[petalas]

Summary

Remove exactly 7,982,993 of 19,651,114 scalar field-product terms, plus the same number of field additions, from the complete PR5692 raw-pencil decomposition replay. The replacement retains 11,668,121 terms and preserves every returned matrix product. Claim types: finite arithmetic-work certificate and scoped implementation-preservation proof. Wall-time improvement remains inconclusive; no new kappa is claimed.

- extract columns and sum every scalar product term
+ update rows directly and skip zero left coefficients
+ handle zero-width and single-column shapes explicitly

The scoped proof gives the exact bills mkn and n*nnz(A). The new counter visits every actual input matrix, validates both shapes and sums those bills. This counts field arithmetic, not processor instructions or paid recursive calls. All changes are isolated in research/raw-pencil-row-products/; predecessor files and the selected frontier remain unchanged.

This contribution was produced by Nick Petalas. Credit Ali Romman / #5692 for the decomposition, field source, proof, complete caller and faster-extraction question; the retained #5664/#5637 and #5250/#5672/#5683/#5689 pencil lineage; David Ganz's tensor-atlas lineage; and classical direct matrix multiplication. No literature priority is claimed for skipping zero terms.

Evidence and reproduction

All 2,779,301 actual matrix products agree exactly with the predecessor routine. The unchanged complete caller retains its five exhaustive raw-pencil domains, literal basis-map equations, polynomial divisor oracle, constructed examples, no-invertible-slice cases and all original failure controls. Their scientific claims belong to the predecessor. This contribution checks preservation of the new implementation, rather than independently auditing that mathematics.

There are 2,000 additional product comparisons over F2/F3/F4/F9, with all dimension triples from 0 through 4, input preservation and distinct output rows. Three defective product implementations and an actual changed source file reject. Source bytes are authenticated before execution, and explicit checks remain active under -O.

Both direct normal/optimized checks pass locally in 24.88 and 25.75 seconds. Both registered commands pass, with total launcher time 52.72 seconds. Each command is well below its 240-second bound. Syntax, JSON, function-size and whitespace checks pass. Committed-byte/mode and source-history preflight passes at 7cb0b0ce7bf81f40dad23d3711b6a041818812bb for all fifteen package/input files and the reachable source base. Official export and integrity checking pass. The read-only Git-free optimized replay passes with empty PATH in 25.26 seconds. All sixteen exported files retain their bytes and read-only modes; its generated receipt equals both registered receipts.

From the repository root, CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/raw-pencil-row-products/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/raw-pencil-row-products/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/raw-pencil-row-products
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/raw-pencil-row-products --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/raw-pencil-row-products/benchmark.py

Another CPython 3.12 path can replace the interpreter path. Expected markers are PASS_RAW_PENCIL_DECOMPOSITION, PASS_RAW_PENCIL_ROW_PRODUCTS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Generated verifier receipts bind the package files. The optional benchmark emits all samples and package hashes.

The final five alternating fresh-process pairs have observed medians 22.613577 and 20.655674 seconds, with zero failures and identical complete scientific output bytes. Reference range is 21.130680 to 23.452057 seconds; new range is 20.180521 to 22.918939 seconds. Their overlap and variation prevent a reliable wall-time claim. An earlier unspecialized variant was also inconclusive. All final samples and the exact environment are in the README. Timing includes startup, authentication, imports and the complete caller on CPython 3.12.15 / macOS arm64. No timing threshold is registered in CI.

Canonical main was refreshed before submission and remains 2485805a6d47f36ea83ddde6327d453ac82d44dc, priority revision 8. The selected binding orientation passes with conditional kappa 781451/250000000. Live claims, local reservations, open work, main formulas and merged searches were checked; they are not a global novelty certificate. The advance CLAIM predicted a replay-speed experiment. The progress finding explicitly narrows delivery to the exact arithmetic reduction and retains the inconclusive timing result.

Limitations and unrun obligations

The proof covers valid rectangular matrices over the retained explicit field. The complete source fixture uses dimensions at most 18. The exact term reduction does not establish fewer processor instructions, lower paid recursion, a faster whole replay or a practical multiplication speedup. Other callers and hosts need their own coverage and measurements.

No native supplier, physical caller, rounded implementation, complete recursive invoice or complete multiplication theorem is supplied. All five selected framework assumptions remain. Lean, Docker, other pencil callers and the selected scientific cold replay were not run. No registered check is deferred solely to CI. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Does a quiescent host confirm a wall-time gain? Run the same complete five-pair benchmark and retain every sample and output hash. Current timing variation is the blocker.
• Does the separate raw-pencil SL transporter benefit? Profile one complete phase and name its input/output and coverage boundary before installing the kernel. No such measurement or preservation argument is supplied here.

Optimization stopped at the first exact arithmetic reduction. Neither follow-on is attempted or reserved.

Related open work

• Reduce complete extension-field tensor replay time by 16.56% (no κ) #6303 optimizes finite-field row elimination in the complete PR5637 tensor-recognition caller. It uses a related retained field implementation, but a different arithmetic kernel and scientific caller. The two results are complementary; no timing or work-count transfer is claimed.

• Reduce complete Frobenius recognition caller time by 65.44% (no κ) #6310 reduces associativity contractions in the separate PR5502 Frobenius-recognition caller. Its algebra-law checks and contraction count differ from this matrix-product kernel and complete raw-pencil workload; no same-instance numerical comparison is made.

• Skip 113,164 redundant modular products in stage-pair replay (no κ) #6313 removes modular products in the historical h=6 stage-pair rank kernel. It is related work on exact arithmetic workload, but its matrices, field, caller and product count differ. Neither work count transfers to this replay.

The completion refresh read all sixteen open account PRs created since the CLAIM at 2026-10-30T18:14:40Z, including this PR. The other fifteen use different source callers or mathematical quantities. None computes this raw-pencil product-term bill, and no shared-instance numerical disagreement was identified. No extra computation checked another contribution. Merged #5692 supplies this caller, and merged #5712 studies its separate SL transport consumer.

CI notes

Both hosted checks, plan and replay (research/raw-pencil-row-products), passed at exact head 7cb0b0ce7bf81f40dad23d3711b6a041818812bb in Registered research packages run 41107442355. The completed checks were read once; separate run metadata confirms completed status, success and the matching head SHA. The registered replay runs the normal and optimized verifier commands.

All local direct, registered, committed-byte/source-history and portable-export checks above also passed. Hosted replay confirms execution of the declared finite checks and controls. It does not establish a reliable wall-time improvement, independent human review, formal proof-kernel acceptance or a complete multiplication theorem. The five framework assumptions remain.

Merge danger

Door: two-way. Blast radius: isolated package. Nine new files and an opt-in replay; shared implementations, frozen evidence and the selected result are unchanged.