[jmbarrancoml]

I implement a complete normalized Gaussian principal-window compression core on fixed tapes. The retained word multiplier and local inverse still needed a concrete global coefficient producer, old-matrix Schur preparation, and both local passes around the coupled boundary solve. This package supplies that producer/consumer pair, C selection, D normalization and the returned complex core.

I preserve one global rounded H. During Schur setup I cancel the exact similarity weights, invert the original truncated Toeplitz interior, and restore the weights in the correction. For online local applications I use Chafik Boukhalfa's PR489 two-convolution factorization. I keep its dense q-model separate from the old-H Schur calculation and test a case where substituting that model changes the answer.

For m source rows I charge online time O((m+m*w²/d+w²)*p*log²(p+2)) and space O((m+m*w²/d+w²)*p). I charge polynomial(t,p) exact setup time and scratch separately. PROOF.md gives the production domain, coefficient enclosures, guard bounds, numerical error and conditional cost argument. The complete outer tensor must still pay the catalogue, halo/router and setup schedule.

Evidence and publication status

I execute interior, wrapped and clipped finite windows against separate staged and dense exact old-H models. I check nonzero complex outputs, the complete word-call invoice, source/head restoration, 15 malformed-packet and ownership controls, positive reuse and production-domain metadata. VERIFY.json groups 12 checks into eight bounded commands with a 1200-second total allowance.

I passed all 12 checks locally, from the official export of 7ab67b4a84d28cd562b9711aa315ca5b47822b37 outside Git, and in CI39176785123. I verified source history and all 42 committed package/input files; the export replay preserved all 43 files including its receipt. I checked the twelve hosted results, the tested merge commit and CI's unchanged-input step. On Darwin arm64, Python 3.12.13, the numerical, ownership and joined-window groups took 64.279, 105.970 and 174.333 seconds.

• Export SHA256: 2d97e093634a4119d9990dcc5d175917df3eabda5e8d1c204f640381e92ab4a2.
• Receipt SHA256: d387e4d0a1a43b9b24104f76f8992603d4265bbe447601bea476d492e92e5340.
• Head tree: 056ec03cc00126f58728dbf57d7e3bccf4f6b985; PR merge f0c06f037b7aed33f5ebc9397aaab90c9bdda48c has the same tree.

I bound coefficient preparation with a certified early Taylor-tail stop and exact reflected-coefficient reuse. Seven cases check zero, tiny-positive and continuing terms against a separate oracle and full-length directed reference. I retain the complete-window and ownership fixtures, eight commands and the 1200-second total allowance. EVIDENCE.md records the earlier replay timeouts; I count only completed checks as passes.

Limits and attribution

The finite grids do not satisfy the asymptotic production domain. The large production case checks metadata without materializing its payload. The finite Gaussian packets call the actual word API's Karatsuba branch; PR3787 separately exercised the FFT branch. I retain the inherited Gaussian and word-recovery premises. The scoped argument still needs independent human or formal-kernel review.

The full expansion/compression tensor, physical CRT conversions and integer recovery remain open. For the outer restoration I distinguish D, the number of resampled axes, from d, the phase-spacing parameter: the combined scale is gamma=sum_i(2u_i+j_i+1), or D(2u+j+1) with common parameters. The pinned bulk caller has D=d−1. A future caller must also establish the existing chirp consumer's 0.6 input norm; one expansion axis only gives the source bound 0.75. I leave the selected frontier unchanged.

I started from canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6, and recorded the expansion decision and claim in Discussion3664. TASK.json records the overlap coverage and next questions.

Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance in mathematics, implementation, source analysis, verification and prose. I retain Chafik Boukhalfa's PR481/486/489 credit, the RaD project (hipotures) reports and CC0 notice, Mason Hale's numerical reference, Dugongue's scalar ABI precedent, and the inherited FFT/Karatsuba/q-calculus, Harvey–van der Hoeven, OpenAI manuscript and Douglas Colkitt attribution. SOURCES.json and the complete source archives retain exact pins and notices.