[petalas]

Summary

Scoped written mathematical proof with exact finite corroboration. Cyclic-band elimination can use monotone, row-dependent factor grids while preserving its variable-gap solution guarantee:

common-grid residual: N K 2^(-q)
row-grid residual:    max_i m_i K 2^(-q_i)
bulk row count:      m_i <= min(N,3W+1)
border row count:    m_i <= N

The first N-W factor rows lose the full-length logarithmic guard. The final W rows retain it. For the first strict 65-row cyclic tridiagonal fixture, bulk precision is 34 rather than the matched uniform allocation's 39 fractional bits. The last row uses 39 and the common solve grid uses 35. Abstract stored factor fraction bits fall from 12,441 to 11,171. Both allocations meet the same 2^-16 target. This count includes symbolic zero positions; it is not a measurement of Python memory or a physical tape encoding.

Only research/row-adaptive-band-precision/ is added. The contribution was produced by Nick Petalas. Ali Romman / PR5284 supplies the variable-gap method, one-round factor entries, diagonal-dominance estimates and cyclic fill. The new step is the mixed-grid identity, row-support residual and reduced bulk allocation. Dugongue / PR4554, Ryanson Jonathan / PR4556 and the retained numerical/analytic lineage keep their credit. No predecessor scientific suite is replayed.

Advance CLAIM. Base canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a was unchanged at the submission refresh. The uniform-transfer-audit card is explicitly changed to this original numerical-interface result.

Evidence

• Before: a matched common-grid allocation prices every factor row using N positions.
• After: exact mixed-grid construction prices bulk rows using at most 3W+1 positions. On the strict fixture, 254 bulk factor positions each save five fractional bits, for 1,270 bits total. The unchanged factor schedule performs 253 update products.
• Exact original-matrix residual 1443/4398046511104; sufficient solution-error bound 1793/268435440 < 2^-16.
• 252 small signed/unsigned, generally nonsymmetric systems agree with an independent pivoted rational solver. All 1,125 symbolic fill patterns pass, covering 1,171,245 row positions. Eight invalid-input/control cases reject. Explicit errors remain active under optimized Python.
• Both registered warning-strict CPython 3.12.15 checks pass at 0dfca71b00de4bc059f7ce3e9f320f741f7dbc01, with identical generated JSON and package-file hashes. Local elapsed times are 0.498 and 0.503 seconds, each under its 60-second limit. These times establish replay-budget compliance, not a speedup.
• The package runner and exact committed-byte/source-history preflight pass. The worktree is clean. Source-bound metadata orientation passed separately; it is not a supplier replay.

From the repository root, the exact local reproduction commands are:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/row-adaptive-band-precision
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/row-adaptive-band-precision --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/row-adaptive-band-precision/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/row-adaptive-band-precision/verify.py

The absolute interpreter path records the local environment; another CPython 3.12 executable can replace it. The verifier uses the standard library and writes its generated receipt to stdout.

Limitations

This is a written all-size theorem with finite corroboration, not formal-kernel or independent human acceptance. The matrix is real dyadic, row diagonally dominant with positive diagonal; factor grids must be nondecreasing. Complex right-hand sides are covered by the written two-real-solve argument, while executable fixtures are real. The finite implementation caps N at 257, W at 8 and q at 512.

No optimal precision, measured runtime gain, Gaussian production-family admission, native instruction tariff, fixed-tape representation, selected assumption discharge or new kappa is claimed. A common machine-word implementation can still pay its maximum width. Coefficient formation, final-grid rounding, physical delivery and the complete recursive invoice remain caller obligations. No selected cold replay, predecessor scientific replay, native/tape replay, Lean or Docker run was performed.

Related open work

• PR6459 improves a finite-section Gaussian inverse spectral-norm bound; this result changes factor grids for supplied cyclic-band matrices and does not compute that norm.

• PR6478 studies per-addition precision in an exact Gaussian Newton graph under residue-lift errors; this result uses row-gap numerical elimination and a different error contract.

• PR6483 studies best fixed-rank approximation of the full Gaussian inverse; this result preserves a complete band-system solve and supplies no rank compression.

• PR6481 reduces a complete orthogonal-witness detector's numerical guard; its norm/error contract is separate from cyclic-band factor residuals.

• PR6482 reduces Gaussian residue-packing precision; it changes a paid arithmetic lane rather than the row-grid elimination schedule.

• PR6490 uses increasing grids to reduce fixed-source Horner oracle precision volume; this result prices stored factor entries for an explicit band matrix, with no oracle-volume comparison.

• PR6496 relaxes a bounded-dirty QQ16 producer tolerance; this result supplies a different row-dominance solve contract and does not evaluate that producer ledger.

The complete 432-open-PR snapshot at 00:45:53 UTC contained no other cyclic-band row-grid contribution. The initial local overlap search was sparse and incomplete; arbitrary diffs and unpublished work remain outside the search. The merged cyclic-band solver PR5284 is the credited predecessor, not an open audit target. The final account-wide open-PR refresh listed 94 open PRs, including 14 created since the CLAIM; their titles and bodies were checked for related work. No other result in that refresh computes this cyclic-band fixture or its factor-bit count.

Open next questions

1. Can an actual Gaussian production caller convert the row grids into a smaller paid precision/storage bill? Freeze its coefficient error, row gap, fill and record layout, then charge the border rows and solve buffers. Source generation, native/tape realization and the complete invoice remain open.
2. Can a different elimination order reduce high-precision rows without increasing fill or breaking the monotone-grid identity? Derive its row supports and an exact rescaling schedule first. No further optimization is included here.

CI notes

Both plan and replay (research/row-adaptive-band-precision) passed in run 41555594401 at head 0dfca71b00de4bc059f7ce3e9f320f741f7dbc01. The completed checks were confirmed with one gh pr checks read. Every registered command also passed locally. Hosted execution certifies the registered replay and planning checks; it does not certify the written all-size proof, independent human review or the full multiplication theorem.

Merge danger

Two-way door. The change is one isolated six-file research package with one authenticated predecessor proof input. Blast radius: research. Shared implementations, frozen predecessors and the selected frontier are unchanged.