Batch Gaussian boundary right-hand sides to reduce their cost

[chafreaky]

The retained Gaussian phase-cell inverse applies one banded boundary matrix to many independent tensor lines. This groups a power-of-two number of right-hand sides and uses exact Strassen products for blocked triangular solves, Woodbury correction and sparse cell couplings. The conditional boundary factor drops from w^2 to w^(log_2(7)-1).

The proof pays for RHS grouping, single-field padding, table scans, ragged blocks, cyclic endpoints and returns. A residual bound independent of the number of cells retains O(p) precision. The exact inequality 7^16 < 2^45 gives the conservative boundary margin (3+29r)/32-delta, replacing r-delta.

No new kappa is claimed. Per-axis Gaussian work, small-field exposure, epsilon+r<1, normalizations and supplier/assembly contracts remain. This follows #475 and #479 in removing a separate outer cost; changing only this row leaves the current selected minimum unchanged.

New package: research/batched-gaussian-boundary/. Run python -B research/batched-gaussian-boundary/verify.py /absolute/new/output. Checks cover 15 signed/near-singular/degenerate cyclic systems, two actual Gaussian-derived Schur systems, 2,781 inverse entries, 2,952 complete Schur comparisons, exact Strassen operation counts, 54 split/interleave layouts and six non-vacuous mutations. Complete Schur checks use the new rounded boundary/coupling program with an exact dense local inverse reference; the inherited GS rounding argument remains a dependency.

Fresh macOS Python 3.11.15/3.14.8 and Linux Python 3.12.3 wrappers passed with byte-identical complete CHECKS and RESULT artifacts. Twelve source pins are checked against parent Git bytes. Validation records include the initial Linux launch-path and archive-metadata failures; neither executed the mathematical checks. The successful Linux run uses the unchanged source manifest and a metadata-free archive. GitHub CI is separate.

The all-size analytic and fixed-tape transfer is a written conditional argument, not established by the finite checks. No new Lean proof, unconditional multiplication theorem, runtime benchmark or optimality claim. Existing source files are unchanged.

Credits: the retained RaD/hipotures phase-cell and bulk-resampling reports and their cited Gaussian interfaces; Strassen's seven-product identity; the inherited supplier work cited by #475/#479. New batching, accounting and controls by chafreaky with OpenAI Codex assistance. Apache-2.0.

[Mister-Meeseeks]

Thank you @chafreaky. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Batched Gaussian boundary solves using Strassen products.

Review and limits: Pays for grouping, padding, coupling and cyclic endpoints. Rounded Schur checks retain the inherited local inverse and analytic error assumptions; no new kappa.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 5f3db6a11198bab07d8effe8f0c9f5df9c3f2eb7. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.