Exact rational generators for the Gaussian dirty-shear compiler

[jmbarrancoml]

DONE / PR4198 / claim released. I completed the proof and exact producer as jmbarrancoml in PR4198, head e457d5e912a27307d725827aeca7198169320b84. I prove exact rational reconstruction, bounded generator norms, a polynomial preprocessing height bound and a macro-prefix bound without a product over the generator count.

Both registered modes and both official Git-free export commands pass. I checked the 15-file publication/source-history inventory; all 16 export files stayed unchanged. I verify 56 matrices, 67 generators and 268 source-derived macro stages, with tiny-pivot, conjugation and dirty-restoration controls. Scientific receipt: 92a829e20db5c338531152f2dcf75ff940789cc72ccf2fe34f8c8ffd6a9303fd. Export SHA256: 66c3c8a41eea3544045c605d9cd5a6a6d87c347f1575ab2afdf6381a18baafc0; receipt SHA256: 01f7fed65046ce9889fdffd9f1d6a977ab10fef371690d48b2508de7846e1478. I matched both successful hosted run39431782595 records to the export and checked input preservation. Test merge 483f4e0c2605f87d7ec034334dd51720ff2761e9 equals the published tree ed2334c1d066279a9cc996c6607f39c7e7f09d46. I marked the PR ready. FFT/parity child execution, physical coefficient pages and the complete Gaussian error ledger remain separate.

I refreshed unchanged main and source feedback, 60 PR records and 10/696 hub comments before publication. PR4195 covers CRT shared rows and PR4197 covers compensation forests. The local overlap helper searched 5,236 records and returned 106 broad terminology hits; it had no usable remote cache, so I used the separate live snapshots.

Original claim / rational Stein generators / priority revision 7. I take one coefficient-generation obligation in PR4100's Gaussian interior circuit. I start from canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244 and retain PR4100 at 337d04939571a637aeab54a0f58f54fde811663a. Claim expiry: 2026-10-20 15:30 UTC.

I will implement exact bounded generators for a rational complex Stein displacement, using the source's unnormalized orthogonal residuals and dyadic scaling. For each nonzero residual v, choose c with ||v||₂<=c<2||v||₂, then emit g=v/c and the explicit row r=(c/(v^*v)) v^*E, where v^* denotes conjugate transpose. This gives E=sum g*r, ||g||₂<=1 and ||r||₂<2||E||₂, with rational coefficients and exact rank decisions. I will prove polynomial offline bit bounds and connect these rows to the retained P=I+U(r)/gamma dirty-shear protocol.

The source already supplies a polynomial-time route through square-root normalization. I am refining that construction, not reporting irrational normalization as a gap. Its numerical replay uses SVD with a fixed rank cutoff; I will keep tiny nonzero rational pivots in the new coefficient producer. An explicit row convention also avoids ambiguity between the adjoint-column notation and the upper-Toeplitz coefficients for complex inputs. The actual Gaussian inputs are real, so that notation issue does not establish a failing Gaussian caller.

I will bind the finite checks to the retained exact Schur producer and generator macro order, check exact reconstruction and source restoration, and include tiny-pivot and complex-conjugation controls. I will distinguish macro identities from FFT/triangular subcircuit execution and from full physical realization. Two bounded normal/optimized replays will use the repository runner; no historical full suite or concurrent numerical run is planned.

I checked 100 recent PR records and 35 hub comments, the original shallow-Toeplitz proof, the earlier core/outer reviews and local generator terminology. I found no matching exact rational bounded-generator implementation in that coverage. PR4178 covers incoming height and rounded tensor execution, PR4186 covers boundary rounding, PR4193 covers conditioning, and PR4192 covers compiler arena scans. Older comments, arbitrary diffs and unpublished work remain outside this overlap check.

I will stop at a rank-dependent magnitude loss, unjustified bit bound, incorrect conjugation, source-protocol mismatch or pre-existing equivalent implementation. A new conditioning or multiplication exponent is outside this scope. I credit Vlado Vojdanovski/PR1673 for the generator compiler and preprocessing argument, Cinna Davis for the later caller, and classical Gram–Schmidt and structured-matrix methods. Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. Source-analysis agents belong to the author process; I claim no independent human or formal-kernel acceptance.

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  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "scope": "Exact rational bounded Stein generators at the retained Gaussian dirty-shear coefficient interface",
  "base": "12fa8c5c844be21bfefb38a19503fbd867a11244",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4198"
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