Conditional κ = 0.00310644468 (+1.8133e-7, +0.00583756% over #5644): combine helper cleanup with the A2 exchange

research/p104-a2-neutral-cleanup/CLAIM.json

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+{
+  "schema_version": 1,
+  "status": "LOCAL author-checked conditional composition; normal and registered isolated optimized complete replays passed",
+  "kappa": "77661117/25000000000",
+  "decimal": "0.00310644468",
+  "previous_pr": 5644,
+  "previous_kappa": "62125267/20000000000",
+  "gain": "18133/100000000000",
+  "gain_decimal": "1.8133e-07",
+  "gain_percent": "0.0058375604244887995",
+  "other_comparison": {
+    "pr": 5643,
+    "kappa": "155229/50000000",
+    "gain": "46617/25000000000",
+    "gain_percent": "0.06006223063989332"
+  },
+  "raw_crossing": [
+    "310644469581/100000000000000",
+    "155322234791/50000000000000"
+  ],
+  "raw_headroom": [
+    "1581/100000000000000",
+    "791/50000000000000"
+  ],
+  "local_paid_delta_vs_pr5644": {
+    "2": 1,
+    "3": -1,
+    "5": -1,
+    "6": 1
+  },
+  "local_paid_delta_vs_pr5643": {
+    "1": -3,
+    "5": -3,
+    "6": 3
+  },
+  "target_delta": {},
+  "local_invocations": 18,
+  "complete_paid_calls": 66492,
+  "complete_productive_targets": 12958,
+  "rank_mass_surplus": 760,
+  "local_ordinary_allowance": 220220,
+  "publication": {
+    "published": false,
+    "selected": false,
+    "hosted_ci": "not run"
+  },
+  "scope": "Actual common chronological word, retaining every independent input, productive target and owned operation; conditional on inherited native, erasure, compiler, precision/tape, analytic recovery and helper endpoint contracts. No formal-kernel or independent human verification."
+}

research/p104-a2-neutral-cleanup/COMPONENT-TRIAGE.md

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+# Which construction component is ready to remove?
+
+The best immediate opportunity is [PR #5644](https://github.com/CrocSwap/integer-mult-bounds/pull/5644)'s obsolete late helper cleanup, combined with [PR #5643](https://github.com/CrocSwap/integer-mult-bounds/pull/5643)'s A2 exchange. Its exact input-basis preparation and last-read cancellation remove contacts that force otherwise unused native stops. The resulting common word supports **conditional κ = 0.00310644468** in the fresh complete replay. This is an actual functional replacement with a recomputed invoice.
+
+The original summarized list ran through PR5612; later PR5627 consumed its compatible improvements. Current exact sources are PR5643 `a3ff5c628bfc8825553d62d913f3e91f75d6825f`, PR5644 `c5871955d89bf73506fd68b0a5bf1cf2b3153c02` and their common PR5627 source `ed4dc88a09a7d201971f6603b8c2074e95dd9c32`. All screens below bind the emitted common word, compressed SHA256 `e94809cecf9e9cba56dd7a8a11ff1edeedc4c34013ff6caebdaf3b021351cf8c`.
+
+| Rank | Component and lineage | Complete accounting and decision |
+|---|---|---|
+| 1 — positive | Late helper cleanup, [#5644](https://github.com/CrocSwap/integer-mult-bounds/pull/5644), plus A2, [#5643](https://github.com/CrocSwap/integer-mult-bounds/pull/5643) | The functional common word retains all targets. Against PR5643 it replaces 54 complete H1+H5 paths by H6; against PR5644 it retains 18 A2 rank exchanges. Extra ordinary work is charged. κ increases over both. |
+| 2 — changed interface needed | Owned delivery replaced by a complementary public relay, [#5605](https://github.com/CrocSwap/integer-mult-bounds/pull/5605)/[#5612](https://github.com/CrocSwap/integer-mult-bounds/pull/5612) | Eight paid H7 deliveries remain. All eight fail free reconstruction from current public values at their first read's literal chart and time. A new preparation or source window could still replace one, but must charge storage, routing, cleanup and any native access. |
+| 3 — changed graph needed | Vacating an intermediate frame, [#5561](https://github.com/CrocSwap/integer-mult-bounds/pull/5561)/[#5601](https://github.com/CrocSwap/integer-mult-bounds/pull/5601)/[#5580](https://github.com/CrocSwap/integer-mult-bounds/pull/5580) | Their earlier compatible retimings are already consumed. A2 shows a changed noncommuting dependency graph can escape a fixed-contact screen. The next useful object is one specific last read that can be removed or moved by an exact identity. |
+| 4 — fixed-word route exhausted in tested scope | Independent helper split, [#5599](https://github.com/CrocSwap/integer-mult-bounds/pull/5599)/[#5602](https://github.com/CrocSwap/integer-mult-bounds/pull/5602), with dyadic completion [#5611](https://github.com/CrocSwap/integer-mult-bounds/pull/5611)/[#5615](https://github.com/CrocSwap/integer-mult-bounds/pull/5615) | The complete rational prefix/suffix pairing vanishes at every tested common-chart cut for all 411 ZERO-to-FULL helpers. There is no hidden dyadic unit ideal in this scope. A changed word or charged mixed-boundary interface is required before another basis search can pay. |
+| 5 — no recursive saving by deletion alone | Prepared zero-prefix storage, [#5606](https://github.com/CrocSwap/integer-mult-bounds/pull/5606), and inverse/main wrappers | Prepared role 751's three lifetimes have zero paid MOVEs, and inverse/main adapters already add zero recursive jobs. Removing them alone changes ordinary constants. Their usefulness would be enabling a different charged word, not deleting a recursive bill they do not contain. |
+
+The older [#5413](https://github.com/CrocSwap/integer-mult-bounds/pull/5413) coupled restoration adds 144 H8 paid calls and 144 H8 productive targets. Deleting both is rankwise neutral; retaining the productive credit while deleting its paid realization would need a new source-compatible correction. The larger [#5440](https://github.com/CrocSwap/integer-mult-bounds/pull/5440) headline remains outside this ready list because its priced native primitive and actual physical caller are not matched.
+
+## Exact negative screens and their limits
+
+[Owned-delivery evidence](analysis/owned-deliveries/COMPOSED-CENTRE-RECONSTRUCTION.json) comes from a separate rational interpreter that checks all 752 scalar endpoint rows and owned zero exits. The eight H7 source rows have rational rank eight. At their first public use, the available same-frame public spans have ranks 1, 2, 1, 1, 1, 1, 1, 1; each delivery value has a nonzero residual. This excludes an ADD-only substitution at those exact frames/times. It does not exclude changed windows, preconditioning, paid transport or joint native maps.
+
+[Joint-cut evidence](analysis/helper-cuts/ALL-EVENT-CUT-SCREEN.json) covers 8,847 event/chart states at 7,253 distinct cuts, all actual cohorts of sizes 1–16 and all 411 ZERO-to-FULL helpers. Only 18 states have nonzero prefix and suffix spaces; every full rational boundary pairing matrix is zero. The [scope proof](analysis/helper-cuts/README.md) explains the exact kernels, unaffected-chart state compression and COPY handling. A separate backward-prefix/forward-suffix checker verifies all 18 witnesses; it does not independently recompute every empty kernel.
+
+Reproduce the lightweight checks from the repository root:
+
+```sh
+python3.12 -I -O -B research/p104-a2-neutral-cleanup/analysis/owned-deliveries/check_composed_centres.py
+python3.12 -I -O -B research/p104-a2-neutral-cleanup/analysis/helper-cuts/check_report.py research/p104-a2-neutral-cleanup/WORD.json.gz
+```
+
+The discovery scanner can explicitly regenerate its own receipt:
+
+```sh
+python3.12 -B research/p104-a2-neutral-cleanup/analysis/helper-cuts/scan_all_cuts.py research/p104-a2-neutral-cleanup/WORD.json.gz
+```
+
+## Why apparent accounting deletions are not automatically gains
+
+The retained caller's 3,120 whole idle jobs supply part of its actual endpoint action. Its 15 isolated query shears supply 1,560 paid H1 jobs; literal deletion changes the exact formal caller endpoint. They are not disposable line items. The BIT side is currently slack above this complex ceiling, so a stronger BIT component alone does not raise the complete κ.
+
+The useful distinction is whether a functional replacement removes paid jobs **after** surrendering any lost productive targets and charging new ownership/native access. PR5644's last-read identity and PR5643's changed A2 graph satisfy that test together. Merely dropping a wrapper, deleting both halves of an equal paid/target entry, or treating an independently prepared input as free does not.
+
+Scientific credit remains with the linked contributors and the complete source notices. The coordinating agent and three subagents produced this triage with substantial OpenAI Codex assistance. The [skeptical composition review](analysis/owned-deliveries/COMPOSITION-SKEPTICAL-REVIEW.md) is scoped agent scrutiny, not independent human or proof-kernel acceptance.
+
+## Open next questions
+
+The strongest remaining bounded task is one changed complementary-public-value window for a named H7 delivery. First show the source survives to its consumer under the changed chronology; then emit its whole restoration and native bill. A changed scalar dependency that makes a fixed helper boundary pairing nonzero is another distinct route. The current screens do not supply either construction.

research/p104-a2-neutral-cleanup/COMPOSITION-AUDIT.json

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+{
+  "candidate_events": 25449,
+  "commuting_edit_scope": "After all three cleanup replacements, the unique original A2 motif remains; replacing it and independently reconstructing all typed monotone paths gives the identical complete emitted program.",
+  "delta_vs_pr5643": {
+    "1": -3,
+    "5": -3,
+    "6": 3
+  },
+  "delta_vs_pr5644": {
+    "2": 1,
+    "3": -1,
+    "5": -1,
+    "6": 1
+  },
+  "scalar_events": 22015,
+  "scope": "Exact common scalar skeleton, three-column A2 identity for arbitrary prefix values, reverse-order full chart-path reconstruction, fixed physical boundary data and complete paid histogram. Native phases and caller/compiler evidence remain separate.",
+  "status": "PASS_INDEPENDENT_COMPOSITION_ORDER_AND_FRAME_AUDIT"
+}

research/p104-a2-neutral-cleanup/COMPOSITION-PROOF.md

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+# A common functional word for the A2 exchange and three helper cleanups
+
+This scoped composition combines [zxv](https://github.com/zxv)'s [PR #5643](https://github.com/CrocSwap/integer-mult-bounds/pull/5643), exact head `a3ff5c628bfc8825553d62d913f3e91f75d6825f`, with [Ali Romman](https://github.com/aliromman)'s [PR #5644](https://github.com/CrocSwap/integer-mult-bounds/pull/5644), exact head `c5871955d89bf73506fd68b0a5bf1cf2b3153c02`. Both consume the exact [PR #5627](https://github.com/CrocSwap/integer-mult-bounds/pull/5627) word at `ed4dc88a09a7d201971f6603b8c2074e95dd9c32`. This is a compatibility argument for the emitted program, not an addition of headline κ gains. All source mechanisms and notices retain their attribution. New composition and this scoped audit used substantial OpenAI Codex assistance; Apache-2.0.
+
+## The two substitutions are functional before they are priced
+
+For the A2 substitution, write A = register 715, B = register 388 and C = register 340 at its prefix. The old scalar pair `B -= 4 A; C += B` and new scalar triple `C -= 4 A; C += B; B -= 4 A` both produce `(A, B−4A, C+B−4A)` for arbitrary prefix values. This is a matrix identity on the three full input columns; it does not assume a zero register or a special incoming relation. The new gates retain their specified charts 396, 825 and 127.
+
+For a cleanup helper h, the exact checked prefix relation has the form
+
+```
+h(cut) − Σ_j b_j z_j(cut) = s u_h + Σ_{j ≠ h} a_j u_j.
+```
+
+Here s is an invertible dyadic unit, all initial inputs on the right share h's input chart, and every cancellation register on the left shares h's cut chart. Prepare `u_h' = u_h − Σ(a_j/s)u_j`. Execute the prefix on those prepared inputs, then replace the helper state by `(h(cut) − Σ b_j z_j(cut))/s`. The relation gives exactly `u_h` for every arbitrary input. There is no later read of h, and the removed suffix consists only of its own declared scalar cleanup gates. The original helper's scalar endpoint row is its independent input, while all other scalar endpoint rows have zero dependence on that helper input. Its physical endpoint retains the productive native map Φ_final Φ_initial⁻¹. Thus the preparation and replacement preserve every other endpoint and restore h's exact endpoint. This is the original cleanup proof applied to the actual composed prefix; the verifier reevaluates each relation after the preceding edits.
+
+The three helpers are 291, 321 and 369. All input preparations, dyadic scalings, cancellation gates and resulting transports are present in the emitted common word. They do not use free helper output credits or a new owned-erasure rule.
+
+## Actual composition, in both orders
+
+Applying the three retained cleanup witnesses directly to the emitted PR5643 word passes the complete rational scalar/frame replay after every cleanup. The witnesses' exact anchors, prefix relations, absence of later helper reads, fixed productive endpoint charts and retained owned-event sequence are checked on the evolving word. This supplies an actual common program.
+
+The separate audit reads the saved source, A2, cleanup and common words as data. After deleting MOVE instructions, it finds the unique old A2 scalar pair in the cleanup-only word. Replacing that pair by the specified triple gives exactly the common word's entire scalar skeleton. It independently rebuilds every typed monotone transport path from the cleanup-only initial charts and that skeleton. The resulting full chronological instruction list is exactly the common word, including every MOVE. Thus applying A2 after cleanup gives the same emitted program as applying cleanup after A2; compatibility does not follow merely from disjoint headline histograms.
+
+`audit_composition.py` also checks the three-column A2 identity, all boundary data and the complete paid histograms. A missing final A2 shear fails the functional matrix identity despite retaining all helper cleanup identities. The complete verifier separately rejects a wrong A2 frame, omitted direct A2 correction, false cleanup cancellation, wrong cleanup cut, omitted initial preparation, missing owned CLEAR, extra target credit and an unpaid query.
+
+## From scalar equality to physical action
+
+Let Φ_f be the actual native chart operator at chart f, including its global phase. The frame invariant represents each physical register by `Φ_f z`, where z is its common-coordinate value. A coframed scalar gate obeys `Φ_f z_r + q Φ_f z_s = Φ_f(z_r + qz_s)`. A paid native chart change a→b is `Φ_b Φ_a⁻¹`, so it sends `Φ_a z` to `Φ_b z`. Induction through every chronological instruction transfers the proved scalar endpoint equality to physical endpoint equality. This argument neither commutes Gaussians nor replaces a tensor power of a sum by a sum of tensor powers. All owned instructions and their inherited contracts are retained in order.
+
+The fresh native-map and full caller replays remain necessary evidence for the actual emitted chart transitions; the scalar audit alone does not claim to supply those checks. The inherited all-width realization, compiler, precision/tape, analytic recovery and helper-endpoint assumptions are unchanged.
+
+## Complete bill and limits
+
+Relative to PR5644, the common word's complete local paid change is exactly `H2 + H6 − H3 − H5`. Relative to PR5643, it is exactly `3(H6 − H1 − H5)`. Every productive target is retained. The full caller multiplies each local delta by 18. It has 66,492 paid calls, 12,958 productive fields and rank-mass surplus 760. The local ordinary allowance is 220,220, including the cleanup's extra 54 scalar units above the A2 word.
+
+Both signed moment changes are strictly favorable for `0 < t < 1`: strict concavity gives `3^t−2^t > 6^t−5^t`, and subadditivity gives `1+5^t > 6^t`. The resulting root is recomputed from the common complete invoice. Scientific admission, hosted CI, proof-kernel verification and independent human review remain separate obligations. No global optimality claim follows.