[petalas]

Summary

A scoped algebraic proof gives the exact generic weighted tariff for a separately F2-linear operation evaluated only on independently dirty borrowed input rows. A reflected Gray walk attains the bound and restores every borrowed row. The first bilinear improvement is an eight-call word, with a matching lower bound.

S_j: z_j ^= x_j                  complete price c_j > 0
E:   y   ^= B(z_1,...,z_d)       complete price b > 0

c_1 >= ... >= c_d
OPT = 2^d b + 2 c_1 + sum_(j=2)^d 2^(j-1) c_j

bilinear word: E S_1 E S_2 E S_1 E S_2
matched nested word: 10 calls; new word: 8 calls

Every source, borrowed row and target has arbitrary independent entry data. No clean row or extra public workspace is used. The theorem covers words of arbitrary length, including detours and cancellations. It concerns a word correct for every B, and already follows from the fixed scalar product over F2. Special maps can be cheaper.

Only research/multilinear-dirty-cube/ is added. The connector-alphabet-change card is refined to a multilinear evaluation operation beyond the original linear connector alphabet. This contribution was produced by Nick Petalas. Credit Ross Richardson's PR934/943 complete-call connector lineage; finite differences, reflected Gray codes, Boolean-lattice inversion and summation by parts are classical. Global literature priority is unclaimed.

Evidence

Before, the explicitly defined nested commutator uses 3*2^d-2 calls. After, the Gray word uses 2^(d+1). Boolean polynomial coefficients force every cube vertex to be evaluated, and projected closed-walk bounds give the exact unequal-price lower bound. No predecessor result is audited or replayed as the deliverable.

Local CPython 3.12.15 checks pass:

• Formal endpoints and inverse words in dimensions 1 through 8; all 276 read-parity vectors in dimensions 1 through 3.
• All 2,728 scalar dirty states in dimensions 1 through 5, including the matched comparator and inverse.
• All 65,536 two-source/two-borrowed-row tuples over 2-by-2 binary matrices, with ordered noncommuting multiplication and forward/inverse replay. The target starts at the identity; the proof covers every target baseline.
• Independent shortest-walk searches for all 84 price tuples from {1,2,3,4}^d, d=1,2,3, settling 25,948 states.
• Nineteen stated semantic, price and comparator controls.

Normal and optimized direct checks took 0.74s and 0.73s locally; the registered pair took 1.70s. These are check durations, not runtime speedup claims. Both registered commands, syntax/function bounds, exact committed bytes/source history, official export integrity and a read-only Git-free optimized replay pass at 0c9fbfe4ff3a7691be28755bf51c813a84232465. Receipts are generated and bind the six files by SHA256.

From the repository root:

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

Another CPython 3.12 path can replace the absolute interpreter. The direct verifier has no external dependencies, Git requirement or write path.

Limitations and unrun obligations

This is a complete abstract call invoice, not a native implementation or a favorable supplier recurrence. A physical use must supply E, source access, coefficient preparation, private restoration, movement, instruction construction, stock and all-size compilation with their actual prices. No selected occurrence, new kappa or complete multiplication theorem is claimed. The selected frontier is unchanged. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and a selected scientific cold replay were not run; every registered package command ran locally.

Base main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, was refreshed before commitment. Advance CLAIM. The cache covered 331 open PRs and 6,214 Discussion records, supplemented by main/merged searches and fresh claims. Sparse notes and unpublished work remain coverage limits.

Open next questions

1. Identify an actual BIT bilinear response with legal borrowed-row lifetimes and complete S/E prices, then compare the full paid and productive profiles. A native query and selected consumer are missing.
2. Find a named non-generic map with a proved relation among dirty-input response functions that permits fewer cube evaluations. The relation and its physical implementation must be supplied first.

Related open work

No open PR with the same generic multilinear borrowed-input alphabet was found in the inspected corpus. Related boundaries:

• Implement staged544 rank-five residual with reversible dirty scratch #4017 implements a fixed rounded native macro with dirty scratch and a nonlinear substitution obstruction; it does not supply this separately linear query alphabet or tariff.

• Add five-source arity-four connector cost bound #3909 adds linear arity-four source events on one connector, rather than a multilinear evaluation on distinct borrowed input rows.

• Research: seven-source weight-at-most-four connector boundary (L<=18 excluded, 216-call floor) (no new kappa) #4079 studies bounded-weight linear source events and paired readouts on one connector, a different demand and observation contract.

• Certify PR6141 characteristic-five connector exclusion (independent verification; no κ) #6216 certifies a characteristic-five linear connector exclusion, a different alphabet and field.

• Prove exact tariffs for carriers visiting distinct entrance lines (no κ) #6388 proves a weighted tariff for carrier itineraries in the subspace-distance graph. Its required frames, allowed calls and prices differ from the Boolean cube walk forced here by multilinear response coefficients.

The open-work refresh after CI included the nine other open contributions created since the advance CLAIM at 2026-11-02T19:35:29Z: #6385, #6386, #6387, #6388, #6389, #6390, #6391, #6392 and #6394. None studies this generic multilinear borrowed-input alphabet. No numerical comparison with them is claimed, and no additional computation was run to check them.

CI notes

Both hosted checks passed at contribution head 0c9fbfe4ff3a7691be28755bf51c813a84232465 in Registered research packages run 41512547275: plan and registered package replay.

The checks were read once after completion. Separate run metadata confirms completed status, success and the matching head. Hosted replay executes both registered normal and optimized commands. Every registered command also passed locally, along with committed-byte/source-history and export checks. CI certifies execution of the finite checks, not independent acceptance of the written universal proof. The stated native, physical and formal obligations remain open.

Merge danger

Two-way change confined to one new research package. No shared code, frozen predecessor, selected certificate or review ledger changes. Review the scoped proof and its exact operation contract; mathematical acceptance is separate from execution of the finite checks.