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+# Independent review of open κ claims, 11 October 2026
+
+This review checks the open CrocSwap pull requests that claim a conditional
+exponent above the reviewed gen4 result. It follows the verifier-first practice
+of Georgiev, Gómez-Serrano, Tao and Wagner, *Mathematical exploration and
+discovery at scale* (arXiv:2511.02864): assume a search may exploit its
+verifier, replay every claim from a fresh extract, and recompute the score by
+code that the claim does not ship. Negative results are recorded in
+[NEGATIVE-RESULTS.md](NEGATIVE-RESULTS.md).
+
+Scope: finite replay and exact arithmetic only. The original #109 framework, the
+all-size compiler, chart, routing, restored-row, prime-supply, precision/recovery
+and analytic interfaces, and every unmerged predecessor these PRs vendor, remain
+assumptions. Nothing here is a formal proof or a human peer review. PR replays
+for #355–#365 were performed by delegated Claude subagents in this session and
+spot-checked as noted; the gen4 and #366 replays and the independent #366
+score were run directly.
+
+## Results
+
+All runs used Python 3.13, a fresh `git archive` extract of the PR head, and a new
+output directory. "Scorer" is [independent_bit_score.py](independent_bit_score.py):
+it recomputes every MOVE/COPY rank from frame dimensions, rejects a word whose
+rank fields disagree, and certifies the largest 10^-18 grid bit saving with its
+own exact rational log/exp enclosures.
+
+| PR | Head | Claimed κ | Replay | Scorer | Verdict |
+| --- | --- | --- | --- | --- | --- |
+| main (gen4) | `12a4934` | 90446448750933/1.25e17 ≈ 7.23572e-4 | PASS, 7m13s | — | reproduced |
+| #366 | `d3d8420` | 160987957846297/2e17 ≈ 8.04940e-4 | PASS, 8m42s | exact agreement | reproduced, caveats |
+| #364 | `f8c6a47` | 201219842699733/2.5e17 ≈ 8.04879e-4 | PASS, 6m21s | exact agreement | reproduced, caveats |
+| #362 | `f46ddb5` | 804849251275843/1e18 | PASS, 6m22s | exact agreement | reproduced, caveats |
+| #358 | `e76d7a9` | 80367971186626189473809/1e26 ≈ 8.03680e-4 | PASS, 744s | exact agreement | reproduced |
+| #356 | `c3d16f4` | 401580942485227026650141/5e26 ≈ 8.03162e-4 | PASS, 481s | exact agreement | reproduced |
+| #360 | `edef1a7` | 80351298055091/1e17 ≈ 8.03513e-4 | PASS after packaging fix | exact agreement | reproduced, packaging defect |
+| #355 | `e59117a` | 801967436473393/1e18 ≈ 8.01967e-4 | PASS after packaging fix | exact agreement | reproduced, packaging defect |
+| #365 | `a222705` | complex b = 876412559609473/1e18 | PASS, 2.7s | b reproduced | reproduced; κ unchanged |
+| #363 | `9df55da` | complex b = 932783231884153/1e18 | PASS, 9.9s | b reproduced | reproduced, new unproved layout; κ unchanged |
+
+Every bit-side claim is bound by the bit saving. The largest reproduced claim
+is #366, 11.25% above the reviewed gen4 result.
+
+## Findings
+
+**#366.** The final word's records hash, `b99fa242…`, equals the PR's
+`EXPECTED.json`. The scorer reproduces the bit saving 402794119831239/5e17, and
+the bit→κ map recomputed separately gives the claimed κ with the bit side binding
+(complex headroom ≈ 7.07e-5). Exact checks cover move/copy ranks against frames
+(`stagelib.check`), frame validity and `I-J/9` nondegeneracy, and rational moment
+enclosures with adjacent-grid rejection. Two scope points:
+
+- `price.py` uses a hard-coded default complex saving `b`; `complex_b.py`
+  recomputes b from the E8 certificate but only prints it. We confirmed the
+  printed value equals the default exactly. An assertion would close this.
+- `stagelib`'s column replay is over F2. Integer correctness of the signed lift
+  rests on the columns checker's cancellation-free bound and inherited scalar
+  evidence.
+
+The 47 outer inequalities are asserted by the PR's own `outer.py`; they were not
+recomputed independently.
+
+**#364 / #362.** `verify.sh` asserts the recomputed complex b. In this lineage
+`data/parity` is the PR's own regenerated base word, so "new frames" counts are
+relative to it, not to #325, and the README's "1,064 frames new relative to
+#325" is copied from #359. `stagelib.full_check` still checks every used frame
+of the final word, and nondegeneracy rechecked against the true #325 frames
+(8838 differing frames) found none degenerate. Several `cmd && echo` lines under
+`set -e` would not stop on failure; all printed in this replay. The producer
+regeneration scripts reference paths outside the package.
+
+**#358 / #356.** `legality.cpp` rejects a MOVE/COPY rank that differs from its
+frame dimension; the native 24-stage admission is actually executed. The audit's
+`assembly.py` reads its root from `REFINEMENT.json`; an independent 1e-27 root
+reproduced the claimed κ exactly. The exact integer operator check excludes
+MOVEs, which the native geometry check covers.
+
+**#360 / #355.** Both packages' `MANIFEST.json` pin a `.gitattributes` that is not
+committed, so the documented `verify.py` fails on a fresh checkout with
+"Package source missing, changed or unlisted". Replays used a copy with only
+that manifest entry removed. Integer correctness is checked over Z against the
+source word (int128 with overflow guards); the source words are compacted forms
+of unmerged #351/#354 words whose conversion is not replayed.
+
+**#365 / #363.** Both start from Sussman's identical `gcert1-e8-r783` certificate.
+Neither changes κ: running #366's `outer.assembly` with b equal to the old value,
+#365's or #363's gives the same κ, since the bit side binds. #363's seven-stage
+bridged layout B₃ has no written or Lean proof (Sussman's bridge certificate covers
+the five-stage layout) and inherits five-stage row constants. If admitted, it raises
+the complex cap to ≈ 9.33e-4, so bit-side gains up to that cap would count.
+
+## Reproduce
+
+```sh
+git fetch https://github.com/CrocSwap/integer-mult-bounds.git pull/366/head:pr366
+git archive pr366 research/w3-design-t-mc-coupled-retiming | tar -x -C /tmp/pr366src
+python3 -B /tmp/pr366src/research/w3-design-t-mc-coupled-retiming/verify.py /tmp/pr366out
+python3 -I research/independent-review/independent_bit_score.py /tmp/pr366out/candidate \
+    402794119831239/500000000000000000
+```
+
+Requires numpy, mpmath, a C++17 compiler and Boost headers. The receipt is
+[receipts/pr366.json](receipts/pr366.json).