",\n    \"starting_points\": [\n      \"research/r439-bsection-stars-seed01/README.md\",\n      \"certificates/selected-result.json\"\n    ],\n    \"methods\": [\n      \"Derive the changed legal frame itinerary a"
+    },
+    {
+      "kind": "local-task",
+      "title": "research/staged544-a2-graph-rewrite/TASK.json",
+      "path": "research/staged544-a2-graph-rewrite/TASK.json",
+      "sha256": "a0fdbd910ce782e1cd8ca271f32f9521a5b226ea371209e02e1ff79eca77ac96",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439",
+        "staged544-a2-graph-rewrite"
+      ],
+      "score": 7,
+      "excerpt": "{\n  \"task_id\": \"scalar-to-moment\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can one known scalar simplification be converted into an actual paid-profile improvement?\",\n    \"starting_points\": [\n      \"research/r439-bsection-stars-seed01/README.md\",\n      \"certificates/selected-result.json\"\n    ],\n    \"methods\": [\n      \"Derive the changed legal frame itinerary and exact child histogram.\",\n      \"Prove a scoped obstruction showing why that scalar gain cannot change the p"
+    },
+    {
+      "number": 2223,
+      "title": "Cycle 39 claim: R439 frame 509 source-22 paid-climb cut",
+      "kind": "discussion",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/2223",
+      "updated_at": "2026-10-15T00:38:13Z",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439"
+      ],
+      "score": 6,
+      "excerpt": "*\n\n**Assignment pins.** `incumbent-optimization`, OS seed `5ffa1b4c0c1d31ab3d7c4bd324631b22`, task `scalar-to-moment` (scope key `9dbfd943f23ef75bae31`), priority revision 4, catalogue SHA-256 `d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7`, exact base `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`, selected-record SHA-256 `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.\n\n**Question / changed scope.** In the exact 11-wire PR1015 B-section block at frame 509, can"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/r439-frame508-inverse-cut-seed3b/README.md",
+      "path": "research/r439-frame508-inverse-cut-seed3b/README.md",
+      "sha256": "39331ecd51179c63aec5eb388d5e7f7594ebad8b1e4b5adff324bb286ec2122e",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439"
+      ],
+      "score": 6,
+      "excerpt": "# R439 frame 508: inverse-column cut forces the exceptional source to rank four\n\n**Scoped exact obstruction; no new selected \u03ba.** For the pinned 11-wire linear map at R439 frame 508, register 19 must visit the full rank-four block frame in any monotone same-wire implementation whose operation frames stay inside that block frame. The result allows arbitrary ADD sequences, arbitrary proper intermediate frames, and arbitrary circuit length. It closes the source-19 exception left by the predecessor'"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/staged544-a1858-contraction-audit/README.md",
+      "path": "research/staged544-a1858-contraction-audit/README.md",
+      "sha256": "2205ee04f76fe750a93f24d0b3263a290e9096531b927fe2525586326fd18155",
+      "matched_terms": [
+        "3998",
+        "4019",
+        "4112",
+        "staged544-a1858-contraction-audit"
+      ],
+      "score": 6,
+      "excerpt": "efined**. This is not the #2979\nrank-7 group audit and not the separated seven-block contraction in PR #4019\n(upstream #4112). Discussion #1166 comment 19035159 asks for an independent\nchallenge of the consecutive A1858 contraction and its cleanup in\n[PR #3998](https://github.com/CrocSwap/integer-mult-bounds/pull/3998), head\n`389268a6b5e0373a3125d5c6fcd6995f4ea324f5`.\n\nOn all four frozen words vendored from that head, zero-based `A[1858:1861]` is\n\n```\n457 <- 457 - 475\n456 <- 456 + 457\n456 <- 456"
+    },
+    {
+      "kind": "local-task",
+      "title": "research/r439-b509-inverse-footprint-seed19/TASK.json",
+      "path": "research/r439-b509-inverse-footprint-seed19/TASK.json",
+      "sha256": "c6098287988a3b87fdd6e521c3e502160a72a5dc56a1816b5e48a2d1ff063d06",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439"
+      ],
+      "score": 6,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"priority_revision\": 2,\n  \"base_commit\": \"ef5f39403610d231a0c27e8a6b4f08c10209bcd6\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can one known scalar simplification be converted into an actual paid-profile improvement?\",\n    \"starting_points\": [\n      \"research/r439-bsection-stars-seed01/README.md\",\n      \"certificates/selected-result.json\"\n    ],\n    \"methods\": [\n      \"Derive the changed legal frame itinerary a"
+    },
+    {
+      "kind": "local-task",
+      "title": "research/r439-bblock-frame-barrier/TASK.json",
+      "path": "research/r439-bblock-frame-barrier/TASK.json",
+      "sha256": "f55a20d2d0a1cc7859f367005c3b7e27644ca9aa465d95a634bfb41ba002a43b",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439"
+      ],
+      "score": 6,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"priority_revision\": 2,\n  \"base_commit\": \"bce54fb3f895cbe5cd4c0eb477c3afeed100190f\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can one known scalar simplification be converted into an actual paid-profile improvement?\",\n    \"starting_points\": [\n      \"research/r439-bsection-stars-seed01/README.md\",\n      \"certificates/selected-result.json\"\n    ],\n    \"methods\": [\n      \"Derive the changed legal frame itinerary a"
+    },
+    {
+      "kind": "local-task",
+      "title": "research/r439-dirty-workspace-frame-bound/TASK.json",
+      "path": "research/r439-dirty-workspace-frame-bound/TASK.json",
+      "sha256": "890664557680b9d9c14c5b29018a1abcd08e42081458722b0cd06667db50abe0",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439"
+      ],
+      "score": 6,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"priority_revision\": 3,\n  \"base_commit\": \"0b1cccf86e2f5132ed286a26ef0a681a495740ea\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can one known scalar simplification be converted into an actual paid-profile improvement?\",\n    \"starting_points\": [\n      \"research/r439-bsection-stars-seed01/README.md\",\n      \"certificates/selected-result.json\"\n    ],\n    \"methods\": [\n      \"Derive the changed legal frame itinerary a"
+    },
+    {
+      "kind": "local-task",
+      "title": "research/r439-donor466-enlarged-seam/TASK.json",
+      "path": "research/r439-donor466-enlarged-seam/TASK.json",
+      "sha256": "f28786fa3fa9ffb0cb742a50c3e097bf2bd0f1207ca41f9dcbcac6eb747edfda",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439"
+      ],
+      "score": 6,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"priority_revision\": 4,\n  \"base_commit\": \"0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can one known scalar simplification be converted into an actual paid-profile improvement?\",\n    \"starting_points\": [\n      \"research/r439-bsection-stars-seed01/README.md\",\n      \"certificates/selected-result.json\"\n    ],\n    \"methods\": [\n      \"Derive the changed legal frame itinerary a"
+    },
+    {
+      "kind": "local-task",
+      "title": "research/r439-frame508-inverse-cut-seed3b/TASK.json",
+      "path": "research/r439-frame508-inverse-cut-seed3b/TASK.json",
+      "sha256": "1abf0316a6b89b7b7cb82c640e7cf66afbd46d2b0bf9212c7def190223729801",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439"
+      ],
+      "score": 6,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"scope_key\": \"9dbfd943f23ef75bae31\",\n  \"seed\": \"3b2676afb1d8667edc131585f741d4b0\",\n  \"track\": \"incumbent-optimization\",\n  \"priority_revision\": 3,\n  \"catalog_sha256\": \"6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075\",\n  \"base_commit\": \"0b1cccf86e2f5132ed286a26ef0a681a495740ea\",\n  \"selected_record_sha256\": \"10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\""
+    },
+    {
+      "kind": "local-task",
+      "title": "research/r439-scalar-histogram/TASK.json",
+      "path": "research/r439-scalar-histogram/TASK.json",
+      "sha256": "9988beb2ff777a12c399b7bb5fe9521a4214bbbb318b782c9c75927c95702114",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439"
+      ],
+      "score": 6,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"priority_revision\": 6,\n  \"base_commit\": \"20ecf2891d80b5acf18d980b2164fb44ea7faa90\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can one known scalar simplification be converted into an actual paid-profile improvement?\",\n    \"starting_points\": [\n      \"research/r439-bsection-stars-seed01/README.md\",\n      \"certificates/selected-result.json\"\n    ],\n    \"methods\": [\n      \"Derive the changed legal frame itinerary a"
+    },
+    {
+      "kind": "local-task",
+      "title": "research/staged544-a1858-contraction-audit/TASK.json",
+      "path": "research/staged544-a1858-contraction-audit/TASK.json",
+      "sha256": "95ac752ceb5be3d92918d56533f6bbbd70b97a9f1fcee4c81f994d351c4d02b9",
+      "matched_terms": [
+        "3998",
+        "4019",
+        "4112",
+        "staged544-a1858-contraction-audit"
+      ],
+      "score": 6,
+      "excerpt": "pe\": \"Independently derive the cleanup edit of the consecutive A1858 contraction on the four frozen PR3998 words, without importing that package and without replaying its frame bill.\",\n  \"scope_change_reason\": \"The card's #2979 rank-7 reconstruction is a different object. Comment 19035159 asks for one source-bound check of the PR3998 A1858 contraction and cleanup. Upstream #4112 already accepted the separated PR4019 blocks, so this package does not repeat that audit.\",\n  \"bottleneck_link\": \"A sc"
+    },
+    {
+      "kind": "local-task",
+      "title": "research/staged544-a2-cone-composition/TASK.json",
+      "path": "research/staged544-a2-cone-composition/TASK.json",
+      "sha256": "7b11ef5972bd54d7df7105326561d7ae942a6673391445f8494485fada5ae214",
+      "matched_terms": [
+        "scalar-to-moment",
+        "scalar simplification",
+        "paid profile",
+        "r439",
+        "3998",
+        "4019"
+      ],
+      "score": 6,
+      "excerpt": "{\n  \"schema_version\": 1,\n  \"task_id\": \"scalar-to-moment\",\n  \"priority_revision\": 6,\n  \"base_commit\": \"20ecf2891d80b5acf18d980b2164fb44ea7faa90\",\n  \"original_card\": {\n    \"id\": \"scalar-to-moment\",\n    \"question\": \"Can one known scalar simplification be converted into an actual paid-profile improvement?\",\n    \"starting_points\": [\n      \"research/r439-bsection-stars-seed01/README.md\",\n      \"certificates/selected-result.json\"\n    ],\n    \"methods\": [\n      \"Derive the changed legal frame itinerary a"
+    },
+    {
+      "number": 1166,
+      "title": "Handoffs and requests for independent checks",
+      "kind": "discussion-comment",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19057496",
+      "updated_at": "2026-10-20T03:31:44Z",
+      "matched_terms": [
+        "3998",
+        "4019",
+        "4112",
+        "4152",
+        "staged544-a1858-contraction-audit"
+      ],
+      "score": 5,
+      "excerpt": "source-bound A1858 contraction and its cleanup (comment 19035159). This is the consecutive block in PR #3998. It is not the separated PR #4019 contraction accepted in #4112.\n\nOrdered eligible list. Chronological open #1166 next tasks, comment-id order, after dropping tasks performed by a later DONE, an open PR, or this account\u2019s round through #4149:\n\n1. Largest physical child rank on one support alphabet (comment 19021222).\n2. T488 ownership/output invoice, question 1 (comment 19022532; question"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/p104-a2-paid-rank-exchange/README.md",
+      "path": "research/p104-a2-paid-rank-exchange/README.md",
+      "sha256": "8618db96c83e1e0dbfc1527adad0e8c54b31912df80fb030d4d52ce39b9c9358",
+      "matched_terms": [
+        "paid profile",
+        "3998",
+        "4019",
+        "4112",
+        "4152"
+      ],
+      "score": 5,
+      "excerpt": " emitted\nPR5627 word is changed by one two-to-three scalar rewrite and a legal frame\nitinerary. Its paid profile changes, every productive target is retained, and\nthe complete native/caller/compiler/outer calculation supports the higher \u03ba.\nThis is not a selected-frontier promotion, formal multiplication theorem,\nindependent human review or global optimum claim.\n\n| Quantity | PR5627 | This construction |\n|---|---:|---:|\n| Claimed conditional \u03ba | 0.00310440546 | **0.00310458** |\n| Increase | \u2014 | *"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/r439-donor466-enlarged-seam/README.md",
+      "path": "research/r439-donor466-enlarged-seam/README.md",
+      "sha256": "b59b4d69b2d56a3c95553beba54314b048903e83d62a10ddec242ac2dededd7f",
+      "matched_terms": [
+        "scalar-to-moment",
+        "paid profile",
+        "r439"
+      ],
+      "score": 5,
+      "excerpt": "# R439 B508: moving the left boundary does not remove the eleven-wire floor\n\n**Scoped proof and exact finite certificate; no new supplier or kappa.**\nIn the retained PR1062 word, enlarge the interval from the isolated B508 block\nto **A[1991:2075], the nine COPY/scatter/ERASE observations, and B[0:195]**.\nThe left cut precedes helper 466's rank-one-to-rank-four promotion. Nevertheless,\nall eleven B508 wires must contain the complete rational frame F508 by the\nright cut. Their exact minimum contri"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/staged544-a2-composition-audit/README.md",
+      "path": "research/staged544-a2-composition-audit/README.md",
+      "sha256": "7dbbe8ec43f9b72b1f384244384000d55dbcc02c87c026e4dce6b78bd5da64db",
+      "matched_terms": [
+        "3998",
+        "4019",
+        "staged544-a2-composition-audit"
+      ],
+      "score": 5,
+      "excerpt": "iscussioncomment-19036786):\nindependently accept the composition and projected-cleanup argument in\n[PR #4019](https://github.com/CrocSwap/integer-mult-bounds/pull/4019), head\n`f408e09df300b575235f7f77f81948071397e16d`.\n\nThe two rational identities, for distinct registers `x, y, z` and nonzero rationals `a, b`, are\n\n```text\n[S(a), T(b), U(-ab)] = [T(b), S(a)]\n[U(ab), T(b), S(a)]  = [S(a), T(b)]\n```\n\nwith `S(a): y += a*z`, `T(b): x += b*y`, and `U(c): x += c*z`. Both sides of the first identity en"
+    },
+    {
+      "kind": "local-readme",
+      "title": "research/staged544-separated-a2/README.md",
+      "path": "research/staged544-separated-a2/README.md",
+      "sha256": "69b1e9054227f707a63c952f213d7894fda0c77ef47a0c30223ee70a1df4eb23",
+      "matched_terms": [
+        "paid profile",
+        "3998",
+        "staged544-separated-a2"
+      ],
+      "score": 5,
+      "excerpt": "# Seven separated A2 contractions with an unchanged paid profile\n\nOn each exact PR3734 provider, seven separated three-ADD blocks contract to\ntwo ADDs without crossing a native event. Beyond PR3998's credited consecutive\nA1858 contraction, this removes **seven forward and five inverse-cleanup ADDs**.\nAgainst the original PR3734 words the combined saving is eight forward and six\ncleanup ADDs. The entire recursive histogram remains\n`{1:2440,2:725,3:224,4:102,5:20,8:8}`: 3,519 children and rank mas"
+    }
+  ]
+}
diff --git a/research/p104-a2-paid-rank-exchange/PUBLICATION-REPLAY.log b/research/p104-a2-paid-rank-exchange/PUBLICATION-REPLAY.log
new file mode 100644
index 0000000000..3079bd89f0
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/PUBLICATION-REPLAY.log
@@ -0,0 +1,13 @@
+CHECK Source hashes, full candidate regeneration, exact arbitrary-input/frame replay and unchanged productive targets.
+PASS exact source foundation bindings 12 and all arbitrary-input/frame rows
+PASS_P104_A2_QUICK
+CHECK Dense transpose, every signed native map/global phase, full caller, two exact moment engines, complete compiler/BIT stock/58 outer conditions and rejection controls.
+PASS exact source foundation bindings 12 and all arbitrary-input/frame rows
+PASS all dense transpose columns 752
+PASS all signed native maps and exact phases 2916
+PASS complete physical caller
+PASS two exact moment engines, compiler, BIT stock and58 outer predicates ['77614664639/25000000000000', '310458658557/100000000000000']
+PASS_P104_A2_KAPPA_00310458
+CHECK Separately implemented three-column A2 identity, complete scalar-word delta, fixed boundaries and exact child-profile change.
+PASS_INDEPENDENT_A2_IDENTITY_AND_PRICE_AUDIT
+PASS_REGISTERED_RESEARCH_PACKAGE research/p104-a2-paid-rank-exchange outcome=completed
diff --git a/research/p104-a2-paid-rank-exchange/PUBLICATION.md b/research/p104-a2-paid-rank-exchange/PUBLICATION.md
new file mode 100644
index 0000000000..9f51c88b8a
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/PUBLICATION.md
@@ -0,0 +1,46 @@
+# Publication checkpoint — October 25, 2026 (America/Detroit)
+
+The package is prepared on canonical main `df0137efd35e11b4445734bf612bdd4c7ebc330c`, which selects conditional κ = 0.00310045. Priority revision remains 7. Current AGENTS.md, CONTRIBUTING.md, autoresearch.md, priorities, selected result, package guide and invoice preflight guidance were read. The scientific predecessor remains [PR #5627](https://github.com/CrocSwap/integer-mult-bounds/pull/5627), head `ed4dc88a09a7d201971f6603b8c2074e95dd9c32`, confirmed unchanged before publication.
+
+The original research checkout's protected Git/network limitations are historical. Publication uses a separate sparse worktree with GitHub access and current canonical ancestry. Every original constructor, verifier, retained input, word, scientific receipt, source manifest and historical replay log is preserved byte for byte. README.md and PR-DRAFT.md are updated for publication; this record, a fresh replay log, refreshed overlap report and derived invoice comparison are added. Historical publication booleans and unrun-obligation lists describe the original session, not this later checkpoint.
+
+## Fresh validation
+
+Environment: Python 3.12.8, macOS 26.5.1, arm64; standard library only. Run from the publication worktree root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-a2-paid-rank-exchange
+```
+
+All three registered commands pass: quick source/interface replay, isolated optimized full replay and the separately implemented scoped algebra audit. [PUBLICATION-REPLAY.log](PUBLICATION-REPLAY.log) records the result. Coverage is 12 exact foundation bindings, 752 dense transpose columns, 2,916 signed native maps with full phases, complete physical caller, two exact rational moment engines, compiler completion, ordinary allowances, BIT stock, 54 strict plus four functional outer predicates and malformed-input controls. The saved RESULT.json and INDEPENDENT-AUDIT.json match the recomputed results.
+
+The current signed-invoice preflight also reports `uniformly_better`:
+
+```sh
+python3.12 -B scripts/compare_recursive_invoices.py \
+  research/p104-a2-paid-rank-exchange/publication-invoices/source.json \
+  research/p104-a2-paid-rank-exchange/publication-invoices/candidate.json
+```
+
+The two inputs are direct copies of `paid` and `targets` from the retained PR5627 and candidate RESULT.json `complete_cost` objects, under the same explicitly declared mixed-target accounting contract. [COMPARISON.json](publication-invoices/COMPARISON.json) gives Δpaid = 18(H2 + H6 − H3 − H5), Δtargets = 0, unchanged rank mass, and a strictly negative cost change for every power 0 < t < 1. This arithmetic screen does not supply physical legality; the full replay supplies the separately scoped native/caller evidence.
+
+The 41-file committed-byte check passes:
+
+```sh
+python3.12 -B scripts/check_research_commit.py \
+  --package research/p104-a2-paid-rank-exchange
+```
+
+The original provenance base `f626db5c3a3d781d8f7695038776c621651f75e1` is retained. The pending PR5627 scientific dependency is a self-contained authenticated archive/input closure rather than an undeclared foreign Git object. The optional `--check-source-history` mode was attempted and declined because this repository is shallow. A direct `git merge-base --is-ancestor f626db5c3a3d781d8f7695038776c621651f75e1 HEAD` check passes, but this is not the full-history preflight. The byte pass is separate from the mathematical replay and hosted CI.
+
+## Refreshed overlap
+
+The fresh [PUBLICATION-OVERLAP.json](PUBLICATION-OVERLAP.json), searched at 2026-10-26 01:58 UTC, covers 335 open PRs and 4,924 Discussion records with complete remote pagination and no refresh warnings. Local coverage is explicitly the original main `f626db5c` working-tree inventory: 5,863 records, with no omitted HEAD note paths. The report retains 20 ranked excerpts from 1,135 possible matches; no absence-of-matches or global novelty claim follows. Closed work is represented only through local notes and the ledger, and arbitrary code diffs are outside this search.
+
+Relevant current bodies were read. [PR #5633](https://github.com/CrocSwap/integer-mult-bounds/pull/5633) independently audits the unchanged PR5627 interface and invoice, not this new A2 word. [PR #5639](https://github.com/CrocSwap/integer-mult-bounds/pull/5639) and [PR #5641](https://github.com/CrocSwap/integer-mult-bounds/pull/5641) concern ordinary BIT components and explicitly do not claim a complete multiplication κ increase. [PR #5640](https://github.com/CrocSwap/integer-mult-bounds/pull/5640) supplies a distinct borrowed-space shear tariff with physical/caller obligations still separate. The A2 predecessor investigations and their credit remain recorded in README.md and PR-DRAFT.md.
+
+## Remaining obligations
+
+Hosted CI, independent human scientific review and proof-kernel verification remain separate. The same five framework assumptions and helper-endpoint contract remain; the enormous uniform word and complete rounded multiplier are algorithm-defined rather than materialized. This publication does not change the selected result or resolve others' PRs. Open next questions remain in README.md and TASK.json.
+
+Prepared for [zxv](https://github.com/zxv) with substantial OpenAI Codex assistance; all retained source credit and Apache-2.0 notices remain.
diff --git a/research/p104-a2-paid-rank-exchange/README.md b/research/p104-a2-paid-rank-exchange/README.md
new file mode 100644
index 0000000000..c881246e18
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/README.md
@@ -0,0 +1,163 @@
+# Conditional κ = 0.00310458 from a paid A2 rank exchange
+
+**Author-checked conditional construction.** The exact emitted
+PR5627 word is changed by one two-to-three scalar rewrite and a legal frame
+itinerary. Its paid profile changes, every productive target is retained, and
+the complete native/caller/compiler/outer calculation supports the higher κ.
+This is not a selected-frontier promotion, formal multiplication theorem,
+independent human review or global optimum claim.
+
+| Quantity | PR5627 | This construction |
+|---|---:|---:|
+| Claimed conditional κ | 0.00310440546 | **0.00310458** |
+| Increase | — | **0.00000017454**, about **0.00562%** |
+| Raw crossing | (0.00310440547057, 0.00310440547058) | **(0.00310458658556, 0.00310458658557)** |
+| Tested saving | 0.003104405465 | 0.0031045805 |
+| Complete paid calls | 66,546 | 66,546 |
+| Complete productive targets | 12,958 | 12,958 |
+| Rank-mass surplus | 760 | 760 |
+| Public inputs / owned roles / peak owned | 743 / 9 / 2 | 743 / 9 / 2 |
+| Local ordinary scalar allowance | 220,154 | 220,166 |
+
+The raw ceiling is strictly above the claimed κ by between
+6.58556×10⁻⁹ and 6.58557×10⁻⁹. The changed signed moment has a rigorously
+negative derivative throughout [0, 0.0034], so this is the unique raw funding
+ceiling for this fixed invoice under the retained BIT cap. The gain is a
+physical paid-profile change, not a refinement of the predecessor's root.
+
+## Mechanism and proof
+
+At source events 14975 and 14977, write A=715, B=388 and C=340. Replace
+
+```
+B -= 4 A     at frame 825
+C += B       at frame 825
+```
+
+by
+
+```
+C -= 4 A     at frame 396
+C += B       at frame 825
+B -= 4 A     at frame 127
+```
+
+Both give `(A, B−4A, C+B−4A)` on arbitrary inputs. The extra ordinary ADD moves
+one dependency earlier and another later. Source A can visit dimension three
+instead of dimension six before its full-dimensional final contact. Its old
+paid itinerary has ranks 5 and 3; the new itinerary has ranks 2 and 6. The
+frames form the actual nested chain 598⊂396⊂825⊂127, with dimensions 1,3,6,9.
+No new frame, register, input, COPY, INIT or CLEAR is introduced.
+
+The complete local change is
+
+\[
+\Delta P=H_2+H_6-H_3-H_5,\qquad \Delta T=0.
+\]
+
+For every 0<t<1, strict concavity gives
+`3^t−2^t > 6^t−5^t`. Thus the paid moment falls strictly, although one ADD is
+added and rank mass is unchanged. The complete caller invokes the local word
+18 times, so its paid change is 18 times this expression. The correction's
+ordinary cost is included in the revised allowance shown above.
+
+This changes the local debt in zxv's
+[PR5554 price-frontier theorem](https://github.com/CrocSwap/integer-mult-bounds/pull/5554).
+It does not contradict that theorem's fixed-provider ceiling. Earlier A2
+contractions in PR3998/4019 and their PR4112/4152 audits retained their paid
+profiles; those frozen-word exclusions are not claims about this changed P104
+word and itinerary. The elementary A2 identity itself is credited prior work.
+
+[INDEPENDENT-AUDIT.md](INDEPENDENT-AUDIT.md) gives the separate agent's
+three-column proof and all-column frame-induction argument. Actual native
+maps include their full phases. The argument does not commute Gaussians or
+treat tensor powers of sums as sums of tensor powers.
+
+## Verification
+
+Python 3.12 and the standard library suffice. From the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-a2-paid-rank-exchange
+```
+
+For a standalone exported directory, run `python3.12 -I -O -B verify.py` from
+any working directory. `--quick` checks the exact source, regenerated word and
+full rational scalar/frame interface without the native replay. `audit.py`
+independently checks the scalar identity, entire scalar-program delta,
+unchanged boundaries/owned instructions, nesting and complete paid histogram.
+
+The full verifier checks all 752 arbitrary-input rows and dense transpose
+columns; 2,916 distinct signed native maps with exact global phases; the full
+physical caller; both exact rational moment engines; compiler completion,
+precision/ordinary allowances and BIT stock; all 54 strict and four functional
+outer predicates. Controls reject an omitted new shear, wrong contact frame,
+undercharged child and missing central translation.
+
+`RESULT.json` and `REPLAY.log` retain the complete successful source-bound
+calculation. `PRICING-EDITS.json` binds the price adaptation to three literal
+claim/comparison edits; no mathematical predicate is weakened. The historical
+`previous_open_pr5601` output-key spelling is inherited; its value in this
+package is the explicitly declared PR5627 comparison, 0.00310440546.
+
+The historical hosted nine-stage packing cap does not apply to this mixed-target
+productive-helper construction: independent helper targets and the complete
+caller replace that coverage law. The replacement bound is the unique complete-
+invoice raw crossing above, proved on [0, 0.0034]. The claimed κ realizes about
+99.9997879% of that fixed-invoice ceiling; this is not a ceiling on changed words.
+
+## Sources, scope and attribution
+
+The exact scientific predecessor is
+[PR5627](https://github.com/CrocSwap/integer-mult-bounds/pull/5627), head
+`ed4dc88a09a7d201971f6603b8c2074e95dd9c32`. Its word, checker, invoice and
+source manifest are retained unchanged. The PR5612 authenticated archive
+reader supplies the unchanged framework closure. All 12 PR5627-declared
+foundation files present in that closure match its recorded SHA256 values
+before use. `SOURCES.json` additionally authenticates every retained input
+byte. There is no foreign-branch object-cache dependency.
+
+During the original research session, canonical Git fetch failed because
+that checkout's `.git/FETCH_HEAD` was
+protected. Exact source files were instead downloaded from their pinned
+GitHub pages. Local task provenance remains main `f626db5c`, priority 7; it is
+not represented as a refreshed canonical checkout. The live open-PR list was
+checked through PR5635, including PR5633's narrower source-interface/invoice
+audit. The older broad overlap cache has incomplete freshness, and no absence
+of matches is asserted as proof of novelty. The submission-time browser check
+subsequently synchronized the publication fork to canonical main
+`09062e91a942f205a529779b74c13c412afdb483`, read current `autoresearch.md`,
+priorities (still revision 7) and `CONTRIBUTING.md`, and checked the live PR
+list through PR5639. PR5639 explicitly does not claim a new best whole
+multiplication κ. The scientific source pins and earlier replay evidence
+above remain frozen; shell Git refresh and committed-byte publication
+preflight remain unavailable. No source frontier, frozen package
+or others' PR was changed. The later publication checkout resolves the Git
+refresh and committed-byte limitations; see [PUBLICATION.md](PUBLICATION.md)
+for the fresh replay, source-history scope and overlap coverage. Historical
+`CLAIM.json`, `TASK.json`, `VALIDATION.json` and their logs retain the original
+session's status and limitations. Publication/CI/maintainer acceptance remain
+separate.
+
+The same five framework assumptions remain: native all-column realization,
+owned erasure, uniform compiler closure, precision/fixed-tape execution and
+analytic recovery, together with the inherited helper-endpoint contract.
+The enormous uniform word and complete rounded multiplier are not materialized.
+
+Prepared for **zxv**, with substantial OpenAI Codex assistance and complementary
+subagent searches. Ali Romman supplies the PR5627 helper/basis composition and
+checker; flyingviper supplies the relays; Scratch-net the prepared prefix;
+Pierre-Loic Doulcet/hexapode the commuting cascade; Joel Pulikkan/GamingPuzzled,
+DreamingOfClouds and thu-nmrc the predecessor handoffs and contacts. Retain
+David Arias, Gareth Davidson, Cinna Davis, Jacob Alan Sussman, Cao Yifan, zxv
+and all native, geometry, BIT, compiler and analytic source notices. Chris R /
+Clear1nsight's PR3998, ToikY's PR4019 and the scoped audits retain credit for
+the A2 mechanism and boundary discipline. Apache-2.0.
+
+## Open next questions
+
+1. Independently review the three literal gates, source chart chain and new
+   paid maps before scientific admission; use the isolated replay above.
+2. Test a different noncommuting scalar motif on the emitted word. A combined
+   gain needs a reconstructed common word and complete invoice; local headline
+   gains are not assumed additive. No further positive candidate is claimed.
diff --git a/research/p104-a2-paid-rank-exchange/REGISTERED-REPLAY.log b/research/p104-a2-paid-rank-exchange/REGISTERED-REPLAY.log
new file mode 100644
index 0000000000..3079bd89f0
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/REGISTERED-REPLAY.log
@@ -0,0 +1,13 @@
+CHECK Source hashes, full candidate regeneration, exact arbitrary-input/frame replay and unchanged productive targets.
+PASS exact source foundation bindings 12 and all arbitrary-input/frame rows
+PASS_P104_A2_QUICK
+CHECK Dense transpose, every signed native map/global phase, full caller, two exact moment engines, complete compiler/BIT stock/58 outer conditions and rejection controls.
+PASS exact source foundation bindings 12 and all arbitrary-input/frame rows
+PASS all dense transpose columns 752
+PASS all signed native maps and exact phases 2916
+PASS complete physical caller
+PASS two exact moment engines, compiler, BIT stock and58 outer predicates ['77614664639/25000000000000', '310458658557/100000000000000']
+PASS_P104_A2_KAPPA_00310458
+CHECK Separately implemented three-column A2 identity, complete scalar-word delta, fixed boundaries and exact child-profile change.
+PASS_INDEPENDENT_A2_IDENTITY_AND_PRICE_AUDIT
+PASS_REGISTERED_RESEARCH_PACKAGE research/p104-a2-paid-rank-exchange outcome=completed
diff --git a/research/p104-a2-paid-rank-exchange/REPLAY.log b/research/p104-a2-paid-rank-exchange/REPLAY.log
new file mode 100644
index 0000000000..18694175a1
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/REPLAY.log
@@ -0,0 +1,6 @@
+PASS exact source foundation bindings 12 and all arbitrary-input/frame rows
+PASS all dense transpose columns 752
+PASS all signed native maps and exact phases 2916
+PASS complete physical caller
+PASS two exact moment engines, compiler, BIT stock and58 outer predicates ['77614664639/25000000000000', '310458658557/100000000000000']
+PASS_P104_A2_KAPPA_00310458
diff --git a/research/p104-a2-paid-rank-exchange/RESULT.json b/research/p104-a2-paid-rank-exchange/RESULT.json
new file mode 100644
index 0000000000..b20672f0a1
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/RESULT.json
@@ -0,0 +1,342 @@
+{
+  "complete_cost": {
+    "compiler": {
+      "H": 707182000000000,
+      "N": "2^707182000000000",
+      "eta": {
+        "lower": "0.000000134111266725226492278714",
+        "upper": "0.000000134111266725226492278715"
+      },
+      "gap": "1/14000000000",
+      "integer_test_exponent": 50513,
+      "logarithm_power": 1024,
+      "theta": "63/64",
+      "uniform_saving": "21732063/7000000000",
+      "word_materialized": false
+    },
+    "full_forward_calls": 18,
+    "gain": "8727/50000000000",
+    "kappa": "155229/50000000",
+    "kappa_decimal": ".00310458",
+    "margin_at_previous": {
+      "lower": "0.044045194632554451118597166502",
+      "upper": "0.044045194632554451118597166503"
+    },
+    "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": 5986,
+      "local_routes": 5175000,
+      "local_scalar": 220166,
+      "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": "3485999999253/3500000000000",
+        "fixed_halo_explicit_metadata_branch": "269500000001/3500000000",
+        "new_BIT_row_stock_sublinear_in_active_group": "13957999997009/14000000000000",
+        "new_semantic_group_guard": "3/14000000000"
+      },
+      "margins": {
+        "boundary_algebraic_matrix": "608497735956535877/196000000000000000000",
+        "fused_gaussian": "608497735869607631/196000000000000000000",
+        "global_coordinate_router": "10866031/3500000000",
+        "group_bit_router": "152124437467401907/49000000000000000000",
+        "group_scans": "6999999999/7000000000",
+        "main_transform": "608497721956535877/196000000000000000000",
+        "microbox_face_corrections": "269500000001/3500000000",
+        "polynomial_products": "13999999999/14000000000",
+        "sparse_union_movement": "80233497735956535877/196000000000000000000"
+      },
+      "parameters": {
+        "a": "10866031/3500000000",
+        "b": "347713/112000000",
+        "beta": "13999999997/14000000000",
+        "bit_saving": "17/5000",
+        "cap_xi": "178233497721956535877/196000000000000000000",
+        "complex_uniform_saving": "21732063/7000000000",
+        "d": "1/14000000000",
+        "epsilon": "13999999999/14000000000",
+        "kappa": "155229/50000000",
+        "lambda_": "3987581679/4000000000",
+        "lambda_prime": "13956535877/14000000000",
+        "layout": "1/14000000000",
+        "q": "43464123/14000000000",
+        "r0": "1/140000000000",
+        "stop": "1/14000000000",
+        "tested_saving": "6209161/2000000000"
+      },
+      "strict_checks": {
+        "actual_bit_above_effective": "1033969/3500000000",
+        "actual_logarithmic_gamma_guard": "1/14000000000",
+        "actual_logarithmic_group_guard": "17766502299043464123/98000000000000000000",
+        "alpha_logarithm_absorption": "1/140000000000",
+        "boundary_algebraic_matrix_above_kappa": "55956535877/196000000000000000000",
+        "cap_above_band": "80233497721956535877/196000000000000000000",
+        "cap_below_cell": "17766502264043464123/196000000000000000000",
+        "cap_below_phase_cell_power": "213766502278043464123/196000000000000000000",
+        "complex_above_effective": "1/14000000000",
+        "complex_below_one_over_32": "3152287/112000000",
+        "complex_endpoint": "1/7000000000",
+        "epsilon_below_one": "1/14000000000",
+        "epsilon_positive": "13999999999/14000000000",
+        "fine_prime_halo_umbrella": "7/2",
+        "fused_gaussian_above_kappa": "55869607631/196000000000000000000",
+        "gaussian_above_main": "6956535877/98000000000000000000",
+        "global_coordinate_router_above_kappa": "1/3500000000",
+        "global_gamma_power_slack": "9/140000000000",
+        "group_below_dimension": "1/7000000000",
+        "group_bit_router_above_kappa": "17467401907/49000000000000000000",
+        "group_gauge_power_slack": "17766502298343464123/98000000000000000000",
+        "group_key_fits_record": "1/7000000000",
+        "group_positive": "13999999997/14000000000",
+        "group_scans_above_kappa": "6978267939/7000000000",
+        "halo_covers_generic_inverse_radius": "5",
+        "improvement_over_user_baseline": "686804790400418148631/500000000000000000000000",
+        "lambda_above_internal": "1/28000000000",
+        "lambda_above_sigma": "3/28000000000",
+        "lambda_above_tau": "1/28000000000",
+        "lambda_prime_above_lambda": "1/28000000000",
+        "lambda_prime_above_leaf": "223652287/1568000000000000000",
+        "leaf_saving_above_effective": "111652287/1568000000000000000",
+        "main_transform_above_kappa": "41956535877/196000000000000000000",
+        "membership_arithmetic_fits_record": "1/14000000000",
+        "microbox_face_corrections_above_kappa": "269489133971/3500000000",
+        "microbox_halo_volume": "1106000000003/14000000000",
+        "new_complex_uniform_saving_above_b": "1/14000000000",
+        "new_semantic_guard_with_C1_one": "1/14000000000",
+        "polynomial_products_above_kappa": "13956535879/14000000000",
+        "positive_layout_slack": "1/14000000000",
+        "q_below_effective": "1/14000000000",
+        "q_below_internal": "1/14000000000",
+        "q_below_leaf": "223652287/1568000000000000000",
+        "q_positive": "43464123/14000000000",
+        "short_record_fallback": "111652287/112000000",
+        "source_axis_superpolynomial": "1/14000000000",
+        "sparse_union_movement_above_kappa": "79625000055956535877/196000000000000000000",
+        "stop_below_one": "13999999999/14000000000",
+        "stop_positive": "1/14000000000",
+        "strassen_integer_power_gap": "1951441519231",
+        "support_router_span": "3/14000000000",
+        "theta_positive_polynomial_power": "2",
+        "union_header_fits_record": "3/14000000000",
+        "union_tag_work_per_visit_fits_record": "3/14000000000"
+      }
+    },
+    "paid": {
+      "1": 41160,
+      "16": 416,
+      "2": 8226,
+      "24": 416,
+      "3": 8874,
+      "32": 416,
+      "4": 1638,
+      "40": 416,
+      "48": 416,
+      "5": 2916,
+      "56": 416,
+      "6": 342,
+      "64": 416,
+      "7": 270,
+      "72": 208
+    },
+    "paid_calls": 66546,
+    "previous_open_pr5601": "155220273/50000000000",
+    "productive_fields": 12958,
+    "query_shears": 15,
+    "rank_mass_surplus": 760,
+    "raw_ceiling": {
+      "derivative_upper": "-740652532917121353648727598109847061213602736923586290417876337/3138550867693340381917894711603833208051177722232017256448",
+      "interval": [
+        "0",
+        "17/5000"
+      ]
+    },
+    "raw_headroom": [
+      "164639/25000000000000",
+      "658557/100000000000000"
+    ],
+    "raw_root_interval": [
+      "77614664639/25000000000000",
+      "310458658557/100000000000000"
+    ],
+    "relative_gain_percent": "290900/51740091",
+    "strictification": {
+      "alpha": {
+        "lower": "0.999999993807112591662250764905",
+        "upper": "0.999999993807112591662250764906"
+      },
+      "cutoff": 17,
+      "delta": {
+        "lower": "0.967529933442697782667099686380",
+        "upper": "0.967529933442697782667099686381"
+      },
+      "floor": 18,
+      "gamma": {
+        "lower": "0.968063886259541319614126023064",
+        "upper": "0.968063886259541319614126023065"
+      },
+      "normalized_paid_upper": "1048576000976562501/1048576001073741824",
+      "power": 1024,
+      "reserve": "1/1000000000000",
+      "tail_block": 128,
+      "tail_decay": "39/40",
+      "tail_offset": "1/5",
+      "tested": "6209161/2000000000"
+    },
+    "targets": {
+      "1": 36,
+      "145": 208,
+      "17": 416,
+      "2": 72,
+      "25": 416,
+      "33": 416,
+      "4": 1638,
+      "41": 416,
+      "49": 416,
+      "5": 108,
+      "57": 416,
+      "65": 416,
+      "7": 90,
+      "73": 208,
+      "8": 288,
+      "9": 7398
+    },
+    "whole_idle_jobs": 3120
+  },
+  "controls": {
+    "missing_A2_correction": "All public and owned rows",
+    "missing_central_translation": "Rejected by the complete Laurent endpoint",
+    "undercharged_child": "Complete rank surplus",
+    "wrong_new_contact_frame": "Scalar destination and coefficient"
+  },
+  "local": {
+    "endpoint_coefficients": 565504,
+    "helpers": {
+      "1": 2,
+      "2": 4,
+      "4": 91,
+      "5": 6,
+      "7": 5,
+      "8": 16,
+      "9": 411
+    },
+    "owned_fields": 9,
+    "owned_lifetimes": 11,
+    "paid": {
+      "1": 2200,
+      "2": 457,
+      "3": 493,
+      "4": 91,
+      "5": 162,
+      "6": 19,
+      "7": 15
+    },
+    "paid_rank_mass": 5986,
+    "peak_owned_fields": 2,
+    "public_inputs": 743,
+    "real_scalar_allowance": 220166
+  },
+  "native": {
+    "distinct_maps": 2916,
+    "helper_normalizers": 1070,
+    "helpers": {
+      "1": 2,
+      "2": 4,
+      "4": 91,
+      "5": 6,
+      "7": 5,
+      "8": 16,
+      "9": 411
+    },
+    "main_adapters": 208,
+    "proof": "Every signed Pauli image and exact global phase; frame induction at every positive tensor width",
+    "recipe_payload_sha256": "e2b50e4067abb6ce5255be46754c1a3996d179cd95f949a86249762539138ef3",
+    "wrong_phase_rejected": true
+  },
+  "physical": {
+    "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
+    ]
+  },
+  "source_foundation_matches": [
+    "research/p104-productive-controller/caller.py",
+    "research/p104-productive-controller/CONTROLLER.json",
+    "research/joint-helper-lattice/CONTROLLER.json",
+    "research/p104-native-channel/vendor/clifford_exact.py",
+    "research/p104-native-channel/vendor/native_verify.py",
+    "research/pr4359-p7-padding/native_inventory.py",
+    "research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json",
+    "research/higher-moment-strictification/verify.py",
+    "research/kappa-rebuild/architecture/mixed_target_probe.py",
+    "research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py",
+    "research/query-thinned-controllers/RESULT.json",
+    "research/p104-productive-controller/local.py"
+  ],
+  "source_head": "ed4dc88a09a7d201971f6603b8c2074e95dd9c32",
+  "source_pr": 5627,
+  "status": "PASS_P104_A2_KAPPA_00310458",
+  "transpose": {
+    "arithmetic": "Dense integer rows with binary denominators; reverse chronology",
+    "coefficients": 565504,
+    "columns": 752
+  },
+  "wall_seconds": "156.498"
+}
diff --git a/research/p104-a2-paid-rank-exchange/SOURCES.json b/research/p104-a2-paid-rank-exchange/SOURCES.json
new file mode 100644
index 0000000000..40857a52ba
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/SOURCES.json
@@ -0,0 +1,102 @@
+{
+  "schema_version": 1,
+  "base_commit": "f626db5c3a3d781d8f7695038776c621651f75e1",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ed33caef5c5827693e9572d252a66f99cbb73d40",
+      "sha256": "5732707cc1e460b090b53f2463162749ad860de878a18f7408938671d295a9d9"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    }
+  ],
+  "source_pr": 5627,
+  "source_head": "ed4dc88a09a7d201971f6603b8c2074e95dd9c32",
+  "comparison_kappa": "0.00310440546",
+  "retained_inputs": [
+    {
+      "local_path": "inputs/pr5612/SOURCES.json",
+      "size": 5008,
+      "sha256": "20a101adf25437cfa63110f0b8cb63b31001af0d4a6d7488ceffd27e63f4b0b4",
+      "git_blob": "4e7fa4ba58f8316ef9aba90d53d197172f65dbb6"
+    },
+    {
+      "local_path": "inputs/pr5612/inputs/pr5601-CHANGES.json",
+      "size": 637,
+      "sha256": "cc9c3088d509e56362b552cd79875bce45531c2183f3ac0b83a6d75e0d462467",
+      "git_blob": "a46098f911d6483fcba9e3c89ac2d24e24cb1a74"
+    },
+    {
+      "local_path": "inputs/pr5612/inputs/pr5601-construct.py",
+      "size": 2507,
+      "sha256": "40aa1dd9aa9faca1a4faae7f923e90f216dc74267a92922e02d495222f07abdc",
+      "git_blob": "2506f4bdc82be48ea3e97e3eabc1f96cd0f61329"
+    },
+    {
+      "local_path": "inputs/pr5612/inputs/pr5606-construct.py",
+      "size": 1522,
+      "sha256": "037d5f00f1c92b6f59020f90f11345251b429ba2bb2c39a65966b5e73094756a",
+      "git_blob": "e2c1473be00e78b91b4139493e31e47b591ac0df"
+    },
+    {
+      "local_path": "inputs/pr5612/inputs/pr5606-prepared.py",
+      "size": 5872,
+      "sha256": "166d3362f5a776d422412efc5e905fb3ace13caa1e715b870b175295e145a4eb",
+      "git_blob": "6eae5764287d6325ad372400480a0447a12568a7"
+    },
+    {
+      "local_path": "inputs/pr5612/inputs/pr5606-pricing.py",
+      "size": 9765,
+      "sha256": "c7e2a95c1a285982e0db9a9b881154ecda1ee76031cf216909e30b4939dd841a",
+      "git_blob": "02efc04ddec40b9cec9f785b00fd20841be6009d"
+    },
+    {
+      "local_path": "inputs/pr5612/inputs/pr5608-source.tar.gz",
+      "size": 1030364,
+      "sha256": "dbded26ad3b6f7800c6e3b76aaee4df3e86dcb2278bb882d97fe3007f76e0712",
+      "git_blob": "404e95f089c10ea0589934ea45160d8604468b68"
+    },
+    {
+      "local_path": "inputs/pr5612/sources.py",
+      "size": 2664,
+      "sha256": "a4524e1597f794fb071351d2fd192c0df296c89b08f1cc73c80adb218283c6d8",
+      "git_blob": "fce090a6b72958277da8716a16aceec5e8b3e04c"
+    },
+    {
+      "local_path": "inputs/pr5627-RESULT.json",
+      "size": 15990,
+      "sha256": "f24925b2f549dce6ff404bd9d35fb0a19e5ebc5fa5cbf67996fd4064e450137b",
+      "git_blob": "1e90afdd0372980c68c1a444a6c0fe5f1e7da7c2"
+    },
+    {
+      "local_path": "inputs/pr5627-SOURCES.json",
+      "size": 10676,
+      "sha256": "3f530d4409fa3b55211bb1ae7a616369cf6e9cd85f8eec7a74293c3dcd2b04af",
+      "git_blob": "bb8cb5eca312aa302fe442885e028d04a822f6e4"
+    },
+    {
+      "local_path": "inputs/pr5627-WORD.json.gz",
+      "size": 107885,
+      "sha256": "162e6f13a6e3805517946309bd6247eaa45b337ab9c39621806b254b415075ec",
+      "git_blob": "b60955f312f76d08ee4f86ac2ad664e3ab4e46ea"
+    },
+    {
+      "local_path": "inputs/pr5627-local.py",
+      "size": 9392,
+      "sha256": "716d01a342467f87b14d67853d8de30740f9f2fa881afe271db0df5ef8b19697",
+      "git_blob": "e72cb02dba9b68d3bd5934371b5d25a44589c482"
+    },
+    {
+      "local_path": "inputs/pr5627-pricing.py",
+      "size": 10146,
+      "sha256": "2e783d890833dd8c12320964965c91c38a82a3b3943acb6e5cd1a6246a20bb00",
+      "git_blob": "430934914b1cc653e8ea1da91e90f5da9192d41f"
+    }
+  ],
+  "source_routing": "Exact-commit GitHub Download raw file; retained bytes bound here. Foreign Git objects are not available in this checkout."
+}
diff --git a/research/p104-a2-paid-rank-exchange/TASK.json b/research/p104-a2-paid-rank-exchange/TASK.json
new file mode 100644
index 0000000000..86176b9c8d
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/TASK.json
@@ -0,0 +1,118 @@
+{
+  "schema_version": 1,
+  "task_id": "scalar-to-moment",
+  "priority_revision": 7,
+  "base_commit": "f626db5c3a3d781d8f7695038776c621651f75e1",
+  "original_card": {
+    "id": "scalar-to-moment",
+    "question": "Can a changed frame itinerary remove a paid transition beyond the already covered scalar A2 contractions?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/staged544-a2-graph-rewrite/README.md",
+      "research/staged544-separated-a2/README.md",
+      "research/staged544-a2-composition-audit/README.md",
+      "research/staged544-a1858-contraction-audit/README.md"
+    ],
+    "methods": [
+      "Derive the changed legal frame itinerary and exact child histogram.",
+      "Prove a scoped obstruction showing why that scalar gain cannot change the paid contract."
+    ],
+    "acceptance_gate": "An unchanged histogram is not a kappa gain; record any separate runtime or proof benefit explicitly.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read the A2 and separated-A2 constructions and their composition/contraction audits. Scalar contractions can preserve the complete recursive histogram; fewer ADDs alone do not justify further scans.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Identify an actual paid transition to remove and construct a legal replacement frame itinerary, charging restoration, source windows and changed helpers. Compare complete before/after histograms. Stop if only scalar count changes; report that benefit honestly, without expanding the same scalar-only census.",
+    "bottleneck_link": "A legal change to the histogram, not addition count alone, is the bridge to an exponent gain.",
+    "allowed_variations": [
+      "One existing scalar simplification and one proposed frame-path change."
+    ],
+    "scope_change_triggers": [
+      "Submitting only another scalar count as completion of a paid-profile task."
+    ],
+    "overlap_terms": [
+      "scalar simplification",
+      "paid profile",
+      "R439",
+      "3998",
+      "4019",
+      "4112",
+      "4152",
+      "staged544-a2-graph-rewrite",
+      "staged544-separated-a2",
+      "staged544-a2-composition-audit",
+      "staged544-a1858-contraction-audit"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #3998: consecutive A2 scalar contraction",
+        "evidence": "research/staged544-a2-graph-rewrite/README.md",
+        "remaining": "A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.",
+        "status": "author-checked",
+        "source_commit": "389268a6b5e0373a3125d5c6fcd6995f4ea324f5",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4019: separated A2 sites with unchanged recursive histogram",
+        "evidence": "research/staged544-separated-a2/README.md",
+        "remaining": "A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.",
+        "status": "author-checked",
+        "source_commit": "f408e09df300b575235f7f77f81948071397e16d",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4112: composition audit",
+        "evidence": "research/staged544-a2-composition-audit/README.md",
+        "remaining": "A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.",
+        "status": "author-checked",
+        "source_commit": "205b7380656ba4b6316a8a493b1a3c679b8f7a0e",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4152: A1858 contraction audit",
+        "evidence": "research/staged544-a1858-contraction-audit/README.md",
+        "remaining": "A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.",
+        "status": "author-checked",
+        "source_commit": "caaa14cde5962adc372fdca71f63d0f29e73c8b2",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Apply one literal two-to-three A2 graph rewrite to PR5627 and reassemble legal native frame paths to replace H3+H5 by H2+H6, retaining every productive target and checking the full conditional kappa.",
+  "scope_change_reason": "The user assigned the next kappa improvement; this is an actual paid-profile change on the newer P104 word rather than another scalar-only contraction on the older staged544 word.",
+  "bottleneck_link": "The exact local paid moment drops at every exponent between zero and one; the full eighteen-call supplier and outer transfer certify a higher conditional kappa.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at one exact paid-profile improvement, full source-bound replay, explicit raw ceiling and headroom; do not claim a global optimum or selected promotion.",
+  "outcome": "completed",
+  "evidence": "RESULT.json, REPLAY.log, INDEPENDENT-AUDIT.md and verify.py. Full native, physical caller, both exact moment engines, compiler and 58 outer predicates pass.",
+  "author": "zxv, with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance. The main agent ran the source and complete candidate replays; five subagents explored complementary mechanisms, and a separate native-factor agent audited the successful A2 identity and frame implication.",
+  "open_questions": [
+    {
+      "question": "Does another compatible noncommuting shear rewrite improve this emitted word?",
+      "why_it_matters": "The positive step escapes fixed-graph commuting retiming screens.",
+      "first_step": "Screen a bounded different scalar motif, construct the combined word, and compare its fully charged moment.",
+      "blocked_by": "No further positive candidate is claimed."
+    },
+    {
+      "question": "Can an independent reviewer reproduce the new native maps and all-column frame invariant?",
+      "why_it_matters": "Agent cross-checks do not replace independent scientific acceptance.",
+      "first_step": "Run the isolated verifier and audit.py, then inspect the frame induction in INDEPENDENT-AUDIT.md.",
+      "blocked_by": "The same five inherited framework hypotheses remain."
+    }
+  ],
+  "source_comparison": {
+    "pr": 5627,
+    "head": "ed4dc88a09a7d201971f6603b8c2074e95dd9c32",
+    "kappa": "0.00310440546"
+  },
+  "refresh_limitation": "Canonical fetch was attempted but .git/FETCH_HEAD is protected. Source commits were instead read and downloaded through exact-commit GitHub pages. The current live open-PR list was checked through PR5635; cached broad overlap is older."
+}
diff --git a/research/p104-a2-paid-rank-exchange/VALIDATION.json b/research/p104-a2-paid-rank-exchange/VALIDATION.json
new file mode 100644
index 0000000000..2ced9bd72f
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/VALIDATION.json
@@ -0,0 +1,30 @@
+{
+  "status": "PASS",
+  "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",
+  "registered_replay": "PASS_REGISTERED_RESEARCH_PACKAGE research/p104-a2-paid-rank-exchange outcome=completed",
+  "source_bound_full_replay_log": "REPLAY.log",
+  "isolated_optimized_full_replay_log": "OPTIMIZED-REPLAY.log",
+  "registered_replay_log": "REGISTERED-REPLAY.log",
+  "source_foundation_matches": [
+    "research/p104-productive-controller/caller.py",
+    "research/p104-productive-controller/CONTROLLER.json",
+    "research/joint-helper-lattice/CONTROLLER.json",
+    "research/p104-native-channel/vendor/clifford_exact.py",
+    "research/p104-native-channel/vendor/native_verify.py",
+    "research/pr4359-p7-padding/native_inventory.py",
+    "research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json",
+    "research/higher-moment-strictification/verify.py",
+    "research/kappa-rebuild/architecture/mixed_target_probe.py",
+    "research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py",
+    "research/query-thinned-controllers/RESULT.json",
+    "research/p104-productive-controller/local.py"
+  ],
+  "scope": "Actual cold source-export reader and complete candidate native/caller/moment/compiler/outer replay; independent agent scalar/frame/paid-delta audit. No historical suites or Lean replay.",
+  "not_run": [
+    "Hosted CI",
+    "Publication committed-byte preflight (checkout .git is protected and package is not committed)",
+    "Independent human or proof-kernel audit"
+  ],
+  "source_files_unchanged_during_registered_replay": true
+}
diff --git a/research/p104-a2-paid-rank-exchange/VERIFY.json b/research/p104-a2-paid-rank-exchange/VERIFY.json
new file mode 100644
index 0000000000..f17d47c258
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/VERIFY.json
@@ -0,0 +1,41 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py",
+        "--quick"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Source hashes, full candidate regeneration, exact arbitrary-input/frame replay and unchanged productive targets."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 600,
+      "checks": "Dense transpose, every signed native map/global phase, full caller, two exact moment engines, complete compiler/BIT stock/58 outer conditions and rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/audit.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Separately implemented three-column A2 identity, complete scalar-word delta, fixed boundaries and exact child-profile change."
+    }
+  ]
+}
diff --git a/research/p104-a2-paid-rank-exchange/WORD.json.gz b/research/p104-a2-paid-rank-exchange/WORD.json.gz
new file mode 100644
index 0000000000..7cf851189d
Binary files /dev/null and b/research/p104-a2-paid-rank-exchange/WORD.json.gz differ
diff --git a/research/p104-a2-paid-rank-exchange/audit.py b/research/p104-a2-paid-rank-exchange/audit.py
new file mode 100644
index 0000000000..ab7f44cb5b
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/audit.py
@@ -0,0 +1,52 @@
+"""Independent scoped audit of the proposed P104 A2 paid exchange.
+
+Prepared for zxv with substantial OpenAI Codex assistance. Apache-2.0.
+The transport agent supplies the construction; all source P104 credit remains.
+This checks the exact local rewrite and invoice, not the full inherited theorem.
+"""
+from fractions import Fraction as Q
+from collections import Counter
+from pathlib import Path
+from hashlib import sha256
+import gzip,json
+HERE=Path(__file__).resolve().parent
+SOURCE=HERE/'inputs/pr5627-WORD.json.gz'
+CANDIDATE=HERE/'WORD.json.gz'
+def need(ok,why):
+ if not ok:raise ValueError(why)
+def read(p):return json.loads(gzip.decompress(p.read_bytes()))
+def add(row,other,q):
+ for k,v in other.items():
+  z=row.get(k,0)+q*v
+  if z:row[k]=z
+  else:row.pop(k,None)
+def matrix(events):
+ rows={r:{r:Q(1)} for r in (340,388,715)}
+ for e in events:add(rows[e[1]],rows[e[2]],Q(e[-2]))
+ return rows
+def span(b):
+ s={0}
+ for v in b:s|={x^v for x in tuple(s)}
+ return s
+w,v=read(SOURCE),read(CANDIDATE)
+old=[['ADD',388,715,'-4',825],['ADD',340,388,'1',825]]
+new=[['ADD',340,715,'-4',396],['ADD',340,388,'1',825],['ADD',388,715,'-4',127]]
+a=[e for e in w['program'] if e[0]!='MOVE'];b=[e for e in v['program'] if e[0]!='MOVE'];matches=[i for i in range(len(a)-1) if a[i:i+2]==old]
+need(len(matches)==1,'Unique adjacent scalar motif');k=matches[0]
+need(b==a[:k]+new+a[k+2:],'Only the exact three-register scalar rewrite')
+need(matrix(old)==matrix(new),'All three arbitrary-input columns')
+need(all(w[k]==v[k] for k in ('public','owned_fields','initial','final','ports','live','frames')),'Every boundary and physical inventory retained')
+chain=[598,396,825,127]
+need(all(span(w['frames'][a])<span(w['frames'][b]) for a,b in zip(chain,chain[1:])),'Strict nested retimed source chain')
+need([len(w['frames'][x]) for x in chain]==[1,3,6,9],'Exact source itinerary dimensions')
+def price(word):
+ return Counter(abs(len(word['frames'][e[3]])-len(word['frames'][e[2]])) for e in word['program'] if e[0]=='MOVE')
+d=price(v);d.subtract(price(w));delta={r:c for r,c in d.items() if c}
+need(delta=={2:1,3:-1,5:-1,6:1},'Complete paid H3+H5 to H2+H6 exchange')
+need([e for e in a if e[0]in('COPY','INIT','CLEAR')]==[e for e in b if e[0]in('COPY','INIT','CLEAR')],'No changed owned erasure or copy contract')
+controls={}
+for name,bad in [('omitted_direct_shear',new[1:]),('omitted_last_shear',new[:-1]),('wrong_direct_sign',[['ADD',340,715,'4',396]]+new[1:])]:
+ controls[name]=matrix(old)!=matrix(bad);need(controls[name],'Control rejects '+name)
+result={'status':'PASS_INDEPENDENT_A2_IDENTITY_AND_PRICE_AUDIT','source_sha256':sha256(SOURCE.read_bytes()).hexdigest(),'candidate_sha256':sha256(CANDIDATE.read_bytes()).hexdigest(),'old_gates':old,'new_gates':new,'old_itinerary_dimensions':[1,6,9],'new_itinerary_dimensions':[1,3,9],'paid_delta':delta,'controls':controls,'scope':'Exact three-column scalar identity; only scalar motif replacement; unchanged physical boundaries/owned instructions; strict source frame chain; complete paid-profile delta. Native, physical caller and outer replays supplied separately by root.'}
+expected=json.loads((HERE/'INDEPENDENT-AUDIT.json').read_bytes())
+need(json.loads(json.dumps(result))==expected,'Exact saved independent receipt');print(result['status'])
diff --git a/research/p104-a2-paid-rank-exchange/construct.py b/research/p104-a2-paid-rank-exchange/construct.py
new file mode 100644
index 0000000000..9500a34a53
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/construct.py
@@ -0,0 +1,64 @@
+"""Construct a complete P104 A2 paid-rank exchange against exact PR5627 bytes.
+
+Retains Ali Romman's P104 local checker and all predecessor notices. The A2
+identity and complete-boundary discipline are credited to Chris R/Clear1nsight
+(PR3998), ToikY(PR4019), and the scoped audits(PR4112/4152). New P104 witness,
+construction and checks with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import Counter
+from copy import deepcopy
+from hashlib import sha256
+import argparse,gzip,importlib.util,json,sys
+HERE=Path(__file__).resolve().parent
+SOURCE_WORD='162e6f13a6e3805517946309bd6247eaa45b337ab9c39621806b254b415075ec'
+SOURCE_CHECKER='716d01a342467f87b14d67853d8de30740f9f2fa881afe271db0df5ef8b19697'
+EXPECTED='c7545eab4e1394ba942a7ac8e4d70860d172cabda151a559fd566478aa38204b'
+def need(ok,why):
+ if not ok:raise ValueError(why)
+def construct():
+ for rel,digest in [('inputs/pr5627-WORD.json.gz',SOURCE_WORD),('inputs/pr5627-local.py',SOURCE_CHECKER)]:
+  need(sha256((HERE/rel).read_bytes()).hexdigest()==digest,'Exact source pin '+rel)
+ spec=importlib.util.spec_from_file_location('retained_p104_checker',HERE/'inputs/pr5627-local.py');L=importlib.util.module_from_spec(spec);spec.loader.exec_module(L)
+ w=json.loads(gzip.decompress((HERE/'inputs/pr5627-WORD.json.gz').read_bytes()));v=deepcopy(w)
+ need(w['program'][14975]==['ADD',388,715,'-4',825] and w['program'][14977]==['ADD',340,388,'1',825],'Literal old pair')
+ need(w['program'][14976]==['MOVE',340,396,825,False],'Only the declared intermediate transport')
+ replacement=[['ADD',340,715,'-4',396],['ADD',340,388,'1',825],['ADD',388,715,'-4',127]]
+ need([w['frames'][f]for f in[396,825,127]]==[[13,70,416],[7,10,32,65,144,272],[1,2,4,8,16,32,64,128,256]],'Exact existing frames')
+ scalar=[]
+ for i,e in enumerate(w['program']):
+  if i==14975:scalar.extend(replacement)
+  if i not in(14975,14977) and e[0]!='MOVE':scalar.append(e[:])
+ frames=[L.span(b)for b in w['frames']];n=w['public'];current=w['initial']+[None]*w['owned_fields'];active=set();out=[]
+ def move(r,f):
+  a=current[r]
+  if a==f:return
+  private=r in active
+  need(a is not None and (frames[f]<frames[a]if private else frames[a]<frames[f]),'Strict nested typed transport')
+  out.append(['MOVE',r,a,f,private]);current[r]=f
+ for e in scalar:
+  op,r=e[:2]
+  if op=='INIT':need(current[r]is None,'Owned slot available');active.add(r);current[r]=e[2]
+  elif op=='CLEAR':move(r,e[2]);active.remove(r);current[r]=None
+  else:
+   move(r,e[-1])
+   if op!='SCALE':move(e[2],e[-1])
+  out.append(e)
+ for r in range(n):move(r,w['final'][r])
+ v['program']=out;v['schema']='p104-a2-paid-rank-exchange/1'
+ before,after=L.replay(w),L.replay(v)
+ delta=Counter(after['paid']);delta.subtract(before['paid']);delta={r:c for r,c in delta.items()if c}
+ need(delta=={2:1,3:-1,5:-1,6:1},'Exact full paid-rank exchange')
+ need(before['helpers']==after['helpers'],'Every target retained')
+ need(after['real_scalar_allowance']-before['real_scalar_allowance']==12,'Correction ADD fully counted')
+ payload=(json.dumps(v,separators=(',',':'))).encode()
+ need(sha256(payload).hexdigest()==EXPECTED,'Entire regenerated candidate payload')
+ need(gzip.decompress((HERE/'WORD.json.gz').read_bytes())==payload,'Exact saved candidate')
+ return payload,{'status':'PASS_P104_A2_PAID_RANK_EXCHANGE','paid_delta':delta,'target_delta':{},'local':after,'source_local':before,'candidate_decompressed_sha256':EXPECTED}
+def main():
+ ap=argparse.ArgumentParser(description=__doc__);ap.add_argument('--out',type=Path);args=ap.parse_args();payload,result=construct()
+ if args.out:
+  with args.out.open('xb')as f:f.write(gzip.compress(payload,mtime=0))
+ print(json.dumps(result,indent=2,sort_keys=True))
+if __name__=='__main__':main()
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5612/SOURCES.json b/research/p104-a2-paid-rank-exchange/inputs/pr5612/SOURCES.json
new file mode 100644
index 0000000000..4e7fa4ba58
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5612/SOURCES.json
@@ -0,0 +1,105 @@
+{
+  "schema_version": 1,
+  "base_commit": "5e0052ca0aa3036a3afa8b0438927377c8d97d53",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "fc3d4ba105e1f097d0d7d051a308281e86ab1446",
+      "sha256": "66974efdced4083c00025f97d95d83909a226cdf801edb9301c195c6641f6903"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/p104-delayed-shared-relay/inputs/pr5601-CHANGES.json",
+      "usage": "input",
+      "pin": "head:13e0c9cf5834154ec6a46d93cf3f7283ed780535:research/p104-delayed-shared-relay/inputs/pr5601-CHANGES.json",
+      "sha256": "cc9c3088d509e56362b552cd79875bce45531c2183f3ac0b83a6d75e0d462467",
+      "git_blob": "a46098f911d6483fcba9e3c89ac2d24e24cb1a74"
+    },
+    {
+      "path": "research/p104-delayed-shared-relay/inputs/pr5601-construct.py",
+      "usage": "input",
+      "pin": "head:13e0c9cf5834154ec6a46d93cf3f7283ed780535:research/p104-delayed-shared-relay/inputs/pr5601-construct.py",
+      "sha256": "40aa1dd9aa9faca1a4faae7f923e90f216dc74267a92922e02d495222f07abdc",
+      "git_blob": "2506f4bdc82be48ea3e97e3eabc1f96cd0f61329"
+    },
+    {
+      "path": "research/p104-delayed-shared-relay/inputs/pr5606-construct.py",
+      "usage": "input",
+      "pin": "head:13e0c9cf5834154ec6a46d93cf3f7283ed780535:research/p104-delayed-shared-relay/inputs/pr5606-construct.py",
+      "sha256": "037d5f00f1c92b6f59020f90f11345251b429ba2bb2c39a65966b5e73094756a",
+      "git_blob": "e2c1473be00e78b91b4139493e31e47b591ac0df"
+    },
+    {
+      "path": "research/p104-delayed-shared-relay/inputs/pr5606-prepared.py",
+      "usage": "input",
+      "pin": "head:13e0c9cf5834154ec6a46d93cf3f7283ed780535:research/p104-delayed-shared-relay/inputs/pr5606-prepared.py",
+      "sha256": "166d3362f5a776d422412efc5e905fb3ace13caa1e715b870b175295e145a4eb",
+      "git_blob": "6eae5764287d6325ad372400480a0447a12568a7"
+    },
+    {
+      "path": "research/p104-delayed-shared-relay/inputs/pr5606-pricing.py",
+      "usage": "input",
+      "pin": "head:13e0c9cf5834154ec6a46d93cf3f7283ed780535:research/p104-delayed-shared-relay/inputs/pr5606-pricing.py",
+      "sha256": "c7e2a95c1a285982e0db9a9b881154ecda1ee76031cf216909e30b4939dd841a",
+      "git_blob": "02efc04ddec40b9cec9f785b00fd20841be6009d"
+    },
+    {
+      "path": "research/p104-delayed-shared-relay/inputs/pr5608-source.tar.gz",
+      "usage": "input",
+      "pin": "head:13e0c9cf5834154ec6a46d93cf3f7283ed780535:research/p104-delayed-shared-relay/inputs/pr5608-source.tar.gz",
+      "sha256": "dbded26ad3b6f7800c6e3b76aaee4df3e86dcb2278bb882d97fe3007f76e0712",
+      "git_blob": "404e95f089c10ea0589934ea45160d8604468b68"
+    }
+  ],
+  "external_inputs": [
+    {
+      "local_path": "inputs/pr5601-CHANGES.json",
+      "source_commit": "8f7e870b22885a367b347dc12791da98fd53f83d",
+      "source_path": "research/p104-coupled-frame-step/CHANGES.json",
+      "sha256": "cc9c3088d509e56362b552cd79875bce45531c2183f3ac0b83a6d75e0d462467",
+      "git_blob": "a46098f911d6483fcba9e3c89ac2d24e24cb1a74"
+    },
+    {
+      "local_path": "inputs/pr5601-construct.py",
+      "source_commit": "8f7e870b22885a367b347dc12791da98fd53f83d",
+      "source_path": "research/p104-coupled-frame-step/construct.py",
+      "sha256": "40aa1dd9aa9faca1a4faae7f923e90f216dc74267a92922e02d495222f07abdc",
+      "git_blob": "2506f4bdc82be48ea3e97e3eabc1f96cd0f61329"
+    },
+    {
+      "local_path": "inputs/pr5606-prepared.py",
+      "source_commit": "cd7d89345c6c5d2114625e5ce912a52ecd5e1d27",
+      "source_path": "research/p104-prepared-split-reuse/prepared.py",
+      "sha256": "166d3362f5a776d422412efc5e905fb3ace13caa1e715b870b175295e145a4eb",
+      "git_blob": "6eae5764287d6325ad372400480a0447a12568a7"
+    },
+    {
+      "local_path": "inputs/pr5606-pricing.py",
+      "source_commit": "cd7d89345c6c5d2114625e5ce912a52ecd5e1d27",
+      "source_path": "research/p104-prepared-split-reuse/pricing.py",
+      "sha256": "c7e2a95c1a285982e0db9a9b881154ecda1ee76031cf216909e30b4939dd841a",
+      "git_blob": "02efc04ddec40b9cec9f785b00fd20841be6009d"
+    },
+    {
+      "local_path": "inputs/pr5606-construct.py",
+      "source_commit": "cd7d89345c6c5d2114625e5ce912a52ecd5e1d27",
+      "source_path": "research/p104-prepared-split-reuse/construct.py",
+      "sha256": "037d5f00f1c92b6f59020f90f11345251b429ba2bb2c39a65966b5e73094756a",
+      "git_blob": "e2c1473be00e78b91b4139493e31e47b591ac0df"
+    }
+  ],
+  "source_export": {
+    "local_path": "inputs/pr5608-source.tar.gz",
+    "sha256": "dbded26ad3b6f7800c6e3b76aaee4df3e86dcb2278bb882d97fe3007f76e0712",
+    "commit": "5ec3b5b3ee4738f896e823c1dfc04e7ff1789ffb",
+    "package": "research/p104-relay-split-composition",
+    "files": 24,
+    "source_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/5608"
+  }
+}
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5601-CHANGES.json b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5601-CHANGES.json
new file mode 100644
index 0000000000..a46098f911
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5601-CHANGES.json
@@ -0,0 +1,50 @@
+{
+  "frames": [
+    [
+      30,
+      44,
+      459
+    ]
+  ],
+  "changes": [
+    {
+      "event": 14374,
+      "old": [
+        "ADD",
+        319,
+        85,
+        "1",
+        728
+      ],
+      "new_frame": 1089
+    },
+    {
+      "event": 14399,
+      "old": [
+        "ADD",
+        507,
+        319,
+        "1",
+        728
+      ],
+      "new_frame": 1089
+    },
+    {
+      "event": 16628,
+      "old": [
+        "ADD",
+        307,
+        85,
+        "-1",
+        236
+      ],
+      "new_frame": 235
+    },
+    {
+      "endpoint": "final",
+      "register": 307,
+      "old_frame": 236,
+      "new_frame": 235
+    }
+  ]
+}
\ No newline at end of file
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5601-construct.py b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5601-construct.py
new file mode 100644
index 0000000000..2506f4bdc8
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5601-construct.py
@@ -0,0 +1,65 @@
+"""Change three coupled charts on the PR5565 P104 cascade.
+
+DreamingOfClouds supplied the event-lattice search used for discovery.
+Ali Romman prepared this composition and exact certificate with substantial
+OpenAI Codex assistance. Apache-2.0. See SOURCES.json and README.md.
+"""
+from collections import defaultdict
+from copy import deepcopy
+from pathlib import Path
+import json
+
+
+def need(ok, why):
+    if not ok:
+        raise ValueError(why)
+
+
+def construct(source):
+    declaration = json.loads(Path(__file__).with_name('CHANGES.json').read_text())
+    changes = declaration['changes']
+    need(len(changes) == 4 and len(source['program']) == 25133,
+         'Exact source and edit counts')
+    word = deepcopy(source)
+    need(len(source['frames']) == 1089, 'Exact source frame count')
+    word['frames'].extend(declaration['frames'])
+    old, new = source['program'], word['program']
+    for change in changes:
+        if 'event' in change:
+            i = change['event']
+            need(old[i] == change['old'], 'Exact source event')
+            new[i][-1] = change['new_frame']
+        else:
+            key, r = change['endpoint'], change['register']
+            need(source[key][r] == change['old_frame'], 'Exact source endpoint')
+            word[key][r] = change['new_frame']
+    anchors = {r: [(-1, source['initial'][r], word['initial'][r])]
+               for r in range(source['public'])}
+    moves = defaultdict(list)
+    for i, event in enumerate(old):
+        op, r = event[:2]
+        if op == 'MOVE':
+            moves[r].append(i)
+        elif op in ('ADD', 'COPY', 'SCALE'):
+            registers = event[1:3] if op != 'SCALE' else event[1:2]
+            for r in registers:
+                if r in anchors:
+                    anchors[r].append((i, event[-1], new[i][-1]))
+    remove, before = set(), defaultdict(list)
+    for r, visits in anchors.items():
+        visits.append((len(old), source['final'][r], word['final'][r]))
+        for (i, a, x), (j, b, y) in zip(visits, visits[1:]):
+            if (a, b) == (x, y):
+                continue
+            remove.update(k for k in moves[r] if i < k < j)
+            if x != y:
+                before[j].append(['MOVE', r, x, y, False])
+    program = []
+    for i, event in enumerate(new):
+        program.extend(before[i])
+        if i not in remove:
+            program.append(event)
+    program.extend(before[len(old)])
+    word['program'] = program
+    need(len(program) == 25132, 'One fewer paid transport')
+    return word
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5606-construct.py b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5606-construct.py
new file mode 100644
index 0000000000..e2c1473be0
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5606-construct.py
@@ -0,0 +1,39 @@
+"""Remove two queries from PR5599's independent helper split.
+
+Scratch-net with substantial OpenAI Codex assistance. Apache-2.0.
+Ali Romman's fifteen independent splits and source word remain credited.
+"""
+from copy import deepcopy
+
+
+def need(ok, why):
+    if not ok:
+        raise ValueError(why)
+
+
+def construct(source):
+    w = deepcopy(source)
+    need(w['public'] == 750 and len(w['program']) == 25502, 'Exact source shape')
+    need(w['initial'][573] == 3 and w['final'][573] == 603, 'Rank-two prefix helper')
+    need(w['frames'][3] == [] and len(w['frames'][603]) == 2, 'Prefix target rank')
+    replacements = {
+        13670: (['MOVE',573,3,595,False],
+                [['CLEAR',573,3],['INIT',573,595],
+                 ['COPY',573,572,'1',595],['ADD',573,732,'-2',595]]),
+        13691: (['MOVE',573,595,603,False],
+                [['CLEAR',573,595],['INIT',573,603],
+                 ['COPY',573,572,'1',603],['ADD',573,710,'2',603]])}
+    out = [['INIT',573,3]]
+    for i,event in enumerate(source['program']):
+        if i in replacements:
+            old,new = replacements[i]
+            need(event == old,'Pinned paid site '+str(i))
+            out.extend(deepcopy(new))
+        else:
+            out.append(deepcopy(event))
+    out.append(['CLEAR',573,603])
+    w.update(schema='p104-prepared-split-reuse/1',program=out,
+             retired_public=[573],fields=758,
+             independent_public=[r for r in range(750) if r != 573],
+             owned_roles=[573]+list(range(750,758)))
+    return w
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5606-prepared.py b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5606-prepared.py
new file mode 100644
index 0000000000..6eae576428
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5606-prepared.py
@@ -0,0 +1,134 @@
+"""Exact endpoints for the prepared P104 helper.
+
+Adapted from Ali Romman's P104 local.py; retained native sources are unchanged.
+Changes: one retired helper, arbitrary owned roles, two ordinary reconstructions,
+and 758 total fields. Scratch-net with OpenAI Codex assistance. Apache-2.0.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from common import need
+
+def add(dst, src, q):
+    for k,v in tuple(src.items()):
+        z=dst.get(k,Q(0))+q*v
+        if z:dst[k]=z
+        else:dst.pop(k,None)
+
+def span(basis):
+    out={0}
+    for v in basis:
+        need(type(v) is int and 0<v<512 and v not in out, 'Independent binary basis')
+        out|={x^v for x in tuple(out)}
+    return frozenset(out)
+
+def replay(w):
+    need(len(w['program']) < 26000, 'Fixed operation bound')
+    need(w['fields'] == 758 and w['retired_public'] == [573]
+         and w['independent_public'] == [r for r in range(750) if r != 573]
+         and w['owned_roles'] == [573]+list(range(750,758)), 'Exact independent and owned interface')
+    frames = [span(x) for x in w['frames']]
+    rows = [{r:Q(1)} for r in range(758)]
+    current = list(w['initial']) + [None]*8
+    current[573] = None
+    owned = set(w['owned_roles'])
+    lifetimes = 0
+    paid, scalar, live_owned, peak = Counter(), 0, set(), 0
+    for e in w['program']:
+        op, r = e[:2]
+        need(0 <= r < 758, 'Role index')
+        if op == 'MOVE':
+            _, r, a, b, private = e
+            need(current[r] == a and (r in live_owned) == private, 'Transport source and ownership')
+            ascending = r < 750
+            need(frames[a] < frames[b] if ascending else frames[b] < frames[a],
+                 'Strict nested chronological transport')
+            paid[abs(len(w['frames'][b])-len(w['frames'][a]))] += 1
+            current[r] = b
+        elif op == 'INIT':
+            need(r in owned and r not in live_owned, 'Owned initialization')
+            lifetimes += 1
+            live_owned.add(r)
+            peak = max(peak, len(live_owned))
+            rows[r] = {}
+            current[r] = e[2]
+            scalar += 2
+        elif op == 'CLEAR':
+            need(r in live_owned and current[r] == e[2], 'Owned erasure at declared frame')
+            rows[r] = {}
+            current[r] = None
+            live_owned.remove(r)
+            scalar += 2
+        else:
+            q, f = Q(e[-2]), e[-1]
+            need(q and current[r] == f, 'Scalar destination and coefficient')
+            n, d = abs(q.numerator), q.denominator
+            need(d & (d-1) == 0, 'Dyadic scalar coefficient')
+            need(abs(q)+1 < 2**16, 'Fixed scalar growth bound')
+            if op == 'SCALE':
+                need(n & (n-1) == 0, 'Dyadic unit SCALE')
+                rows[r] = {k:q*x for k,x in rows[r].items()}
+                scalar += 2*(abs(n.bit_length()-d.bit_length()) + int(q < 0))
+            else:
+                s = e[2]
+                need(s != r and 0 <= s < 758 and current[s] == f, 'Common scalar chart')
+                if op == 'COPY':
+                    need(r in live_owned and not rows[r], 'COPY into cleared owned field')
+                    rows[r] = {k:q*x for k,x in rows[s].items()}
+                else:
+                    need(op == 'ADD', 'Scalar opcode')
+                    add(rows[r], rows[s], q)
+                scalar += 2*(3+n.bit_length()-1+n.bit_count()+d.bit_length()-1)
+    need(not live_owned and peak == 2 and all(current[r] == w['final'][r] for r in w['independent_public']), 'All public and owned exits')
+    need(lifetimes == 11, 'Complete owned lifetime census')
+    live = set(w['live'])
+    for r,row in enumerate(rows):
+        expected = {r:Q(1)} if r < 750 and r != 573 else {}
+        if r-112 in live:
+            expected[r-112] = Q(1)
+        need(row == expected, 'All 758 public and owned rows')
+    main = live | {r+112 for r in live}
+    helpers = Counter(len(w['frames'][w['final'][r]])-len(w['frames'][w['initial'][r]]) for r in range(750) if r not in main and r != 573)
+    need(len(live) == len(w['ports']) == 104 and len(main) == 208 and min(helpers) > 0, 'Independent productive inputs')
+    return {'paid':dict(paid), 'helpers':dict(helpers), 'public_inputs':749, 'owned_lifetimes':lifetimes,
+            'peak_owned_fields':peak, 'real_scalar_allowance':scalar, 'endpoint_coefficients':758*758}
+
+
+def reverse_columns(w):
+    """Check the transpose with dense dyadic rows and the reverse chronology."""
+    n = 758
+    rows = [[int(i == j) for j in range(n)] for i in range(n)]
+    exponents = [0]*n
+    for e in reversed(w['program']):
+        op,r = e[:2]
+        if op == 'MOVE':
+            continue
+        if op in ('INIT','CLEAR'):
+            rows[r] = [0]*n
+            exponents[r] = 0
+            continue
+        q = Q(e[-2])
+        qe = q.denominator.bit_length()-1
+        if op == 'SCALE':
+            rows[r] = [q.numerator*x for x in rows[r]]
+            exponents[r] += qe
+        else:
+            s = e[2]
+            level = max(exponents[s], exponents[r]+qe)
+            a, b = 1 << (level-exponents[s]), q.numerator << (level-exponents[r]-qe)
+            rows[s] = [a*x+b*y for x,y in zip(rows[s],rows[r])]
+            exponents[s] = level
+            if op == 'COPY':
+                rows[r] = [0]*n
+                exponents[r] = 0
+            r = s
+        while exponents[r] and not any(x & 1 for x in rows[r]):
+            rows[r] = [x//2 for x in rows[r]]
+            exponents[r] -= 1
+    for r,row in enumerate(rows):
+        expected = [0]*n
+        if r < 750 and r != 573:
+            expected[r] = 1 << exponents[r]
+        if r in w['live']:
+            expected[r+112] = 1 << exponents[r]
+        need(row == expected, 'Independent dense transpose columns')
+    return {'columns':n, 'coefficients':n*n, 'arithmetic':'Dense integer rows with binary denominators; reverse chronology'}
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5606-pricing.py b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5606-pricing.py
new file mode 100644
index 0000000000..02efc04dde
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5606-pricing.py
@@ -0,0 +1,169 @@
+"""Complete mixed-target price, compiler and outer transfer.
+
+The interval and compiler bounds follow zxv and Cao Yifan. The productive
+recursion is by Jacob Alan Sussman. The controller is by Cinna Davis.
+Adapted from Ali Romman's PR5544/5545 for the two coupled contact changes.
+Successor choices were prepared for thu-nmrc with OpenAI Codex assistance.
+Helper-subspace targets were added by Ali Romman with OpenAI Codex assistance.
+Ali Romman's complete caller/price proof and all predecessor credit are retained.
+Apache-2.0.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+import local as L
+
+KAPPA, TESTED = Q('.00309871'), Q('.0030987101')
+PREVIOUS = Q('.00309839')
+
+
+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 assemble(local, native, word, physical):
+    paid, targets = Counter(), Counter()
+    counts = Counter()
+    L.need(local['helpers'] == native['helpers'], 'Every productive helper has a native target')
+    for e in word['physical_events']:
+        counts[e[0]] += 1
+        if e[0] == 'LOCAL':
+            L.need(e[3] is False, 'Every source invocation is forward')
+            paid.update(local['paid'])
+            targets.update(local['helpers'])
+        elif e[0] == 'QUERY_SHEAR':
+            paid[1] += 104
+        elif e[0] == 'IDLE':
+            paid[8*len(e[2])] += 208
+    main = Counter()
+    idle = Counter({8*len(e[2]):0 for e in word['physical_events'] if e[0] == 'IDLE'})
+    for e in word['physical_events']:
+        if e[0] == 'IDLE':
+            idle[8*len(e[2])] += 208
+    L.need(idle == physical['whole_idle_jobs'], 'Every whole idle has a native Gaussian and full rank')
+    for term in word['diagonal']:
+        main[1+8*sum(term[1:])] += 104
+    L.need(main == physical['main_targets'], 'Every main target has its full native phase')
+    targets.update(main)
+    L.need(counts['LOCAL'] == 18 and counts['QUERY_SHEAR'] == counts['IDLE'] == 15, 'Complete call census')
+    # Recount by the two path lengths, without using emitted main diagonals.
+    second_t = Counter({r:18*n for r,n in local['helpers'].items()})
+    second_p = Counter({r:18*n for r,n in local['paid'].items()})
+    second_t[145] += 208
+    second_p[1] += 15*104
+    for length in (7,8):
+        for j in range(1,length+1):
+            second_t[1+8*(j+1)] += 208
+            second_p[8*(j+1)] += 208
+    L.need((targets,paid) == (second_t,second_p), 'Independent full invoice')
+    L.need(max(targets) == 145 and max(paid) == 72, 'Maximum recursive ranks')
+    L.need(sum(targets.values()) == 13066 and sum(paid.values()) == 66816, 'Complete field and call counts')
+    L.need(sum(r*n for r,n in targets.items())-sum(r*n for r,n in paid.items()) == 760, 'Complete rank surplus')
+    return targets,paid,counts
+
+
+def run(local,native,word,physical):
+    T,P,counts = assemble(local,native,word,physical)
+    E = L.load(L.ROOT/'research/higher-moment-strictification/verify.py','p104_productive_intervals')
+    other = L.load(L.ROOT/'research/kappa-rebuild/architecture/mixed_target_probe.py','p104_productive_second_intervals')
+    def margin(k):
+        a,b = E.moment(T,1-k), E.moment(P,1-k)
+        return a[0]-b[1],a[1]-b[0]
+    lo,hi,grid = 0,10**12,10**14
+    while hi-lo > 1:
+        m=(lo+hi)//2
+        if margin(Q(m,grid))[0] > 0:
+            lo=m
+        else:
+            hi=m
+    lower,upper = Q(lo,grid),Q(hi,grid)
+    signed = Counter(T)
+    signed.subtract(P)
+    cap = Q('.0034')
+    derivative_upper = Q(0)
+    for rank, count in signed.items():
+        logarithm = E.log_bounds(rank)
+        low = E.mul(E.power(rank, 1-cap), logarithm)[0]
+        high = rank*logarithm[1]
+        derivative_upper -= count*(low if count >= 0 else high)
+    L.need(derivative_upper < 0, 'Unique raw crossing below the retained BIT cap')
+    L.need(margin(lower)[0] > 0 and margin(upper)[1] < 0, 'Strict rational crossing enclosure')
+    L.need(PREVIOUS < KAPPA < TESTED < lower and margin(PREVIOUS)[0] > 0, 'Changed complete bill improves selected kappa')
+    t,u,v,c,b = 1-TESTED,Q(1,5),1024,17,18
+    target,paid = E.moment(T,t),E.moment(P,t)
+    tl,th = other.moment(T,TESTED)
+    pl,ph = other.moment(P,TESTED)
+    L.need(tl > ph and max(tl,target[0]) <= min(th,target[1]) and max(pl,paid[0]) <= min(ph,paid[1]), 'Two exact moment engines')
+    alpha = E.divide(paid,target)
+    gamma = E.divide(E.moment(P,t+u),E.mul(E.power(c,u),target))
+    delta = E.divide((Q(sum(T.values())*b),)*2,E.mul(E.power(c,TESTED),target))
+    reserve,decay = Q(1,10**10),Q(39,40)
+    L.need(alpha[1] < 1-reserve and gamma[1] < decay and delta[1] < decay, 'All three charged tail ratios')
+    L.need(decay**128 < Q(1,16), 'Both tail decay blocks')
+    normalized = 1/(1+v*reserve)+Q(4,2**32)
+    L.need(normalized < 1 and c**64 < b**63, 'Full completion, splitting and rounding charges')
+    theta,power = Q(63,64),1024
+    eta = E.scale(E.power(145,-v*TESTED),t)
+    L.need(2*(c**v)**64 <= (145**v)**63 and eta[0] > theta**power, 'Shifted exits and complement capacity')
+    L.need(E.power(145,v*TESTED)[1] < 2**power, 'Compiler initial range')
+    gap = (TESTED-KAPPA)/7
+    L.need(gap.numerator == 1, 'Exact compiler integer grid')
+    step = gap.denominator
+    def wins(j):
+        return (step*j+1)**power < 2**j
+    left,right=1,2
+    while not wins(right):
+        left,right=right,2*right
+    while right-left > 1:
+        m=(left+right)//2
+        if wins(m):
+            right=m
+        else:
+            left=m
+    H=step*right
+    L.need(H > 8*v and wins(right) and not wins(right-1), 'Finite uniform threshold with predecessor rejection')
+    local_routes = (sum(local['paid'].values())+native['helper_normalizers']+native['main_adapters']+4*104+4*local['owned_lifetimes'])*1000
+    L.need(local['real_scalar_allowance']+local_routes < 10**7, 'Every local scalar, route and owned operation is funded')
+    boundary = sum(len(word['boundaries'][side+'_program']) for side in ('left','right'))
+    raw = 18*10**7 + 15*1664 + 32*104*(len(word['physical_events'])+boundary)
+    raw += 32*104*(26+1000)
+    L.need(raw < 10**9, 'Complete nonrecursive raw bound')
+    bit = L.read(L.ROOT/'research/query-thinned-controllers/RESULT.json')['ordinary_BIT']
+    atom = Q(bit['atom'])
+    ordinary=(1-atom)*Q(bit['after_cover'])+atom*Q('.0000384599')
+    admitted=Q(bit['admitted'])
+    L.need(atom > ordinary > admitted > TESTED and atom < 1-ordinary, 'Retained BIT admission and stock')
+    outer_module=L.load(L.ROOT/'research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py','p104_productive_outer')
+    params={'kappa':KAPPA,'tested_saving':TESTED,'complex_uniform_saving':KAPPA+6*gap,
+            'b':KAPPA+5*gap,'a':KAPPA+4*gap,'q':KAPPA+3*gap,'bit_saving':admitted,
+            'epsilon':1-gap,'beta':1-3*gap,'r0':gap/10,'stop':gap,'layout':gap,'d':gap}
+    outer=outer_module.ledger(params)
+    outer['functional_checks']['actual_new_BIT_stock_sublinear']=params['beta']*(1-atom)
+    L.need(len(outer['strict_checks']) == 54 and len(outer['functional_checks']) == 4, 'Complete outer predicate inventory')
+    L.need(all(x > 0 for key in ('strict_checks','functional_checks') for x in outer[key].values()), 'All outer inequalities strict')
+    wrong_cutoff = E.divide((Q(sum(T.values())*21),)*2,E.mul(E.power(20,TESTED),target))
+    L.need(wrong_cutoff[0] > 1, 'Old cutoff fails with this helper inventory')
+    bad=outer_module.ledger(dict(params,bit_saving=Q('.0027')))
+    L.need(any(x <= 0 for x in bad['strict_checks'].values()), 'Insufficient BIT contract rejected')
+    return encode({'kappa':KAPPA,'kappa_decimal':'.00309871','previous_open_pr5599':PREVIOUS,'gain':KAPPA-PREVIOUS,
+                   'relative_gain_percent':100*(KAPPA/PREVIOUS-1),'raw_root_interval':[lower,upper],
+                   'raw_headroom':[lower-KAPPA,upper-KAPPA],
+                   'raw_ceiling':{'interval':[Q(0),cap],'derivative_upper':derivative_upper},'margin_at_previous':E.display(margin(PREVIOUS)),
+                   'targets':T,'paid':P,'productive_fields':sum(T.values()),'paid_calls':sum(P.values()),
+                   'rank_mass_surplus':760,'full_forward_calls':18,'query_shears':15,'whole_idle_jobs':3120,
+                   'strictification':{'tested':TESTED,'power':v,'cutoff':c,'floor':b,'tail_offset':u,'reserve':reserve,'tail_decay':decay,'tail_block':128,
+                                     'alpha':E.display(alpha),'gamma':E.display(gamma),'delta':E.display(delta),'normalized_paid_upper':normalized},
+                   'compiler':{'theta':theta,'logarithm_power':power,'eta':E.display(eta),'gap':gap,'H':H,'N':'2^'+str(H),
+                               'integer_test_exponent':right,'uniform_saving':params['complex_uniform_saving'],'word_materialized':False},
+                   'nonrecursive':{'local_scalar':local['real_scalar_allowance'],'local_routes':local_routes,
+                                   'raw_scalar_upper':raw,'raw_budget':10**9,'peak_owned_fields':local['peak_owned_fields'],
+                                   'fixed_scalar_growth_bound':'At most 2^(16*26000) before native factors; a declared constant, not an attained norm.',
+                                   'local_paid_rank_mass':sum(r*n for r,n in local['paid'].items()),
+                                   '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':bit,'outer':outer,
+                   'negative_controls':{'old_cutoff':'Rejected by completion-tail ratio greater than one','old_BIT':'Rejected by outer inequalities'}})
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5608-source.tar.gz b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5608-source.tar.gz
new file mode 100644
index 0000000000..404e95f089
Binary files /dev/null and b/research/p104-a2-paid-rank-exchange/inputs/pr5612/inputs/pr5608-source.tar.gz differ
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5612/sources.py b/research/p104-a2-paid-rank-exchange/inputs/pr5612/sources.py
new file mode 100644
index 0000000000..fce090a6b7
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5612/sources.py
@@ -0,0 +1,42 @@
+"""Authenticate the official PR5608 export before importing its source closure.
+
+Prepared for flyingviper with substantial OpenAI Codex assistance. Apache-2.0.
+Source-export format and payload checks follow the retained repository exporter.
+"""
+from contextlib import contextmanager
+from hashlib import sha256,sha1
+from pathlib import Path,PurePosixPath
+import gzip,importlib.util,io,json,sys,tarfile,tempfile
+HERE=Path(__file__).resolve().parent
+def need(ok,why):
+    if not ok:raise ValueError(why)
+def read(path):
+    raw=Path(path).read_bytes();return json.loads(gzip.decompress(raw) if str(path).endswith('.gz') else raw)
+def load(path,name):
+    spec=importlib.util.spec_from_file_location(name,path);m=importlib.util.module_from_spec(spec);sys.modules[name]=m;spec.loader.exec_module(m);return m
+@contextmanager
+def sources():
+    pins=read(HERE/'SOURCES.json')
+    for p in pins['external_inputs']:
+        raw=(HERE/p['local_path']).read_bytes()
+        need(sha256(raw).hexdigest()==p['sha256'],'Pinned source SHA256 '+p['local_path'])
+        need(sha1(('blob '+str(len(raw))+'\0').encode()+raw).hexdigest()==p['git_blob'],'Pinned source blob '+p['local_path'])
+    pin=pins['source_export'];raw=(HERE/pin['local_path']).read_bytes()
+    need(sha256(raw).hexdigest()==pin['sha256'],'Exact published export archive')
+    with tempfile.TemporaryDirectory(prefix='p104-composed-source-') as directory:
+        root=Path(directory)
+        with tarfile.open(fileobj=io.BytesIO(gzip.decompress(raw))) as bundle:
+            names=[]
+            for m in bundle.getmembers():
+                p=PurePosixPath(m.name)
+                need(m.isfile() and not p.is_absolute() and '..' not in p.parts and '\\' not in m.name and ':' not in m.name,'Regular relative source file')
+                need(m.name not in names,'Unique export path');names.append(m.name)
+                target=root.joinpath(*p.parts);target.parent.mkdir(parents=True,exist_ok=True);target.write_bytes(bundle.extractfile(m).read())
+        manifest=read(root/'.research-export.json')
+        need(manifest['commit']==pin['commit'] and len(manifest['files'])==pin['files'],'Exact published source inventory')
+        need(set(names)=={'.research-export.json'}|{r['path'] for r in manifest['files']},'No omitted or extra source payload')
+        for r in manifest['files']:
+            b=(root/r['path']).read_bytes();need(len(b)==r['size'] and sha256(b).hexdigest()==r['sha256'],'Authenticated source payload '+r['path'])
+        package=root/pin['package']
+        common=load(package/'common.py','retained_source_common')
+        with common.closure() as framework:yield package,framework
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5627-RESULT.json b/research/p104-a2-paid-rank-exchange/inputs/pr5627-RESULT.json
new file mode 100644
index 0000000000..1e90afdd03
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5627-RESULT.json
@@ -0,0 +1,597 @@
+{
+  "authenticated_inputs": 18,
+  "boundaries": [
+    {
+      "cut": 11382,
+      "events": 25181,
+      "helper": 652,
+      "helpers": {
+        "4": 85,
+        "5": 4,
+        "8": 14,
+        "9": 424
+      },
+      "paid": {
+        "1": 2202,
+        "2": 456,
+        "3": 494,
+        "4": 91,
+        "5": 167,
+        "6": 18,
+        "7": 15
+      },
+      "reverse": true
+    },
+    {
+      "cut": 11167,
+      "events": 25178,
+      "helper": 341,
+      "helpers": {
+        "4": 86,
+        "5": 4,
+        "8": 14,
+        "9": 423
+      },
+      "paid": {
+        "1": 2202,
+        "2": 456,
+        "3": 494,
+        "4": 91,
+        "5": 166,
+        "6": 18,
+        "7": 15
+      },
+      "reverse": false
+    },
+    {
+      "cut": 11400,
+      "events": 25180,
+      "helper": 596,
+      "helpers": {
+        "4": 87,
+        "5": 4,
+        "8": 14,
+        "9": 422
+      },
+      "paid": {
+        "1": 2202,
+        "2": 456,
+        "3": 494,
+        "4": 91,
+        "5": 165,
+        "6": 18,
+        "7": 15
+      },
+      "reverse": true
+    },
+    {
+      "cut": 11471,
+      "events": 25182,
+      "helper": 668,
+      "helpers": {
+        "4": 88,
+        "5": 4,
+        "8": 14,
+        "9": 421
+      },
+      "paid": {
+        "1": 2202,
+        "2": 456,
+        "3": 494,
+        "4": 91,
+        "5": 164,
+        "6": 18,
+        "7": 15
+      },
+      "reverse": true
+    },
+    {
+      "cut": 11515,
+      "events": 25184,
+      "helper": 684,
+      "helpers": {
+        "4": 89,
+        "5": 4,
+        "8": 14,
+        "9": 420
+      },
+      "paid": {
+        "1": 2202,
+        "2": 456,
+        "3": 494,
+        "4": 91,
+        "5": 163,
+        "6": 18,
+        "7": 15
+      },
+      "reverse": true
+    }
+  ],
+  "complete_cost": {
+    "compiler": {
+      "H": 80511200000000000,
+      "N": "2^80511200000000000",
+      "eta": {
+        "lower": "0.000000134230972128213080564458",
+        "upper": "0.000000134230972128213080564459"
+      },
+      "gap": "1/1400000000000",
+      "integer_test_exponent": 57508,
+      "logarithm_power": 1024,
+      "theta": "63/64",
+      "uniform_saving": "86923353/28000000000",
+      "word_materialized": false
+    },
+    "full_forward_calls": 18,
+    "gain": "922773/50000000000",
+    "kappa": "155220273/50000000000",
+    "kappa_decimal": ".00310440546",
+    "margin_at_previous": {
+      "lower": "4.488346782018422611631558671661",
+      "upper": "4.488346782018422611631558671662"
+    },
+    "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": 5986,
+      "local_routes": 5175000,
+      "local_scalar": 220154,
+      "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": "6084634707195653832353/1960000000000000000000000",
+        "fused_gaussian": "6084634707186961497059/1960000000000000000000000",
+        "global_coordinate_router": "135817739/43750000000",
+        "group_bit_router": "190144834643342546783/61250000000000000000000",
+        "group_scans": "699999999999/700000000000",
+        "main_transform": "6084634705795653832353/1960000000000000000000000",
+        "microbox_face_corrections": "26950000000001/350000000000",
+        "polynomial_products": "1399999999999/1400000000000",
+        "sparse_union_movement": "802334634707195653832353/1960000000000000000000000"
+      },
+      "parameters": {
+        "a": "135817739/43750000000",
+        "b": "4346167649/1400000000000",
+        "beta": "1399999999997/1400000000000",
+        "bit_saving": "17/5000",
+        "cap_xi": "1782334634705795653832353/1960000000000000000000000",
+        "complex_uniform_saving": "86923353/28000000000",
+        "d": "1/1400000000000",
+        "epsilon": "1399999999999/1400000000000",
+        "kappa": "155220273/50000000000",
+        "lambda_": "79751647563/80000000000",
+        "lambda_prime": "1395653832353/1400000000000",
+        "layout": "1/1400000000000",
+        "q": "4346167647/1400000000000",
+        "r0": "1/14000000000000",
+        "stop": "1/1400000000000",
+        "tested_saving": "620881093/200000000000"
+      },
+      "strict_checks": {
+        "actual_bit_above_effective": "12932261/43750000000",
+        "actual_logarithmic_gamma_guard": "1/1400000000000",
+        "actual_logarithmic_group_guard": "177665365296304346167647/980000000000000000000000",
+        "alpha_logarithm_absorption": "1/14000000000000",
+        "boundary_algebraic_matrix_above_kappa": "5595653832353/1960000000000000000000000",
+        "cap_above_band": "802334634705795653832353/1960000000000000000000000",
+        "cap_below_cell": "177665365292804346167647/1960000000000000000000000",
+        "cap_below_phase_cell_power": "2137665365294204346167647/1960000000000000000000000",
+        "complex_above_effective": "1/1400000000000",
+        "complex_below_one_over_32": "39403832351/1400000000000",
+        "complex_endpoint": "1/700000000000",
+        "epsilon_below_one": "1/1400000000000",
+        "epsilon_positive": "1399999999999/1400000000000",
+        "fine_prime_halo_umbrella": "7/2",
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diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5627-SOURCES.json b/research/p104-a2-paid-rank-exchange/inputs/pr5627-SOURCES.json
new file mode 100644
index 0000000000..bb8cb5eca3
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5627-SOURCES.json
@@ -0,0 +1,223 @@
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+    },
+    {
+      "local_path": "inputs/PR5565-WORD.json.gz",
+      "source_repository": "CrocSwap/integer-mult-bounds",
+      "source_commit": "9af814164b9dfc21f19ec737959d67cb71488c95",
+      "source_path": "research/p104-commuting-contact/WORD.json.gz",
+      "sha256": "3d62828f53e654e3db218bf4611a47aed4a677d48043713dc004db0f8a32990d",
+      "git_blob": "4ebe6fcb7adf319bf6b247cf23d89eb1ca4cd914"
+    },
+    {
+      "local_path": "inputs/PR5601-WORD.json.gz",
+      "source_repository": "aliromman/integer-mult-bounds",
+      "source_commit": "8f7e870b22885a367b347dc12791da98fd53f83d",
+      "source_path": "research/p104-coupled-frame-step/WORD.json.gz",
+      "sha256": "ac81682679d1eb628a2a0a330b55a7a79722531ca163cd565a3eba5d87589979",
+      "git_blob": "3c9a5ce8969f40f03724a73d34f984dd17efde96"
+    },
+    {
+      "local_path": "inputs/PR5602-construct.py",
+      "source_repository": "aliromman/integer-mult-bounds",
+      "source_commit": "6c224aa720203c4f4615e6538d8038a275387db8",
+      "source_path": "research/p104-coupled-helper-bases/construct.py",
+      "sha256": "2360066b0a1390c60a86d9788ada1504478e7e48bbfe3fcdf27a5563379fc001",
+      "git_blob": "ef5936ca89d6502d7f6af1ce6903d8500c117401"
+    },
+    {
+      "local_path": "inputs/PR5602-WITNESS.json",
+      "source_repository": "aliromman/integer-mult-bounds",
+      "source_commit": "6c224aa720203c4f4615e6538d8038a275387db8",
+      "source_path": "research/p104-coupled-helper-bases/WITNESS.json",
+      "sha256": "e4dfc526453eb9d232772ffdb587fff1923e247341405aee031318b0fb9cb638",
+      "git_blob": "8073b3c9772a4a7c94e07d613c2f5c45b1c650fc"
+    },
+    {
+      "local_path": "inputs/PR5605-construct.py",
+      "source_repository": "flyingviper/integer-mult-bounds",
+      "source_commit": "4a57d245ac3918ccc9947a867ece24573605e243",
+      "source_path": "research/p104-shared-residual-relay/construct.py",
+      "sha256": "5831eb6f88cd6ed8200f32d2794dd9535ea06aa0d3ae6dd111a58bc01b611cbd",
+      "git_blob": "31f9b76c9b3d9c410aec1d0945d967ed985154c3"
+    },
+    {
+      "local_path": "inputs/PR5612-delayed.py",
+      "source_repository": "flyingviper/integer-mult-bounds",
+      "source_commit": "faa9f6dbe6b8cc4a996d3edeca71823056453e1a",
+      "source_path": "research/p104-delayed-shared-relay/delayed.py",
+      "sha256": "fcb8b97785ae05db6f60c38d8d75ee4fe856ea41a1327b772eff7a6ee388b3c8",
+      "git_blob": "afd1bd8e084a31071d3ae6002e7183f6e3c8e891"
+    }
+  ],
+  "code_derivation": [
+    {
+      "source_repository": "GamingPuzzled/integer-mult-bounds",
+      "source_commit": "af9f091ac6f30c6eaacb9c27777bfbe2d0e5ffdf",
+      "source_path": "research/p104-priced-handoff/pricing.py",
+      "sha256": "a4df92b46234bb56a966df30d9c1d43251e1346e9a1ad06cf5ee30a79c7ae591",
+      "git_blob": "bb135338f1c57c48147cb214d14f542cdfe4e533",
+      "scope": "Pricing code attribution; this predecessor file is not executed."
+    },
+    {
+      "local_path": "inputs/PR5606-construct.py",
+      "source_repository": "Scratch-net/integer-mult-bounds",
+      "source_commit": "cd7d89345c6c5d2114625e5ce912a52ecd5e1d27",
+      "source_path": "research/p104-prepared-split-reuse/construct.py",
+      "sha256": "037d5f00f1c92b6f59020f90f11345251b429ba2bb2c39a65966b5e73094756a",
+      "git_blob": "e2c1473be00e78b91b4139493e31e47b591ac0df",
+      "scope": "The fixed preparation identities are adapted to the final split word and explicit nine-role owned inventory; this predecessor constructor is not executed."
+    }
+  ]
+}
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5627-WORD.json.gz b/research/p104-a2-paid-rank-exchange/inputs/pr5627-WORD.json.gz
new file mode 100644
index 0000000000..b60955f312
Binary files /dev/null and b/research/p104-a2-paid-rank-exchange/inputs/pr5627-WORD.json.gz differ
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5627-local.py b/research/p104-a2-paid-rank-exchange/inputs/pr5627-local.py
new file mode 100644
index 0000000000..e72cb02dba
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5627-local.py
@@ -0,0 +1,208 @@
+"""Check the complete P104 word with an independent helper inventory.
+
+Adapted from Ali Romman's PR5544/5545. David Arias supplied the word;
+Cinna Davis supplied private reuse; Gareth Davidson supplied composition.
+Prepared by Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+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
+ROOT = HERE.parents[1]
+
+def need(ok, why):
+    if not ok:
+        raise ValueError(why)
+
+def read(path):
+    raw = path.read_bytes()
+    return json.loads(gzip.decompress(raw) if path.suffix == '.gz' else raw)
+
+def load(path, name):
+    spec = importlib.util.spec_from_file_location(name, path)
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[name] = module
+    spec.loader.exec_module(module)
+    return module
+
+def add(dst, src, q):
+    for k, v in tuple(src.items()):
+        value = dst.get(k, 0) + q*v
+        if value:
+            dst[k] = value
+        else:
+            dst.pop(k, None)
+
+def span(basis):
+    out = {0}
+    for v in basis:
+        need(type(v) is int and 0 < v < 512 and v not in out, 'Independent binary basis')
+        out |= {x ^ v for x in tuple(out)}
+    return frozenset(out)
+
+def replay(w):
+    public = w['public']
+    owned = w.get('owned_fields', 8)
+    need(type(owned) is int and 8 <= owned <= 9, 'Owned inventory bound')
+    need(type(public) is int and 735 <= public <= 750, 'Public inventory bound')
+    need(len(w['initial']) == len(w['final']) == public, 'Public boundary size')
+    need(len(w['program']) < 26000, 'Fixed operation bound')
+    frames = [span(x) for x in w['frames']]
+    rows = [{r:Q(1)} for r in range(public+owned)]
+    current = list(w['initial']) + [None]*owned
+    paid, scalar, live_owned, peak = Counter(), 0, set(), 0
+    lifetimes = 0
+    for e in w['program']:
+        op, r = e[:2]
+        need(0 <= r < public+owned, 'Role index')
+        if op == 'MOVE':
+            _, r, a, b, private = e
+            need(current[r] == a and (r in live_owned) == private, 'Transport source and ownership')
+            need(frames[b] < frames[a] if private else frames[a] < frames[b], 'Nested chronological transport')
+            paid[abs(len(w['frames'][b])-len(w['frames'][a]))] += 1
+            current[r] = b
+        elif op == 'INIT':
+            need(r >= public and r not in live_owned, 'Owned initialization')
+            live_owned.add(r)
+            lifetimes += 1
+            peak = max(peak, len(live_owned))
+            rows[r] = {}
+            current[r] = e[2]
+            scalar += 2
+        elif op == 'CLEAR':
+            need(r in live_owned and current[r] == e[2], 'Owned erasure at declared frame')
+            rows[r] = {}
+            current[r] = None
+            live_owned.remove(r)
+            scalar += 2
+        else:
+            q, f = Q(e[-2]), e[-1]
+            need(q and current[r] == f, 'Scalar destination and coefficient')
+            n, d = abs(q.numerator), q.denominator
+            need(d & (d-1) == 0, 'Dyadic scalar coefficient')
+            need(abs(q)+1 < 2**16, 'Fixed scalar growth bound')
+            if op == 'SCALE':
+                need(n & (n-1) == 0, 'Dyadic unit SCALE')
+                rows[r] = {k:q*x for k,x in rows[r].items()}
+                scalar += 2*(abs(n.bit_length()-d.bit_length()) + int(q < 0))
+            else:
+                s = e[2]
+                need(s != r and 0 <= s < public+owned and current[s] == f, 'Common scalar chart')
+                if op == 'COPY':
+                    need(r in live_owned and not rows[r], 'COPY into cleared owned field')
+                    rows[r] = {k:q*x for k,x in rows[s].items()}
+                else:
+                    need(op == 'ADD', 'Scalar opcode')
+                    add(rows[r], rows[s], q)
+                scalar += 2*(3+n.bit_length()-1+n.bit_count()+d.bit_length()-1)
+    need(not live_owned and 0 < peak <= 2 and current[:public] == w['final'], 'All public and owned exits')
+    live = set(w['live'])
+    for r,row in enumerate(rows):
+        expected = {r:Q(1)} if r < public else {}
+        if r-112 in live:
+            expected[r-112] = Q(1)
+        need(row == expected, 'All public and owned rows')
+    main = live | {r+112 for r in live}
+    helpers = Counter(len(w['frames'][w['final'][r]])-len(w['frames'][w['initial'][r]]) for r in range(public) if r not in main)
+    need(len(live) == len(w['ports']) == 104 and len(main) == 208 and min(helpers) > 0, 'Independent productive inputs')
+    return {'paid':dict(paid), 'helpers':dict(helpers), 'public_inputs':public, 'owned_fields':owned, 'owned_lifetimes':lifetimes,
+            'peak_owned_fields':peak, 'real_scalar_allowance':scalar, 'endpoint_coefficients':(public+owned)*(public+owned),
+            'paid_rank_mass':sum(r*count for r,count in paid.items())}
+
+def reverse_columns(w):
+    """Check the transpose with dense dyadic rows and the reverse chronology."""
+    public = w['public']
+    n = public+w.get('owned_fields', 8)
+    rows = [[int(i == j) for j in range(n)] for i in range(n)]
+    exponents = [0]*n
+    for e in reversed(w['program']):
+        op,r = e[:2]
+        if op == 'MOVE':
+            continue
+        if op in ('INIT','CLEAR'):
+            rows[r] = [0]*n
+            exponents[r] = 0
+            continue
+        q = Q(e[-2])
+        qe = q.denominator.bit_length()-1
+        if op == 'SCALE':
+            rows[r] = [q.numerator*x for x in rows[r]]
+            exponents[r] += qe
+        else:
+            s = e[2]
+            level = max(exponents[s], exponents[r]+qe)
+            a, b = 1 << (level-exponents[s]), q.numerator << (level-exponents[r]-qe)
+            rows[s] = [a*x+b*y for x,y in zip(rows[s],rows[r])]
+            exponents[s] = level
+            if op == 'COPY':
+                rows[r] = [0]*n
+                exponents[r] = 0
+            r = s
+        while exponents[r] and not any(x & 1 for x in rows[r]):
+            rows[r] = [x//2 for x in rows[r]]
+            exponents[r] -= 1
+    for r,row in enumerate(rows):
+        expected = [0]*n
+        if r < public:
+            expected[r] = 1 << exponents[r]
+        if r in w['live']:
+            expected[r+112] = 1 << exponents[r]
+        need(row == expected, 'Independent dense transpose columns')
+    return {'columns':n, 'coefficients':n*n, 'arithmetic':'Dense integer rows with binary denominators; reverse chronology'}
+
+def native(w):
+    n = w['public']
+    C = load(ROOT/'research/p104-native-channel/vendor/clifford_exact.py', 'p104_productive_clifford')
+    V = load(ROOT/'research/p104-native-channel/vendor/native_verify.py', 'p104_productive_gaussian')
+    @lru_cache(None)
+    def chart(i):
+        return C.canonical_frame(9, tuple(w['frames'][i]))
+    recipes = {}
+    def check(a,b):
+        if (a,b) not in recipes:
+            recipe,_ = C.lower_child(chart(a).adjoint().then(chart(b)))
+            expected = V.adjoint(V.frame(w['frames'][a],9)) + V.frame(w['frames'][b],9)
+            V.equal(V.recipe(recipe),expected,'Native chart change and exact global phase')
+            need(recipe['rank'] == abs(len(w['frames'][a])-len(w['frames'][b])), 'Native boundary rank')
+            recipes[a,b] = recipe
+        return recipes[a,b]
+    for e in w['program']:
+        if e[0] == 'MOVE':
+            check(e[2],e[3])
+    main = set(w['live']) | {r+112 for r in w['live']}
+    helper_profile = Counter()
+    for r in range(n):
+        if r not in main:
+            helper_profile[check(w['initial'][r],w['final'][r])['rank']] += 1
+    adapters = []
+    for r,q in zip(w['live'],w['ports']):
+        need(w['frames'][w['initial'][r]] == [q], 'Literal central port')
+        need(span(w['frames'][w['final'][r+112]]) == frozenset(x for x in range(512) if not (x&q).bit_count()%2), 'Literal output hyperplane')
+        tests = [
+            (C.cgate(9,q,1).adjoint().then(chart(w['initial'][r])), V.adjoint((('C',q,1),))+V.frame(w['frames'][w['initial'][r]],9)),
+            (chart(w['final'][r+112]).adjoint().then(C.cgate(9,q,-1).then(C.fourier(9))), V.adjoint(V.frame(w['frames'][w['final'][r+112]],9))+(('C',q,-1),)+V.fourier(9))]
+        for actual,expected in tests:
+            recipe,_ = C.lower_child(actual)
+            need(recipe['rank'] == 0, 'Free intrinsic main adapter')
+            V.equal(V.recipe(recipe),expected,'Main endpoint phase')
+            adapters.append(recipe)
+    bad = json.loads(json.dumps(next(iter(recipes.values()))))
+    bad['left']['phase'] = (bad['left']['phase']+1)%4
+    a,b = next(iter(recipes))
+    try:
+        V.equal(V.recipe(bad),V.adjoint(V.frame(w['frames'][a],9))+V.frame(w['frames'][b],9),'Wrong phase control')
+    except ValueError:
+        pass
+    else:
+        raise ValueError('Wrong native phase accepted')
+    payload = [[a,b,recipes[a,b]] for a,b in sorted(recipes)] + adapters
+    return {'distinct_maps':len(recipes),'main_adapters':len(adapters),'helper_normalizers':2*sum(helper_profile.values()),
+            'helpers':dict(helper_profile),'recipe_payload_sha256':sha256(json.dumps(payload,sort_keys=True,separators=(',',':')).encode()).hexdigest(),
+            'wrong_phase_rejected':True, 'proof':'Every signed Pauli image and exact global phase; frame induction at every positive tensor width'}
diff --git a/research/p104-a2-paid-rank-exchange/inputs/pr5627-pricing.py b/research/p104-a2-paid-rank-exchange/inputs/pr5627-pricing.py
new file mode 100644
index 0000000000..430934914b
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/inputs/pr5627-pricing.py
@@ -0,0 +1,174 @@
+"""Complete mixed-target price, compiler and outer transfer.
+
+The interval and compiler bounds follow zxv and Cao Yifan. The productive
+recursion is by Jacob Alan Sussman. The controller is by Cinna Davis.
+Adapted from Ali Romman's PR5544/5545 for the two coupled contact changes.
+Successor choices were prepared for thu-nmrc with OpenAI Codex assistance.
+Helper-subspace targets were added by Ali Romman with OpenAI Codex assistance.
+The source commuting-contact construction is by Pierre-Loic Doulcet (hexapode) in PR5565.
+The source coupled-frame change is by Ali Romman with OpenAI Codex in PR5601.
+Five fixed boundary relations are from Ali Romman's PR5602.
+The shared and delayed relays are from flyingviper's PR5605 and PR5612. The preparation is from
+Scratch-net's PR5606. Their costs and target debit are included.
+Ali Romman's complete caller/price proof and all predecessor credit are retained.
+Apache-2.0.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+import local as L
+
+KAPPA, TESTED = Q('.00310440546'), Q('.003104405465')
+PREVIOUS = Q('.00308595')
+
+
+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 assemble(local, native, word, physical):
+    paid, targets = Counter(), Counter()
+    counts = Counter()
+    L.need(local['helpers'] == native['helpers'], 'Every productive helper has a native target')
+    for e in word['physical_events']:
+        counts[e[0]] += 1
+        if e[0] == 'LOCAL':
+            L.need(e[3] is False, 'Every source invocation is forward')
+            paid.update(local['paid'])
+            targets.update(local['helpers'])
+        elif e[0] == 'QUERY_SHEAR':
+            paid[1] += 104
+        elif e[0] == 'IDLE':
+            paid[8*len(e[2])] += 208
+    main = Counter()
+    idle = Counter({8*len(e[2]):0 for e in word['physical_events'] if e[0] == 'IDLE'})
+    for e in word['physical_events']:
+        if e[0] == 'IDLE':
+            idle[8*len(e[2])] += 208
+    L.need(idle == physical['whole_idle_jobs'], 'Every whole idle has a native Gaussian and full rank')
+    for term in word['diagonal']:
+        main[1+8*sum(term[1:])] += 104
+    L.need(main == physical['main_targets'], 'Every main target has its full native phase')
+    targets.update(main)
+    L.need(counts['LOCAL'] == 18 and counts['QUERY_SHEAR'] == counts['IDLE'] == 15, 'Complete call census')
+    # Recount by the two path lengths, without using emitted main diagonals.
+    second_t = Counter({r:18*n for r,n in local['helpers'].items()})
+    second_p = Counter({r:18*n for r,n in local['paid'].items()})
+    second_t[145] += 208
+    second_p[1] += 15*104
+    for length in (7,8):
+        for j in range(1,length+1):
+            second_t[1+8*(j+1)] += 208
+            second_p[8*(j+1)] += 208
+    L.need((targets,paid) == (second_t,second_p), 'Independent full invoice')
+    L.need(max(targets) == 145 and max(paid) == 72, 'Maximum recursive ranks')
+    L.need(sum(targets.values()) == 12958 and sum(paid.values()) == 66546, 'Complete field and call counts')
+    L.need(sum(r*n for r,n in targets.items())-sum(r*n for r,n in paid.items()) == 760, 'Complete rank surplus')
+    return targets,paid,counts
+
+
+def run(local,native,word,physical):
+    T,P,counts = assemble(local,native,word,physical)
+    E = L.load(L.ROOT/'research/higher-moment-strictification/verify.py','p104_productive_intervals')
+    other = L.load(L.ROOT/'research/kappa-rebuild/architecture/mixed_target_probe.py','p104_productive_second_intervals')
+    def margin(k):
+        a,b = E.moment(T,1-k), E.moment(P,1-k)
+        return a[0]-b[1],a[1]-b[0]
+    lo,hi,grid = 0,10**12,10**14
+    while hi-lo > 1:
+        m=(lo+hi)//2
+        if margin(Q(m,grid))[0] > 0:
+            lo=m
+        else:
+            hi=m
+    lower,upper = Q(lo,grid),Q(hi,grid)
+    signed = Counter(T)
+    signed.subtract(P)
+    cap = Q('.0034')
+    derivative_upper = Q(0)
+    for rank, count in signed.items():
+        logarithm = E.log_bounds(rank)
+        low = E.mul(E.power(rank, 1-cap), logarithm)[0]
+        high = rank*logarithm[1]
+        derivative_upper -= count*(low if count >= 0 else high)
+    L.need(derivative_upper < 0, 'Unique raw crossing below the retained BIT cap')
+    L.need(margin(lower)[0] > 0 and margin(upper)[1] < 0, 'Strict rational crossing enclosure')
+    L.need(PREVIOUS < KAPPA < TESTED < lower and margin(PREVIOUS)[0] > 0, 'Changed complete bill improves the source claim')
+    t,u,v,c,b = 1-TESTED,Q(1,5),1024,17,18
+    target,paid = E.moment(T,t),E.moment(P,t)
+    tl,th = other.moment(T,TESTED)
+    pl,ph = other.moment(P,TESTED)
+    L.need(tl > ph and max(tl,target[0]) <= min(th,target[1]) and max(pl,paid[0]) <= min(ph,paid[1]), 'Two exact moment engines')
+    alpha = E.divide(paid,target)
+    gamma = E.divide(E.moment(P,t+u),E.mul(E.power(c,u),target))
+    delta = E.divide((Q(sum(T.values())*b),)*2,E.mul(E.power(c,TESTED),target))
+    reserve,decay = Q(1,10**12),Q(39,40)
+    L.need(alpha[1] < 1-reserve and gamma[1] < decay and delta[1] < decay, 'All three charged tail ratios')
+    L.need(decay**128 < Q(1,16), 'Both tail decay blocks')
+    normalized = 1/(1+v*reserve)+Q(4,2**32)
+    L.need(normalized < 1 and c**64 < b**63, 'Full completion, splitting and rounding charges')
+    theta,power = Q(63,64),1024
+    eta = E.scale(E.power(145,-v*TESTED),t)
+    L.need(2*(c**v)**64 <= (145**v)**63 and eta[0] > theta**power, 'Shifted exits and complement capacity')
+    L.need(E.power(145,v*TESTED)[1] < 2**power, 'Compiler initial range')
+    gap = (TESTED-KAPPA)/7
+    L.need(gap.numerator == 1, 'Exact compiler integer grid')
+    step = gap.denominator
+    def wins(j):
+        return (step*j+1)**power < 2**j
+    left,right=1,2
+    while not wins(right):
+        left,right=right,2*right
+    while right-left > 1:
+        m=(left+right)//2
+        if wins(m):
+            right=m
+        else:
+            left=m
+    H=step*right
+    L.need(H > 8*v and wins(right) and not wins(right-1), 'Finite uniform threshold with predecessor rejection')
+    local_routes = (sum(local['paid'].values())+native['helper_normalizers']+native['main_adapters']+4*104+4*local['owned_lifetimes'])*1000
+    L.need(local['real_scalar_allowance']+local_routes < 10**7, 'Every local scalar, route and owned operation is funded')
+    boundary = sum(len(word['boundaries'][side+'_program']) for side in ('left','right'))
+    raw = 18*10**7 + 15*1664 + 32*104*(len(word['physical_events'])+boundary)
+    raw += 32*104*(26+1000)
+    L.need(raw < 10**9, 'Complete nonrecursive raw bound')
+    bit = L.read(L.ROOT/'research/query-thinned-controllers/RESULT.json')['ordinary_BIT']
+    atom = Q(bit['atom'])
+    ordinary=(1-atom)*Q(bit['after_cover'])+atom*Q('.0000384599')
+    admitted=Q(bit['admitted'])
+    L.need(atom > ordinary > admitted > TESTED and atom < 1-ordinary, 'Retained BIT admission and stock')
+    outer_module=L.load(L.ROOT/'research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py','p104_productive_outer')
+    params={'kappa':KAPPA,'tested_saving':TESTED,'complex_uniform_saving':KAPPA+6*gap,
+            'b':KAPPA+5*gap,'a':KAPPA+4*gap,'q':KAPPA+3*gap,'bit_saving':admitted,
+            'epsilon':1-gap,'beta':1-3*gap,'r0':gap/10,'stop':gap,'layout':gap,'d':gap}
+    outer=outer_module.ledger(params)
+    outer['functional_checks']['actual_new_BIT_stock_sublinear']=params['beta']*(1-atom)
+    L.need(len(outer['strict_checks']) == 54 and len(outer['functional_checks']) == 4, 'Complete outer predicate inventory')
+    L.need(all(x > 0 for key in ('strict_checks','functional_checks') for x in outer[key].values()), 'All outer inequalities strict')
+    wrong_cutoff = E.divide((Q(sum(T.values())*21),)*2,E.mul(E.power(20,TESTED),target))
+    L.need(wrong_cutoff[0] > 1, 'Old cutoff fails with this helper inventory')
+    bad=outer_module.ledger(dict(params,bit_saving=Q('.0027')))
+    L.need(any(x <= 0 for x in bad['strict_checks'].values()), 'Insufficient BIT contract rejected')
+    return encode({'kappa':KAPPA,'kappa_decimal':'.00310440546','previous_open_pr5601':PREVIOUS,'gain':KAPPA-PREVIOUS,
+                   'relative_gain_percent':100*(KAPPA/PREVIOUS-1),'raw_root_interval':[lower,upper],
+                   'raw_headroom':[lower-KAPPA,upper-KAPPA],
+                   'raw_ceiling':{'interval':[Q(0),cap],'derivative_upper':derivative_upper},'margin_at_previous':E.display(margin(PREVIOUS)),
+                   'targets':T,'paid':P,'productive_fields':sum(T.values()),'paid_calls':sum(P.values()),
+                   'rank_mass_surplus':760,'full_forward_calls':18,'query_shears':15,'whole_idle_jobs':3120,
+                   'strictification':{'tested':TESTED,'power':v,'cutoff':c,'floor':b,'tail_offset':u,'reserve':reserve,'tail_decay':decay,'tail_block':128,
+                                     'alpha':E.display(alpha),'gamma':E.display(gamma),'delta':E.display(delta),'normalized_paid_upper':normalized},
+                   'compiler':{'theta':theta,'logarithm_power':power,'eta':E.display(eta),'gap':gap,'H':H,'N':'2^'+str(H),
+                               'integer_test_exponent':right,'uniform_saving':params['complex_uniform_saving'],'word_materialized':False},
+                   'nonrecursive':{'local_scalar':local['real_scalar_allowance'],'local_routes':local_routes,
+                                   'raw_scalar_upper':raw,'raw_budget':10**9,'peak_owned_fields':local['peak_owned_fields'],
+                                   'fixed_scalar_growth_bound':'At most 2^(16*26000) before native factors; a declared constant, not an attained norm.',
+                                   'local_paid_rank_mass':local['paid_rank_mass'],
+                                   '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':bit,'outer':outer,
+                   'negative_controls':{'old_cutoff':'Rejected by completion-tail ratio greater than one','old_BIT':'Rejected by outer inequalities'}})
diff --git a/research/p104-a2-paid-rank-exchange/pricing.py b/research/p104-a2-paid-rank-exchange/pricing.py
new file mode 100644
index 0000000000..1e92330015
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/pricing.py
@@ -0,0 +1,174 @@
+"""Complete mixed-target price, compiler and outer transfer.
+
+The interval and compiler bounds follow zxv and Cao Yifan. The productive
+recursion is by Jacob Alan Sussman. The controller is by Cinna Davis.
+Adapted from Ali Romman's PR5544/5545 for the two coupled contact changes.
+Successor choices were prepared for thu-nmrc with OpenAI Codex assistance.
+Helper-subspace targets were added by Ali Romman with OpenAI Codex assistance.
+The source commuting-contact construction is by Pierre-Loic Doulcet (hexapode) in PR5565.
+The source coupled-frame change is by Ali Romman with OpenAI Codex in PR5601.
+Five fixed boundary relations are from Ali Romman's PR5602.
+The shared and delayed relays are from flyingviper's PR5605 and PR5612. The preparation is from
+Scratch-net's PR5606. Their costs and target debit are included.
+Ali Romman's complete caller/price proof and all predecessor credit are retained.
+Apache-2.0.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+import local as L
+
+KAPPA, TESTED = Q('.00310458'), Q('.0031045805')
+PREVIOUS = Q('.00310440546')
+
+
+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 assemble(local, native, word, physical):
+    paid, targets = Counter(), Counter()
+    counts = Counter()
+    L.need(local['helpers'] == native['helpers'], 'Every productive helper has a native target')
+    for e in word['physical_events']:
+        counts[e[0]] += 1
+        if e[0] == 'LOCAL':
+            L.need(e[3] is False, 'Every source invocation is forward')
+            paid.update(local['paid'])
+            targets.update(local['helpers'])
+        elif e[0] == 'QUERY_SHEAR':
+            paid[1] += 104
+        elif e[0] == 'IDLE':
+            paid[8*len(e[2])] += 208
+    main = Counter()
+    idle = Counter({8*len(e[2]):0 for e in word['physical_events'] if e[0] == 'IDLE'})
+    for e in word['physical_events']:
+        if e[0] == 'IDLE':
+            idle[8*len(e[2])] += 208
+    L.need(idle == physical['whole_idle_jobs'], 'Every whole idle has a native Gaussian and full rank')
+    for term in word['diagonal']:
+        main[1+8*sum(term[1:])] += 104
+    L.need(main == physical['main_targets'], 'Every main target has its full native phase')
+    targets.update(main)
+    L.need(counts['LOCAL'] == 18 and counts['QUERY_SHEAR'] == counts['IDLE'] == 15, 'Complete call census')
+    # Recount by the two path lengths, without using emitted main diagonals.
+    second_t = Counter({r:18*n for r,n in local['helpers'].items()})
+    second_p = Counter({r:18*n for r,n in local['paid'].items()})
+    second_t[145] += 208
+    second_p[1] += 15*104
+    for length in (7,8):
+        for j in range(1,length+1):
+            second_t[1+8*(j+1)] += 208
+            second_p[8*(j+1)] += 208
+    L.need((targets,paid) == (second_t,second_p), 'Independent full invoice')
+    L.need(max(targets) == 145 and max(paid) == 72, 'Maximum recursive ranks')
+    L.need(sum(targets.values()) == 12958 and sum(paid.values()) == 66546, 'Complete field and call counts')
+    L.need(sum(r*n for r,n in targets.items())-sum(r*n for r,n in paid.items()) == 760, 'Complete rank surplus')
+    return targets,paid,counts
+
+
+def run(local,native,word,physical):
+    T,P,counts = assemble(local,native,word,physical)
+    E = L.load(L.ROOT/'research/higher-moment-strictification/verify.py','p104_productive_intervals')
+    other = L.load(L.ROOT/'research/kappa-rebuild/architecture/mixed_target_probe.py','p104_productive_second_intervals')
+    def margin(k):
+        a,b = E.moment(T,1-k), E.moment(P,1-k)
+        return a[0]-b[1],a[1]-b[0]
+    lo,hi,grid = 0,10**12,10**14
+    while hi-lo > 1:
+        m=(lo+hi)//2
+        if margin(Q(m,grid))[0] > 0:
+            lo=m
+        else:
+            hi=m
+    lower,upper = Q(lo,grid),Q(hi,grid)
+    signed = Counter(T)
+    signed.subtract(P)
+    cap = Q('.0034')
+    derivative_upper = Q(0)
+    for rank, count in signed.items():
+        logarithm = E.log_bounds(rank)
+        low = E.mul(E.power(rank, 1-cap), logarithm)[0]
+        high = rank*logarithm[1]
+        derivative_upper -= count*(low if count >= 0 else high)
+    L.need(derivative_upper < 0, 'Unique raw crossing below the retained BIT cap')
+    L.need(margin(lower)[0] > 0 and margin(upper)[1] < 0, 'Strict rational crossing enclosure')
+    L.need(PREVIOUS < KAPPA < TESTED < lower and margin(PREVIOUS)[0] > 0, 'Changed complete bill improves the source claim')
+    t,u,v,c,b = 1-TESTED,Q(1,5),1024,17,18
+    target,paid = E.moment(T,t),E.moment(P,t)
+    tl,th = other.moment(T,TESTED)
+    pl,ph = other.moment(P,TESTED)
+    L.need(tl > ph and max(tl,target[0]) <= min(th,target[1]) and max(pl,paid[0]) <= min(ph,paid[1]), 'Two exact moment engines')
+    alpha = E.divide(paid,target)
+    gamma = E.divide(E.moment(P,t+u),E.mul(E.power(c,u),target))
+    delta = E.divide((Q(sum(T.values())*b),)*2,E.mul(E.power(c,TESTED),target))
+    reserve,decay = Q(1,10**12),Q(39,40)
+    L.need(alpha[1] < 1-reserve and gamma[1] < decay and delta[1] < decay, 'All three charged tail ratios')
+    L.need(decay**128 < Q(1,16), 'Both tail decay blocks')
+    normalized = 1/(1+v*reserve)+Q(4,2**32)
+    L.need(normalized < 1 and c**64 < b**63, 'Full completion, splitting and rounding charges')
+    theta,power = Q(63,64),1024
+    eta = E.scale(E.power(145,-v*TESTED),t)
+    L.need(2*(c**v)**64 <= (145**v)**63 and eta[0] > theta**power, 'Shifted exits and complement capacity')
+    L.need(E.power(145,v*TESTED)[1] < 2**power, 'Compiler initial range')
+    gap = (TESTED-KAPPA)/7
+    L.need(gap.numerator == 1, 'Exact compiler integer grid')
+    step = gap.denominator
+    def wins(j):
+        return (step*j+1)**power < 2**j
+    left,right=1,2
+    while not wins(right):
+        left,right=right,2*right
+    while right-left > 1:
+        m=(left+right)//2
+        if wins(m):
+            right=m
+        else:
+            left=m
+    H=step*right
+    L.need(H > 8*v and wins(right) and not wins(right-1), 'Finite uniform threshold with predecessor rejection')
+    local_routes = (sum(local['paid'].values())+native['helper_normalizers']+native['main_adapters']+4*104+4*local['owned_lifetimes'])*1000
+    L.need(local['real_scalar_allowance']+local_routes < 10**7, 'Every local scalar, route and owned operation is funded')
+    boundary = sum(len(word['boundaries'][side+'_program']) for side in ('left','right'))
+    raw = 18*10**7 + 15*1664 + 32*104*(len(word['physical_events'])+boundary)
+    raw += 32*104*(26+1000)
+    L.need(raw < 10**9, 'Complete nonrecursive raw bound')
+    bit = L.read(L.ROOT/'research/query-thinned-controllers/RESULT.json')['ordinary_BIT']
+    atom = Q(bit['atom'])
+    ordinary=(1-atom)*Q(bit['after_cover'])+atom*Q('.0000384599')
+    admitted=Q(bit['admitted'])
+    L.need(atom > ordinary > admitted > TESTED and atom < 1-ordinary, 'Retained BIT admission and stock')
+    outer_module=L.load(L.ROOT/'research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py','p104_productive_outer')
+    params={'kappa':KAPPA,'tested_saving':TESTED,'complex_uniform_saving':KAPPA+6*gap,
+            'b':KAPPA+5*gap,'a':KAPPA+4*gap,'q':KAPPA+3*gap,'bit_saving':admitted,
+            'epsilon':1-gap,'beta':1-3*gap,'r0':gap/10,'stop':gap,'layout':gap,'d':gap}
+    outer=outer_module.ledger(params)
+    outer['functional_checks']['actual_new_BIT_stock_sublinear']=params['beta']*(1-atom)
+    L.need(len(outer['strict_checks']) == 54 and len(outer['functional_checks']) == 4, 'Complete outer predicate inventory')
+    L.need(all(x > 0 for key in ('strict_checks','functional_checks') for x in outer[key].values()), 'All outer inequalities strict')
+    wrong_cutoff = E.divide((Q(sum(T.values())*21),)*2,E.mul(E.power(20,TESTED),target))
+    L.need(wrong_cutoff[0] > 1, 'Old cutoff fails with this helper inventory')
+    bad=outer_module.ledger(dict(params,bit_saving=Q('.0027')))
+    L.need(any(x <= 0 for x in bad['strict_checks'].values()), 'Insufficient BIT contract rejected')
+    return encode({'kappa':KAPPA,'kappa_decimal':'.00310458','previous_open_pr5601':PREVIOUS,'gain':KAPPA-PREVIOUS,
+                   'relative_gain_percent':100*(KAPPA/PREVIOUS-1),'raw_root_interval':[lower,upper],
+                   'raw_headroom':[lower-KAPPA,upper-KAPPA],
+                   'raw_ceiling':{'interval':[Q(0),cap],'derivative_upper':derivative_upper},'margin_at_previous':E.display(margin(PREVIOUS)),
+                   'targets':T,'paid':P,'productive_fields':sum(T.values()),'paid_calls':sum(P.values()),
+                   'rank_mass_surplus':760,'full_forward_calls':18,'query_shears':15,'whole_idle_jobs':3120,
+                   'strictification':{'tested':TESTED,'power':v,'cutoff':c,'floor':b,'tail_offset':u,'reserve':reserve,'tail_decay':decay,'tail_block':128,
+                                     'alpha':E.display(alpha),'gamma':E.display(gamma),'delta':E.display(delta),'normalized_paid_upper':normalized},
+                   'compiler':{'theta':theta,'logarithm_power':power,'eta':E.display(eta),'gap':gap,'H':H,'N':'2^'+str(H),
+                               'integer_test_exponent':right,'uniform_saving':params['complex_uniform_saving'],'word_materialized':False},
+                   'nonrecursive':{'local_scalar':local['real_scalar_allowance'],'local_routes':local_routes,
+                                   'raw_scalar_upper':raw,'raw_budget':10**9,'peak_owned_fields':local['peak_owned_fields'],
+                                   'fixed_scalar_growth_bound':'At most 2^(16*26000) before native factors; a declared constant, not an attained norm.',
+                                   'local_paid_rank_mass':local['paid_rank_mass'],
+                                   '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':bit,'outer':outer,
+                   'negative_controls':{'old_cutoff':'Rejected by completion-tail ratio greater than one','old_BIT':'Rejected by outer inequalities'}})
diff --git a/research/p104-a2-paid-rank-exchange/publication-invoices/COMPARISON.json b/research/p104-a2-paid-rank-exchange/publication-invoices/COMPARISON.json
new file mode 100644
index 0000000000..5c14e66639
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/publication-invoices/COMPARISON.json
@@ -0,0 +1,89 @@
+{
+  "assurance": {
+    "contract": "Exact matching user declarations; mathematical applicability is not verified.",
+    "inconclusive": "Mixed hinge signs do not establish a power-moment crossing or failure.",
+    "input_binding": "SHA-256 identifies the exact input bytes read, when loaded by this CLI.",
+    "scope": "Histogram arithmetic only. Histogram correctness, circuit legality, ownership, phases, precision, supplier admission and compiler endpoints are not validated."
+  },
+  "classification": "uniformly_better",
+  "contract": {
+    "ambient_rank": 145,
+    "boundary": "Complete PR5627-derived p16/a7 caller before the same mixed-target compiler",
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r costs r^(1-C); debit surrendered productive fields"
+  },
+  "delta_paid": {
+    "2": 18,
+    "3": -18,
+    "5": -18,
+    "6": 18
+  },
+  "delta_targets": {},
+  "hinges": [
+    {
+      "rank": 0,
+      "value": 0
+    },
+    {
+      "rank": 2,
+      "value": 0
+    },
+    {
+      "rank": 3,
+      "value": -18
+    },
+    {
+      "rank": 5,
+      "value": -18
+    },
+    {
+      "rank": 6,
+      "value": 0
+    }
+  ],
+  "inputs": {
+    "new": {
+      "bytes": 815,
+      "path": "research/p104-a2-paid-rank-exchange/publication-invoices/candidate.json",
+      "sha256": "390d998edd31dbc76996b14e0f8e5e1ddeab479bd40703ce2d96bd66506dd547"
+    },
+    "old": {
+      "bytes": 815,
+      "path": "research/p104-a2-paid-rank-exchange/publication-invoices/source.json",
+      "sha256": "477ec973ed0540d3192828bf60bf54b20a8022e45f3238ab2fbf337739f756e2"
+    }
+  },
+  "method_credit": "Classical concave order/majorization; mightyOil's t16-moment-order-audit and its retained predecessors. No new theorem claim.",
+  "power_moment": {
+    "exponent": "0 < t < 1",
+    "new_minus_old_sign": "strictly_negative",
+    "tariff": "r^t"
+  },
+  "quantity": "(new paid - new targets) - (old paid - old targets)",
+  "rank_mass": {
+    "delta": {
+      "paid": 0,
+      "paid_minus_targets": 0,
+      "targets": 0
+    },
+    "new": {
+      "paid": 240764,
+      "paid_minus_targets": -760,
+      "targets": 241524
+    },
+    "old": {
+      "paid": 240764,
+      "paid_minus_targets": -760,
+      "targets": 241524
+    }
+  },
+  "schema": "integer-mult-recursive-invoice-comparison/v1",
+  "signed_net_delta": {
+    "2": 18,
+    "3": -18,
+    "5": -18,
+    "6": 18
+  },
+  "tariff_class": "increasing concave f on nonnegative ranks, with f(0)=0",
+  "uniform_order": "weak; some concave tariffs can give equality"
+}
diff --git a/research/p104-a2-paid-rank-exchange/publication-invoices/candidate.json b/research/p104-a2-paid-rank-exchange/publication-invoices/candidate.json
new file mode 100644
index 0000000000..1bff83b0b2
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/publication-invoices/candidate.json
@@ -0,0 +1,44 @@
+{
+  "schema": "integer-mult-recursive-invoice/v1",
+  "contract": {
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r costs r^(1-C); debit surrendered productive fields",
+    "ambient_rank": 145,
+    "boundary": "Complete PR5627-derived p16/a7 caller before the same mixed-target compiler"
+  },
+  "paid": {
+    "1": 41160,
+    "16": 416,
+    "2": 8226,
+    "24": 416,
+    "3": 8874,
+    "32": 416,
+    "4": 1638,
+    "40": 416,
+    "48": 416,
+    "5": 2916,
+    "56": 416,
+    "6": 342,
+    "64": 416,
+    "7": 270,
+    "72": 208
+  },
+  "targets": {
+    "1": 36,
+    "145": 208,
+    "17": 416,
+    "2": 72,
+    "25": 416,
+    "33": 416,
+    "4": 1638,
+    "41": 416,
+    "49": 416,
+    "5": 108,
+    "57": 416,
+    "65": 416,
+    "7": 90,
+    "73": 208,
+    "8": 288,
+    "9": 7398
+  }
+}
diff --git a/research/p104-a2-paid-rank-exchange/publication-invoices/source.json b/research/p104-a2-paid-rank-exchange/publication-invoices/source.json
new file mode 100644
index 0000000000..d421d354b5
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/publication-invoices/source.json
@@ -0,0 +1,44 @@
+{
+  "schema": "integer-mult-recursive-invoice/v1",
+  "contract": {
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r costs r^(1-C); debit surrendered productive fields",
+    "ambient_rank": 145,
+    "boundary": "Complete PR5627-derived p16/a7 caller before the same mixed-target compiler"
+  },
+  "paid": {
+    "1": 41160,
+    "16": 416,
+    "2": 8208,
+    "24": 416,
+    "3": 8892,
+    "32": 416,
+    "4": 1638,
+    "40": 416,
+    "48": 416,
+    "5": 2934,
+    "56": 416,
+    "6": 324,
+    "64": 416,
+    "7": 270,
+    "72": 208
+  },
+  "targets": {
+    "1": 36,
+    "145": 208,
+    "17": 416,
+    "2": 72,
+    "25": 416,
+    "33": 416,
+    "4": 1638,
+    "41": 416,
+    "49": 416,
+    "5": 108,
+    "57": 416,
+    "65": 416,
+    "7": 90,
+    "73": 208,
+    "8": 288,
+    "9": 7398
+  }
+}
diff --git a/research/p104-a2-paid-rank-exchange/verify.py b/research/p104-a2-paid-rank-exchange/verify.py
new file mode 100644
index 0000000000..4731427a95
--- /dev/null
+++ b/research/p104-a2-paid-rank-exchange/verify.py
@@ -0,0 +1,69 @@
+"""Complete conditional A2 replay; zxv with substantial OpenAI Codex assistance. Apache-2.0."""
+from pathlib import Path
+from collections import Counter
+from copy import deepcopy
+from hashlib import sha256
+import sys,json,gzip,time
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent
+sys.path.insert(0,str(HERE))
+sys.path.insert(0,str(HERE/'inputs/pr5612'))
+from sources import sources,load,need,read
+L=load(HERE/'inputs/pr5627-local.py','local')
+manifest=read(HERE/'SOURCES.json')
+for item in manifest['retained_inputs']:
+ raw=(HERE/item['local_path']).read_bytes()
+ need(len(raw)==item['size'] and sha256(raw).hexdigest()==item['sha256'],'Retained source '+item['local_path'])
+quick='--quick' in sys.argv
+with sources() as (prior,framework):
+ L.ROOT=framework
+ bindings=[]
+ manifest=read(HERE/'inputs/pr5627-SOURCES.json')
+ for row in manifest['files']:
+  path=framework/row['path']
+  if path.is_file():
+   need(sha256(path.read_bytes()).hexdigest()==row['sha256'],'Latest declared foundation equals authenticated framework: '+row['path'])
+   bindings.append(row['path'])
+ source=read(HERE/'inputs/pr5627-WORD.json.gz')
+ from construct import construct
+ payload,construction=construct()
+ word=json.loads(payload)
+ before=L.replay(source);local=L.replay(word)
+ delta=Counter(local['paid']);delta.subtract(before['paid'])
+ need({r:c for r,c in delta.items() if c}=={2:1,3:-1,5:-1,6:1},'Exact full paid-rank exchange')
+ need(local['helpers']==before['helpers'],'All productive targets retained')
+ print('PASS exact source foundation bindings',len(bindings),'and all arbitrary-input/frame rows',flush=True)
+ if quick:
+  print('PASS_P104_A2_QUICK',flush=True)
+  raise SystemExit(0)
+ original=(HERE/'inputs/pr5627-pricing.py').read_text()
+ for before,after in read(HERE/'PRICING-EDITS.json'):
+  need(original.count(before)==1,'Unique declared pricing edit');original=original.replace(before,after)
+ need(original==(HERE/'pricing.py').read_text(),'Only declared pricing edits')
+ transpose=L.reverse_columns(word)
+ print('PASS all dense transpose columns',transpose['columns'],flush=True)
+ native=L.native(word)
+ print('PASS all signed native maps and exact phases',native['distinct_maps'],flush=True)
+ C=load(framework/'research/p104-productive-controller/caller.py','new_a2_caller')
+ caller=C.construct();need(caller==read(framework/'research/p104-productive-controller/CONTROLLER.json'),'Unchanged complete caller')
+ physical=C.physical(caller,word)
+ print('PASS complete physical caller',flush=True)
+ P=load(HERE/'pricing.py','a2_new_price')
+ price=P.run(local,native,caller,physical)
+ print('PASS two exact moment engines, compiler, BIT stock and58 outer predicates',price['raw_root_interval'],flush=True)
+ controls=C.controls(caller)
+ def reject(f):
+  try:f()
+  except ValueError as e:return str(e)
+  raise ValueError('Malformed control accepted')
+ bad=deepcopy(word);gate=['ADD',340,715,'-4',396];need(bad['program'].count(gate)==1,'Unique new A2 correction');bad['program'].remove(gate)
+ controls['missing_A2_correction']=reject(lambda:L.replay(bad))
+ bad=deepcopy(word);index=bad['program'].index(gate);bad['program'][index][-1]=825
+ controls['wrong_new_contact_frame']=reject(lambda:L.replay(bad))
+ bad=deepcopy(local);bad['paid'][2]+=1;bad['paid'][3]-=1
+ controls['undercharged_child']=reject(lambda:P.assemble(bad,native,caller,physical))
+ result=P.encode({'status':'PASS_P104_A2_KAPPA_00310458','source_pr':5627,'source_head':'ed4dc88a09a7d201971f6603b8c2074e95dd9c32','source_foundation_matches':bindings,'local':local,'transpose':transpose,'native':native,'physical':physical,'complete_cost':price,'controls':controls})
+expected=read(HERE/'RESULT.json')
+for key in ('local','transpose','native','physical','complete_cost','controls'):
+ need(result[key]==expected[key],'Exact saved complete receipt '+key)
+print(result['status'],flush=True)