[bbMisk]

Result and scope

This mathematical audit derives the exact repair density of the retained eight-rotation packed-bit gadget on each fixed-source fiber. With B=2^K and consecutive positions j_t of the used source 1 bits,

d(c) = 1 - product_t(1 - 2/B^(j_(t+1)-j_t)).

The written proof covers every K>=6, f>=1 and allowed offset, with complete y,z fields. It also gives sharp extrema at fixed source weight and recovers PR773's complete-rectangle average. At K=6,f=3, source x=65 has density 1/32 versus the average 1/128. Source patterns 110 and 101 have equal weight but different exact densities. No new kappa is claimed.

This extends Precious (@Kyu)'s PR773 (dbd9da3eed9a0045b77de861099e869259f1ad13) and credits Aryan Sudhir's PR735 (8d36139e49add1e55795e283b3bfa45e1cfe1bf0). PR773 explicitly excludes a per-source guarantee; PR735 counts a sufficient guard envelope. The new evidence is the actual source-pattern law, including cancellation across zero-source segments, its sharp extrema and the conditional-count audit. Neither predecessor theorem is contradicted.

Review and verification

Only research/packed-source-fibers-seed62/ is added. Start with its README proof, SOURCES.json and EXPECTED.json. Base: 49b3157d1d89adcef16df8be98445451672be5ba; upstream manuscript pin: adc7f1241b42e322a6451854ab7e4b4c146bf78a. Track: adversarial mathematical audit; OS-random seed 62. Coordination claim.

python3.12 scripts/research_package.py --package research/packed-source-fibers-seed62

Expected: PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed.

Local CPython 3.12.13 and 3.14.6 checks pass: 873,760 digit cases, 8,188 source patterns, 541,200 complete small-fiber states, 49,792 source-domain simulator comparisons, and exact sorted-payload repair. The optimized relocated replay rejects four mathematical mutations and four packaging faults. Package metadata/pins, all four retained finite-identity tests and make selected-record-check pass. Hosted CI is separate.

The quantified claim rests on the written proof; finite replay is corroboration. A source-restricted physical caller would still need to admit its layout and pay the complete movement/repair bill. All-size/compiler, hosting, precision/tape and analytic transfer remain outside scope. The selected supplier cold replay was not run for this isolated addition.

Prepared by Hanson Sun (@bbMisk) with substantial OpenAI Codex assistance in derivation, implementation, verification and writing. Apache-2.0; predecessor credit is retained in the package.

[Mister-Meeseeks]

100-PR batch review at 44bf3b1d6031e4069f5abe26211f15e52293e1f4.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact fixed-source packed-bit repair density from the gaps between selected ones, sharp fixed-weight bounds and five-state proof; complete dirty fibers required.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.