[aliromman]

The exact production full-cell Gaussian now has a certified dyadic reference and inverse preparation algorithm at any requested accuracy. An explicit row/column inverse guard holds for every integer p>=16. A finite q-product certificate sharpens that guard before choosing bandwidth and precision. This closes the numerical preparation question left by JoyboyBrian's PR #5479. No new complete kappa is claimed.

Proof and costs · construction · exact receipts · verification · source pins · validation. Exact head: b6dfc243c55a124a92a63296221771370f720bce.

The source retains N=floor(1/theta), its half-grid phase, its dilation, the actual alpha ceiling, and 1/(4p^2)<=theta<=1/p^2. The uniform guard is 5p^6; it is effective from p=16 and is not an asymptotic improvement over the predecessor's eventual p^(2+eta) bound. Certified product tails reduce the actual guards from 83,886,080 to 6,550 and from 120,687,845 to 7,049 in the checked cells. The band width is O(sqrt((t+log p)log p)), and every retained coefficient has an exact rational interval.

The algorithm certifies every entry of the integer inverse residual. A Decimal proposal is accepted only after that check; exact rational elimination completes the algorithm if it fails. The proof gives all-parameter error, termination, coefficient-size, arithmetic and working-storage bounds. It charges dense arrays and exact completion. Source-prime selection and alpha setup remain upstream inputs.

python3.12 -B scripts/research_package.py --package research/dyadic-full-cell-gaussian
python3.12 -B scripts/check_research_commit.py --package research/dyadic-full-cell-gaussian

Expected: PASS_DYADIC_FULL_CELL_GAUSSIAN, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. Python 3.12 and the standard library suffice. Normal and optimized checks rebuild the complete 256-column cell and the complete 299-column cell for p=17 and theta=601/599-1. They check requested accuracies 4, 12 and 40 bits. The 40-bit case has certified inverse error below 9.54e-15. Independent rational Taylor enclosures, exact residuals, three small exact inverse cases and 14 rejecting controls pass. Committed-byte, official export, read-only Git-free optimized replay, unchanged exported hashes and actual retained-source corruption rejection pass. The large cases use certified Decimal proposals; a large exact-completion run is not claimed.

Canonical base and final refresh: df0137efd35e11b4445734bf612bdd4c7ebc330c, priority 7. The selected conditional kappa 0.00310045 is unchanged, with the main-transform row still limiting. Only this package is added. The advance claim and final complete metadata search keep this separate from active BIT/P104 late-binding constructions. Source bytes and immediate ancestry pass; full-history preflight is unavailable in the shallow checkout.

Credit JoyboyBrian for the production-scale theorem and the retained source proof, with Jose Manuel, Vlado Vojdanovski, Cinna Davis, Ryanson Jonathan, Nick Petalas and the retained Gaussian/q-LDU lineage. New effective guard, certified preparation, proof and checks: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.

Physical consumer integration, native API admission, page delivery, tape movement, stock reuse, preparation amortization and outer recovery remain open. No selected cold replay, complete historical suite, Lean, Docker, independent human review or formal-kernel acceptance was performed. CI is pending at publication. A numerical inverse table does not discharge those obligations.