[petalas]

Summary

Scoped combinatorial proof, conditional on Hanson Sun / bbMisk's #1245 gap-product theorem. This gives the exact mean repair density of the original eight-rotation packed-bit gadget on uniform source words of length m and weight w, with complete dirty fields. No new κ or physical caller is claimed.

A[m,w] = sum of per-source agreement fractions at length m, weight w
sum A[m,w] z^m u^w
  = (B + (2u-1)z) / (B - (B+1+(B-2)u)z + (1-u)z^2)
slice failure density = 1 - A[m,w]/binom(m,w)

At B=64,m=3,w=2 the exact mean is 43/2048. The inherited fixed-weight maximum is 1/32 and unrestricted-source average is 255/16384. The new contribution is the fixed-weight mean, generating function and coefficient recurrence. The predecessor's digit program, transition table and gap-product law are cited premises and were not re-audited. Credit #1245, Precious / Kyu's #773, Aryan Sudhir's #735 and the retained manuscript lineage. Produced by Nick Petalas.

Advance CLAIM. Scope matches the claim. Research ended at the first verified identity.

Evidence

Before, #1245 supplied each source's fraction and fixed-weight extrema. After, a unique decomposition by leading zeros, trailing zeros and consecutive-one gaps gives the exact aggregate for every length and weight.

Local CPython 3.12.15 verification passes: 131,068 source-pattern cases across 480 weight slices at B=4,6,64,128, 516 boundary rows through length 128, and ten rejection controls. These enumerate the cited product to corroborate the new aggregation; they do not test the predecessor's gadget. Both registered warning-strict normal and optimized commands passed, timed separately at 0.084 s and 0.095 s locally. The complete registered run took 0.526 s. A Git-free, read-only optimized package copy passed with every source hash unchanged; attempted receipt overwrite rejected. The generated receipt binds all six proof/code/metadata hashes.

From the repository root, exact local reproduction commands:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/packed-fixed-weight-repair
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/packed-fixed-weight-repair --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/packed-fixed-weight-repair/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/packed-fixed-weight-repair/verify.py

Expected markers are PASS_PACKED_FIXED_WEIGHT_REPAIR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The publication-byte/source-history check passes at dbaa509. Standard library only; a reviewer can substitute their CPython 3.12 executable. The selected baseline metadata binding also passed; no selected cold replay was run.

Limitations and unrun obligations

The all-parameter conclusion is a written scoped proof, not proof-kernel certification. It depends on #1245's stated theorem. Restricted dirty ranges, nonuniform weighting within a slice and source/dirty correlations are outside the count's hypotheses. A complete source-restricted physical caller, layout admission, movement, sorting, reinsertion, restoration, precision and the complete paid bill remain open. All five selected framework assumptions remain. Native, compiler, Lean, Docker and full multiplication suites were not run for this isolated count.

Related open work

At completion, the 15 other open PRs from this account created since the advance CLAIM were checked. None studies the original eight-rotation gadget's fixed-weight repair count. Before commit, the complete 393-PR overlap snapshot, fresh 100-PR list and focused searches likewise found no matching open work. #1245, #773 and #735 are merged predecessors. Search coverage does not establish novelty.

Open next questions

• Can a paid caller supply uniform fixed-weight source slices while preserving complete dirty fields? Trace one invocation and charge source partition descriptors, movement, correction and return.
• What sparse-weight asymptotic follows from the coefficient formula? This could price a future sparse source caller, but would still leave its physical implementation open.

CI notes

Both hosted checks passed at commit dbaa509654065721bd4acaddc316ffb5421be7c0: plan and replay (research/packed-fixed-weight-repair), in run 40591426759. The completed PR checks were read once and confirmed. Local registered replay, standalone normal/optimized checks and committed-byte/source-history checks also passed. Hosted replay corroborates the scoped aggregation; it does not prove the cited premise or certify a complete multiplication theorem.

Merge danger

Reversible. Scope is one new research package with no shared-code or selected-frontier edits.