Replace local Gaussian inverse branches with two convolutions

[chafreaky]

The retained phase-cell inverse uses a difference of two convolution pairs. This package replaces its online interior operation with a classical q-Gaussian factorization: two triangular convolutions and three diagonal factors inside the existing similarity. The new proof explicitly bounds the difference from the independently rounded, width-truncated source. The global H, exact Schur setup, sparse couplings and periodic boundary system remain unchanged; the complete error still fits the original scalar interface.

The weighted triangular factors also have norm below 3p, and tensoring the central formula produces five stages without branch expansion. No new κ is claimed. A fast joint weighted evaluator, its routing/padding and tensor boundary composition remain unproved. The existing dimension-dependent cost and small-field exposure still bind.

An exact guard also rejects prematurely rounding the unweighted tensor kernel to the scalar P-grid before reweighting. It checks all 4,096 entries of a small three-axis operator and a uniform admissible parameter family where a significant weighted entry disappears. The latter uses symbolic exponent inequalities, not enormous tensor executions. This does not rule out implicit kernels, adaptive precision or other FFT implementations.

Validation: fresh full wrappers on macOS Python 3.11/3.14 and Linux aarch64 Python 3.12 passed with byte-identical complete CHECKS and RESULT files. Checks cover 1,950 LDL identities, 1,950 inverse entries, 2,084 rounded local entries with actual signed integer packing and up to 115 reciprocal guard bits, 3,024 complete original-Schur output entries using the new local operation plus the retained rounded boundary/coupling maps, 756 tensor identity entries, 1,856 weighted-factor entries and five rejected mutants. These finite checks support the conditional written proof; they are not a complete multiplication theorem. GitHub CI is separate.

Reproduce: python3 -B research/branchfree-gaussian-inverse/verify.py /absolute/new/output (Python 3.11+, standard library only). Package manifests, ten Git-checked source pins and fresh receipts are included.

Attribution: the q-calculus identity is classical (Kuznetsov, https://arxiv.org/pdf/1507.02959, with the specialization derived in PROOF.md). The rounded phase-cell/Schur construction comes from the retained RaD/hipotures work; batched boundary and coupling operations come from #481. This adds a separately source-bound package on #486's head without changing inherited sources.

[Mister-Meeseeks]

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

Contribution: Branch-free local Gaussian inverse factorization.

Review and limits: Two convolutions and diagonal weights replace local branches. This does not provide the missing joint weighted evaluator, routing or tensor-boundary composition.

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: 8922e3b3f0fd7772b0fb42028eb2532984c3ab42. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.