[petalas]

Summary

Scoped implementation improvement with an exact finite work-count certificate. The complete historical h=6 stage-pair caller omits 113,164 modular coordinate products while retaining every scientific output field and the existing cache workload.

- Normalize and eliminate every coordinate of each modular row.
+ Retain proved-zero prefixes and update only the remaining suffix.

Row normalization/elimination uses 607,860 -> 494,696 products across 372 cache misses, with 6,764 hits. An opcode observer counts the executed reference products independently of the candidate loop-length counters. Input residue conversion is unchanged. The 64 candidate decisions, six rational fallbacks/local improvements, complete scalar identities, source checks and rank sum 8,684 remain the predecessor's results. The contribution is the smaller arithmetic workload, not an audit or new mathematical search.

The new kernel, adapter, proof, controls and harness were produced by Nick Petalas. Credit Aaron Wang / navinor PR824 for the distinct earlier performance changes, Douglas R. Colkitt for the scientific caller and the retained circuit/frame/upstream authors. Zero-prefix Gaussian elimination is classical. No new kappa, selected-frontier change or supplier admission is claimed.

Evidence and reproduction

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/stage-pair-tail-elimination/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/stage-pair-tail-elimination/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/stage-pair-tail-elimination
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/stage-pair-tail-elimination --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/stage-pair-tail-elimination/benchmark.py --rounds 5

Substitute an equivalent local CPython 3.12 path elsewhere. Direct checks print PASS_STAGE_PAIR_TAIL_ELIMINATION; the registered runner prints PASS_REGISTERED_RESEARCH_PACKAGE. The complete reference/candidate payload digest is e1890d03759c22781d549add2b063119d83a0fb6415a873885412c7b917321a6.

Local evidence includes 1,271 independent kernel fixtures, six invalid-input controls, three arithmetic mutants and every live modular rank in the complete caller. The independent small oracle evaluates determinant minors. All thirteen predecessor modules and every declared data input are SHA256/Git-bound and unchanged. Authenticated compilation prevents stale source bytecode; predecessor assertions remain active even in the optimized control.

Five alternating fresh-process pairs on macOS arm64 / CPython 3.12.15 produced identical complete payloads and cache workloads with zero errors. These are complete uninstrumented process times, including startup, imports, authentication, computation and output serialization. The host was shared, filesystem caches were not flushed and no affinity was set.

Pair	Reference seconds	Tail seconds
1	8.424693	8.471826
2	8.505763	8.494294
3	8.495278	8.519474
4	8.549904	8.526730
5	8.645852	8.546977

Reference median 8.506 s, range 8.425 to 8.646 s. Tail median 8.519 s, range 8.472 to 8.547 s. The runtime result is inconclusive; no complete wall-time gain is claimed. A separate locator profile puts modular elimination at only 0.134 of 25.551 instrumented seconds, with rational subtraction dominating. The exact omitted-product claim is independent of timing noise. Receipts are generated and bound to package hashes.

Limitations and unrun obligations

This is an isolated optional adapter on the fixed historical caller. The new kernel is scoped to rectangular rational/integer matrices of at most 64 rows/columns over a prime field with invertible denominators; the caller uses 36-by-36 matrices and prime 101. It preserves the 65,536-entry modular-rank cache policy. It does not change shared code or the predecessor package.

No selected cold replay, native/physical supplier, compiler, stock, precision/tape, outer theorem, Lean or Docker execution was run. The selected frontier and all five framework assumptions remain unchanged. Independent human and formal proof-kernel review are unclaimed. Hosted CI is recorded separately below.

Related open work

• Reduce complete extension-field tensor replay time by 16.56% (no κ) #6303 prepares finite-field row updates for the separate PR5637 extension-field tensor-recognition caller. This package omits zero-prefix products in the historical h=6 stage-pair rank caller over prime 101. The matrices, field and complete caller differ; no timing or arithmetic-count transfer is claimed.
• Remove 7,982,993 raw-pencil product terms (no κ) #6309 removes zero matrix-product terms in the separate PR5692 raw-pencil decomposition caller. This package changes modular elimination rather than matrix multiplication. Its product count is a different workload and is not compared numerically.
• Reduce complete Frobenius recognition caller time by 65.44% (no κ) #6310 reduces associativity contractions in the separate PR5502 Frobenius recognizer. Its algebra-law kernel and contraction count differ from this modular-rank kernel and stage-pair caller; no same-instance number is compared.

Completion refresh inspected all fifteen open account PRs created since the CLAIM at 2026-10-30T18:19:32Z, including this PR. None targets this exact complete caller or computes its modular row-product bill. No shared-instance numerical disagreement was identified, and no extra computations checked another contribution. PR824 is the merged performance predecessor. Sparse notes and unpublished work remain outside complete coverage.

Open next questions

Would the same optimization matter in a different explicitly pinned modular-rank caller? Profile its complete path and establish the input/cache contracts first. No other caller or shared-code integration is certified or reserved here.

CI notes

Both hosted checks passed at head 38a6cbd7e4371899fcc67f6c1b2eb0812a9d59f4 in Registered research packages run 41109219925: plan in 4m31s and replay (research/stage-pair-tail-elimination) in 3m46s. The completed checks were read once. Hosted replay executes both registered normal and optimized commands.

Local direct normal/optimized and both registered commands pass. Each registered command was timed separately at 19.062 s and 19.103 s, below its 240-second limit. Source metadata, exact committed bytes/modes and declared source ancestry pass. The official export contains 42 declared files and passes input coverage for all 34 inputs. Both read-only Git-free empty-PATH modes pass with the same payload digest, preserving all 43 exported files including the manifest. Actual source-byte corruption rejects in both modes before source execution.

Hosted success covers the bounded registered replay and controls, not a runtime improvement or independent mathematical acceptance. The complete wall-time experiment remains inconclusive. Independent human and formal proof-kernel review are unclaimed; all five selected framework assumptions retain their status.

Merge danger

Two-way door. Scope is the new isolated research package. No existing source, evidence archive or selected record is edited.