[petalas]

Summary

Scoped written proof with exact finite corroboration. For a scalar identity helper word, fix ordered replacement cuts, transport their normalized rows and columns to the input boundary, and write CD=I+L+U, with strictly lower L and strictly upper U. Assume C is a full helper basis. The complete chronological map is

(I+L)t = Ch + Lz
Cf     = (I+U)z - Ut

new-input leakage into early outputs: rank(L)
old-input leakage into final helpers: rank(U)

Without communication between output banks, the prescribed old/new input partition is independent exactly when CD=I. When L=0, final cleanup costs exactly rank(U) cross-bank rank units, even with arbitrary finite restored clean or dirty workspace and free invertible operations within either bank. A sharp family has k individually normalized cuts but needs k-1 repair units for every k>=2.

This chronological helper-replacement application was produced by Nick Petalas. Credit Ali Romman's #5599 for the independent-input replacement contract and retained P104 lineage, and #5615 for single-cut dyadic pairing context. Classical elimination, rank factorization and rank cuts are methods, not novelty claims. Predecessors are cited premises, not replay or audit deliverables.

Only research/chronological-helper-repair/ is added. Advance CLAIM. Canonical base 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. No new kappa or selected-frontier change.

Evidence

Before repair, the two-helper fixture returns t=(h1,h2) and f=(z1+z2-h2,z2), despite both cut pairings being one. One cross-bank addition and a within-bank normalization return f=(z1,z2). The proof determines the exact general repair rank.

Local CPython 3.12.15, macOS arm64 checks pass:

• Direct complete coefficient replay for all 10,752 binary basis/pairing pairs, 729 ternary pairing matrices, 25 quinary pairing matrices and 72 rational fixtures through dimension six. All leakage ranks, reversible endpoints and applicable attaining repairs pass.
• Twenty-five sharp-family fixtures, all 64 candidate binary outer-product repairs of a rank-two defect, eight invalid-domain rejections and four semantic controls pass.
• Both registered normal and optimized commands, package metadata, in-memory syntax checks, committed-byte/source-history/mode preflight, official committed export and input integrity pass.
• A read-only optimized exported replay with Git absent from PATH passes and preserves all seven exported files and their modes. Scientific receipts are generated and bind all six package file hashes.

The timed direct commands took 2.79 s and 2.66 s. These establish the local timeout budget, not a performance improvement. Each registered command has a 120 s limit.

Exact reproduction from the repository root:

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

Another standard-library CPython 3.12 executable may replace the absolute interpreter path. Expected markers are PASS_CHRONOLOGICAL_HELPER_REPAIR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The all-field result rests on the written proof; finite checks corroborate it. The theorem fixes the cut directions and a full helper basis in an identity block. It does not exclude changing that word, those directions or the input partition. The tariff grants unrestricted scalar coefficients and free within-bank bases; it is not a native Gaussian rank tariff or a total scalar-addition optimum.

No actual P104 cut family, physical early-output reader, native supplier replay, complete signed invoice, precision/stock/compiler/outer proof, selected cold replay, Lean or Docker run is supplied. Physical chart compatibility and every omitted implementation charge remain obligations. All five framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Extract one physically eligible cut family from a named changed P104 word and compute its transported pairings. Actual chart and endpoint compatibility are the blockers.
2. Price the required early-output reads, local bases and restoration in a complete signed moment. The exact rank tariff alone cannot certify a profitable physical consumer.

Related open work

• #5994 studies single-cut eligibility on a fixed P104 word. This theorem concerns interaction among multiple prescribed normalized cuts; it does not compute that census or contradict its scoped exclusion.

• #5606 constructs a prepared P104 split-helper reuse. This package gives a general scalar composition/repair criterion and computes no competing invoice or kappa.

• #5983 studies free-carrier single-cut eligibility on G20. No actual G20 cut or competing count is computed here.

• #6003 prices a single operator-valued Pauli pairing normalizer. This theorem uses scalar coefficients and fixed chronological replacements, a different model and resource count.

• #6015 excludes finite-menu adaptivity for exact helper endpoints. Here every cut and linear replacement is fixed in advance; no adaptive selector is studied.

• #6016 optimizes conditioning of a two-role dyadic helper-basis completion. This theorem fixes transported cut directions and determines their chronological leakage and communication rank; no condition number is computed here.

• #6017 extracts linear operator-call profiles from zero-preserving polynomial helper programs. This theorem fixes a linear replacement program and makes no nonlinear-call comparison.

• #6025 classifies independent nonlinear role recodings compatible with ADDs. This package has no role recoding and computes no competing native profile.

• #6023 determines a commuting dyadic repair rank for a fixed H6 kernel. That matrix, coefficient constraint and endpoint differ from chronological helper spill; no common numerical repair rank is computed or compared.

• #6026 supplies a stochastic precision certificate for a selected P104 prefix. This theorem is exact scalar algebra for prescribed replacements and supplies no rounded-error or precision bound.

• #6021 accelerates a selected P104 transpose checkpoint. This package replays synthetic cut fixtures and makes no speedup or selected-word checkpoint claim.

The completion refresh examined 15 other open account PRs created since the advance CLAIM. The related additions are listed above. These descriptions use public PR bodies as context. No additional computation was run to audit or compare those contributions.

CI notes

Both hosted checks passed at exact contribution head 439b7c9b1bada7a3cb4c4d41347e236c32615dcb: plan in 5m15s and replay (research/chronological-helper-repair) in 1m35s, in run 40592081509. The completed check list was read once, and the run head matches the PR head. Hosted replay certified the registered normal and optimized finite checks. Local scientific, metadata, committed-byte/source-history/mode and standalone-export checks also passed. CI does not independently certify the all-field written theorem or discharge any physical supplier or selected framework obligation.

Merge Danger

Door: two-way. This adds an isolated research package.

Blast Radius: package. Shared code, frozen inputs and the selected record are unchanged.