[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. A closed borrowed-anchor tour through every weight-k mask on n unchanged source rows has exact transition cost 2 binomial(n,k)+2k-2, for every n>=1 and 1<=k<=n. If there are R arbitrary dirty targets, the complete one-write-per-target cost is R plus this amount. Every source is restored; arbitrary intermediate masks and repeated visits are allowed.

empty mask -> all weight-k masks in reflected Gray order -> empty mask
k toggles  +  2[binomial(n,k)-1] exchanges  +  k toggles

For five sources, weight three and one target per mask, the complete circuit uses 34 row ADDs versus 40 direct ADDs, including preparation and restoration. The new fixed-weight-layer scope is k>=3. Nick Petalas / PR4899 already covers the pair layer with unequal prices. Cinna Davis / PR2271 supplies the borrowed-source interface and Open next question 2. Classical Gray listings and Hamming distances retain their credit. This contribution was produced by Nick Petalas.

Only research/fixed-weight-borrowed-tours/ is added. The original bit-helper-contract card explicitly changes scope to this local row-register side result. Advance CLAIM. Canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, was refreshed before commitment. The selected frontier is unchanged.

Evidence

Before, independent direct source writes cost 40 row ADDs for the stated fixture. After, 24 anchor toggles plus ten target writes give 34. The first-visit distance argument supplies the lower bound for every legal finite walk; the reflected fixed-weight listing attains it.

Local CPython 3.12.15 normal and optimized replay pass with identical generated receipts:

• 90 mask layers and 8,190 required masks through n=12.
• 20 independent full-hypercube shortest covering walks through n=5, permitting off-layer states and repeated visits. Their distinct discovered state sets sum to 58,616.
• 132 complete circuits through n=8, with one to three targets per mask, checking 301,288 coefficient positions, 532 dirty-data cases and 132 inverse executions.
• Eighteen domain, missing-mask, restoration, illegal-gate, excess-cost and equal-length semantic rejection controls.

Each registered command was timed locally at approximately 0.074 seconds, within its 120-second limit. These are check durations, not runtime-improvement claims. The registered pair passes, as do in-memory AST/function-size checks, whitespace, exact committed bytes, source ancestry and executable modes. A six-file Git-free optimized export replay produces the identical receipt. No predecessor scientific verifier was imported or replayed.

Exact reproduction from the repository root, standard library only:

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

Other machines may substitute their CPython 3.12 path. Expected markers are PASS_FIXED_WEIGHT_BORROWED_TOURS_NO_NEW_KAPPA, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Stdout receipts are generated and bind all six package files by SHA256. Provenance inputs are historical Git pins; the scientific replay has no external live input.

Limitations and unrun obligations

The universal result rests on the written proof; finite checks are corroboration. The allowed gates only toggle one anchor from unchanged sources and write each target once from that anchor. This is not unrestricted CNOT optimality. A physical caller, native frames, storage, routing, tape movement, recursive pricing and complete supplier funding remain unknown. No selected assumption, new kappa, supplier admission or complete multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed.

The selected source-binding orientation passed. The selected scientific cold replay, native suite, Lean, Docker and all-size implementation obligations were not run. Every registered package check ran locally.

Open next questions

1. Locate a source-pinned caller needing one complete fixed-weight parity layer at a common aligned cut; then price its actual native frames, references, storage, routing and restoration. No such caller is supplied here.
2. For an actual nonuniform source-price vector and k>=3, determine the exact restored tariff. Start with its bounded weighted covering-walk instance. The unweighted Gray schedule need not be price-optimal.

Neither follow-on is reserved here.

Related open work

No open PR computing this exact fixed-weight borrowed-anchor tariff was identified in the inspected live and cached corpus. The cache covers 371 open PRs and 6,295 Discussion records with complete remote pagination, supplemented by main/merged searches and fresh claims. Sparse local notes and unpublished work remain coverage limits.

• PR6393 schedules multilinear evaluations on distinct dirty borrowed rows. Its operation alphabet and objective differ from this linear single-anchor parity-delivery boundary; it does not compute this fixed-weight tariff.

• PR6423 studies the stable-lane requirement of the reflected Gray address permutation. It prices a different layout contract and does not compute this row-ADD tariff.

• PR6432 proves repeated-visit matching-balance conditions for closed singleton walks and an attaining hypercube-book family. It studies a different required support and matching-capacity quantity, rather than this complete fixed-weight mask-layer tariff; no common numerical instance is reported.

The completion refresh inspected all 12 open account contributions created since the advance CLAIM, including this PR. The eleven others are PR6421 through PR6426 and PR6428 through PR6432. PR6423 is already distinguished above; PR6432 is the newly adjacent scope. No additional computations were run to check another contribution or compare unrelated numbers.

Ali Romman's merged PR5372 has an all-nonempty-mask optimum under a paid clean-row contract. PR4899 supplies the earlier weighted pair case. No extra computations were run to audit these results or compare unrelated numbers.

CI notes

Both hosted checks passed at exact head 19a139b0f6de2ecf2cd3d931d7193f85ec4023ab in Registered research packages run 41537020626: plan and registered package replay.

The completed check list was read once. Separate run metadata confirms completed status, success and the matching head. Hosted replay executes both registered warning-strict normal and optimized checks. All recorded local scientific, package, committed-byte/source-history/mode and Git-free export checks also passed. CI certifies execution of the finite checks; independent human and formal proof-kernel acceptance remain unclaimed. No failed or waived check remains.

Merge danger

Door: two-way. Blast radius: isolated research package. Six new package files; shared code, frozen predecessor sources, workflows, selected certificates and the review ledger are unchanged. Execution of finite checks is separate from acceptance of the universal proof.