[JoyboyBrian]

The physical Gaussian producer's scalar enclosure left denominator positivity and actual original-coordinate matrix composition open. This package proves those implications for every admitted ProductionPlan/Axis layout tuple, conditional on the scalar theorem in #6085 at 9fc80f1db55d4939c7b53c0799e6840cce35fbf8.

The guards give a_hat_j > 2^(-8p-1) and the unchanged Axis.matrix_entry consumer satisfies ||H-N_w||_infinity < p*2^(16p+4)*eta, stronger than the requested p*2^(16p+7)*eta. The proof handles exact diagonal, zero holes, same-cell ratios, cross-cell coefficients and physical-period cuts. N_w is explicitly the ideal truncated matrix using the source's displacement convention.

Validation: Python 3.12 registered normal/optimized checks pass; 64 actual encoded consumer entries under no_decode, 150 exact geometry pairs, 3,645 denominator-sensitive scalar cases, five failure controls and the original production metadata example with seven domain rejections. Source-byte mutation is rejected. Committed-byte/source-history/mode checks and official 20-file export integrity pass; direct replay in a read-only Git-free export passes in 2.4 seconds. The supplied rational consumer fields are not a complete Axis construction or production coefficient page.

All scientific input bytes are current-main 721c03822c421b7f7fcbb2598993b947c4d88419; #6085 is a cited mathematical dependency, not imported pending bytes or a repeated scalar audit. No existing/frozen source changes.

Ideal infinite tails, global ideal rowgap, uniform Axis.validate_rows admission, inverse/LU, full production pages, online accuracy, tensor/CRT and physical total cost remain open. No new kappa or selected-frontier change. The universal claim rests on the written source-relative proof; finite checks are corroboration by the same AI-assisted contributor, not independent human or formal-kernel acceptance.

Claim: #3664 (comment)

Credit: Jose Manuel/jmbarrancoml #3851 for the physical producer and original precision/phase-cell argument; JoyboyBrian #6085 for the scalar theorem; James Reed's gaussian-gauge-column for its distinct conditional column argument; mihir-s-05 #4449 for abstract cross-cell rounding; RaD/hipotures, Chafik Boukhalfa and the retained Gaussian/physical-word lineage. Prepared by JoyboyBrian with substantial OpenAI Codex assistance, Apache-2.0.