[petalas]

Summary

A scoped written construction with exact finite corroboration removes per-entry denominators from a fixed-source sliding principal-inverse cache. At width m the persistent inverse coefficient state has m^2 integer cofactors and one signed determinant, removing m^2-1 denominator fields. Exact integer deletion/bordering updates and explicit coefficient-height bounds work independently of the number of preceding slides.

clear global source rows once: Z = diag(q) H
cache:                       d = det Z[J,J], A = adj Z[J,J]
inverse output:              A diag(q_J) / d
slide:                       delete first, then border from the same Z
                             2(m-1)^2 exact integer divisions

This contribution was produced by Nick Petalas. Credit David Ganz / Dugongue's PR3872 for the fixed-global-source ordered-window preparation contract, Jose Manuel / jmbarrancoml's PR3851 for its exact old-H boundary, Chafik Boukhalfa, RaD and the retained Gaussian/FFT lineage. Schur complements, adjugates, Bareiss elimination and determinantal condensation are classical. The contribution is the shared-denominator sliding construction, explicit state/height contract and bounded implementation. Predecessor code is not imported or replayed.

No new kappa or selected-frontier change. The denominator-field count is not a total-bit-space or runtime guarantee. The main result is a scoped arithmetic construction, not a complete multiplication theorem.

Evidence

• Before: PR3872 retains an individually reduced rational inverse matrix. After: every steady-state slide uses integer arithmetic and publishes a new adjugate/determinant pair. Optional output materialization is separately charged and creates ordinary rational entries again.
• An independent literal determinant/cofactor oracle checks 10,935 initial states and 19,683 slides across 729 newly enumerated nonsymmetric integer sources; 480 signed rational-source slides; 64 width-sixteen diagonal slides; and the source-dimension boundary at 32.
• The complete two-by-two initialization census accepts 496 nonsingular matrices and rejects 129 singular matrices. Fifteen invalid-input cases, four quotient/singularity controls and four corrupted-state controls reject. A singular first scalar pivot with a legal retained block is a positive control. All counts are generated by enumeration.
• Warning-strict isolated normal and optimized direct checks pass with identical generated hash-bound receipts. The registered hook passes both commands. Measured local command durations were 1.81 s and 1.93 s, below their 120-second limits.
• Committed-byte/source-history preflight passes for all ten package/input files at 2a0cd8ccfb32ac834bfc4bcedd364a5a8c36ac8a. The official export and declared-input integrity checks pass. Both direct checks also pass from a read-only Git-free export with PATH=/nonexistent, identical receipts and unchanged payload bytes.
• In-memory compilation, function-size inspection and staged whitespace checks pass. External lint/type tools were unavailable. The selected binding-only orientation passes; no predecessor scientific suite or selected cold replay was run.

Reproduce from the repository root with CPython 3.12 and the standard library:

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

Any CPython 3.12 interpreter may replace the local path. Direct output contains PASS_SHARED_DENOMINATOR_WINDOW_CACHE; the registered runner reports PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed. Exact source hashes, the unchanged original card, scope change and environment are recorded in the package.

Limitations and unrun obligations

The formulas hold in arbitrary finite dimensions when the old, retained and incoming principal windows are nonsingular. The implementation limits N to 32, m to 16 and reduced input coefficients to 256 bits. Global row clearing and dense initialization are separately paid. Common cofactors can be larger than individually reduced numerators. The height bound is uniform over slide count, not a physical storage claim.

Production protected factor-page use, source row clearing, initialization, integer bit costs, output conversion, ownership/restoration, precision, stock, tape movement and cleanup remain unimplemented/unpriced. No native supplier, complete invoice, five-assumption discharge, selected scientific replay, Lean or Docker result is claimed. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Bind the integer numerator/common-denominator stream to one actual protected factor-page consumer without materializing every inverse entry. Freeze the matrix, page format and access sequence, then price coefficient conversion, ownership and cleanup.
2. Determine whether this representation uses fewer total bits for one authenticated production source family, including global row clearing, initialization and all output conversions. The field count alone does not decide it.

Related open work

• PR6420 studies bandwidth and support of separated circular Schur complements. This package updates exact inverses under principal-window replacement; it computes no competing bandwidth or Gaussian-norm value. No additional computation checked that PR.

No open PR was found implementing the same fixed-source shared-denominator sliding cache in the targeted live searches. PR3872 and the distinct residual-barrier PR5119 are merged context, not replay deliverables. Main, merged bodies, active claims, account submissions and local reservations were checked again before committing. The broad cached snapshot covers 598 open PRs and 6,677 Discussion records with complete pagination; its local notes are sparse. Arbitrary diffs and unpublished work remain coverage limits.

The completion refresh inspected the fourteen other open account PRs created since the advance CLAIM: #6662, #6661, #6660, #6659, #6658, #6656, #6655, #6654, #6653, #6652, #6651, #6650, #6649, #6648. None studies this fixed-source shared-denominator principal-inverse cache or reports a competing value of its denominator-field count. No extra computations checked their mathematical results.

CI notes

Both hosted checks passed for head 2a0cd8ccfb32ac834bfc4bcedd364a5a8c36ac8a in run 41661442248: plan and registered package replay. The completed checks were read once; run metadata confirms the matching head, completed status and success. Hosted replay certifies execution of the registered finite checks and source bindings, not the universal written proof or a multiplication theorem. Both registered local commands, metadata/source pins, committed bytes/history and export checks also pass. Independent human and formal proof-kernel review remain unclaimed.

The advance CLAIM preceded substantial work. The delivered scope matches its proposed representation improvement and stated limitations.

Merge Danger

Door: two-way. The isolated optional package can be removed without changing default commands.

Blast radius: isolated. Only research/shared-denominator-window-cache/ is added. Frozen sources, shared code, predecessor credits and the selected frontier are unchanged.