[petalas]

Summary

Scoped semantics/resource proof with exact finite corroboration. Skip already settled identity columns in PR3851's encoded positive-pivot Gaussian inverse:

 at pivot k, normalize and eliminate over columns
-0 <= j < 2n
+k <= j < 2n

The exact saving is sum_k k(1+m_k) matrix reads, where m_k counts nonzero off-diagonal factors. The sharp all-dimension bounds are n(n-1)/2 and n^2(n-1)/2. Numerical acceptance, exact output records, arithmetic and matrix writes agree with the predecessor. Additional scratch is explicit.

For the complete literal 4I+J physical fixture, matrix reads fall from 236 to 212 and matrix writes remain 80. Including input construction, output consumption and cleanup, pooled tape reads fall from 947301 to 922401, writes from 187399 to 182739, and unit head moves from 869032 to 846336. Peak pool leases rise from 34 to 36. These are finite pooled-record counters, not a complete instruction or recursive supplier invoice. Nine alternating cold-process pairs give identical complete outputs and zero errors, but the wall-time comparison is inconclusive. No speedup or new kappa is claimed.

This contribution was produced by Nick Petalas. Credit Jose Manuel / jmbarrancoml's PR3851 for the exact inverse, physical Gaussian setup and retained rational/tape/word-product lineage. Gauss-Jordan elimination and its identity-column invariant are classical. Predecessor scientific results are cited premises, not audited deliverables. Frozen sources and selected frontier are unchanged.

Evidence

• Before: settled leading columns were repeatedly read and then skipped as zero.
• After: the new suffix iterator avoids exactly those reads. A written induction proves bit-identical augmented states and both sharp bounds.
• Ten paired encoded fixtures, 81 exact rational matrices and eight rejecting executions pass. Every physical output is checked against a separate rational trace and both inverse products. Source preservation, scratch cleanup, the added peak leases and full pooled meters are checked under sealed constructors and a host-decoding ban.

Reproduce from the repository root with CPython 3.12 and the standard library. Another machine may replace this interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gaussian-inverse-settled-columns/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/gaussian-inverse-settled-columns/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gaussian-inverse-settled-columns
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gaussian-inverse-settled-columns --check-source-history --check-modes
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gaussian-inverse-settled-columns/benchmark.py --pairs 9

Warning-strict normal and optimized direct and registered checks pass locally. The direct commands take 1.652 and 1.708 seconds, below their 120-second caps. Committed bytes, modes and declared source ancestry pass for all fifteen package/input files at dc7d0f681a7b326b475863074c825f9824d21a80. Official export integrity and all five declared inputs pass. Both direct modes pass in a read-only Git-free export with empty PATH. Two source-corruption and two semantic-corruption executions reject; all sixteen exported files retain their bytes and modes. The checker prints a generated source-hash-bound receipt and writes no package files. All five live source inputs are declared. Timing is a separate experiment, not a CI threshold.

Limitations

The theorem preserves the positive-pivot numerical domain with adequate scratch. A pool sized only for the original peak need not suffice. Cold source authentication/extraction dominates the tiny benchmark; no full Gaussian preparation or production-size caller was measured. The pooled meters do not count every machine instruction or prove a universal decrease in all tape costs.

Native all-column validity, Gaussian numerical admission, production pages, complete caller, compiler, precision/tape/stock and all five selected framework assumptions remain open. No selected scientific cold replay, Lean, Docker, independent human review or formal proof-kernel review is claimed. The universal result is the written proof, with finite corroboration.

Open next questions

• Implement one fully charged exact streamed row update without increasing peak scratch. Start with its borrowed records, scans and cleanup; no such primitive is supplied.
• Bind this inverse to one actual preparation page and compare complete bytes, ownership and pooled meters. The complete physical caller and production preparation remain blockers.

Related open work

No open PR found in the bounded live searches studies this encoded inverse's settled-column traversal or saved-read quantity. Adjacent work is distinguished without additional computations:

• Conditional κ = 0.001679745508802299: gauge-free (fractional-shift) interiors, structured boundary solves and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1758 proposes structured Gaussian boundary solvers in a historical T74 assembly. This component preserves PR3851's dense positive-pivot inverse and makes no asymptotic structured-solve or assembly claim.

Completion refresh on October 30: this account has no open PR created since the advance CLAIM at 2026-10-29T22:16:43Z. The previously listed #6228 has merged; its rational Stein-generator normalization computes no common saved-read quantity. No common-instance numerical disagreement was identified, and no additional mathematical computations were run for comparison.

The advance CLAIM records the target. Canonical main was refreshed at a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8. Live PR/Discussion, main README, merged-term and local reservation checks found no matching target within their bounded coverage. The cached remote snapshot covers 338 open PRs and 5959 Discussion records; sparse local coverage is incomplete. Unpublished work and arbitrary historical diffs are outside the novelty check.

CI notes

Both hosted checks passed at exact contribution head dc7d0f681a7b326b475863074c825f9824d21a80 in run 40966107580: plan in 4m15s and registered package replay in 1m27s. The completed checks were read once; run metadata confirms this head and a completed successful run. The PR has since been merged.

Hosted replay ran the registered warning-strict normal and optimized conformance commands. All local scientific, publication/source-history and read-only Git-free export checks also passed; no local registered obligation was deferred to CI. CI certifies the bounded executable checks, not independent human or formal proof-kernel acceptance of the written all-dimension proof. The cold-runtime comparison remains inconclusive.

Merge Danger

Door: two-way. Blast radius: isolated. This adds one opt-in package and changes no shared code, frozen predecessor or selected-frontier record.