[jmbarrancoml]

I connect the actual rational FFT and complete conditioned Toeplitz program from PR #4205 to the retained fixed-Hadamard compiler. I preserve the original six-H words and Beneš stage counts, complete mixed singleton/two-wire layers, and execute the stored inverse with its scalar reciprocals and reversed gate order.

Under the retained leading-section and phase contracts, I give one coefficient/data grid with error below 2^-t against the same root or its inverse and signed widths O(F0+t+log N+log lambda+log Q+log R). The new argument includes the final matrix norm in suffix amplification, adds the upstream coefficient defect, and charges rounding after each emitted stage. A scaled zero-shear program gives a suffix more than six times the invalid unit-endpoint allowance. DENSE-PROOF.md derives the corrected bound and states its dependencies.

I preserve completed FFT head 6f17e6208a78cfade27f5c8f03383256f586ab24 and its unchanged scientific receipt. The full extension retains unchanged source modules and proofs from PR #4205 head 78c72e563173f3822fcb4efed9ceeccf05630258. I bind root admission, both actual plans, request bounds and prepared pages; controls reject changes after preparation.

Verification uses four registered normal/optimized commands with an 840-second total allowance:

python3.12 -B scripts/research_package.py --package research/rational-fft-fixed-h

The new receipt covers 11 roots in both directions, 146,262 native stages and 992,742 complex roundings. I compare complete outputs with stored logical execution and exact root/inverse references, including reversed wires and 512-bit incoming data. Four generic cases cover mixed, idle, zero/tiny and empty programs; 24 shear words cover 2,184 segments and inverses; 25 controls check admission, precision and source/request/page integrity. The prior FFT receipt retains 13 cases, 2,525 stages and 23 controls. The canonical handoff records exact committed-export and hosted evidence.

All four registered and Git-free commands pass, with 35 export files unchanged. Hosted run 39508290934 passes and matches all four scientific records and both complete receipt digests. I checked the exact test-merge parents and publication tree. The first attempt failed during temporary Git-directory cleanup in an unchanged helper test, before package execution. The final empty retry commit has the same complete tree and export payload as the verified implementation commit.

I execute arithmetic H/2 stages. Physical T74 calls, complete tensor admission, setup/delivery/tape costs and the outer multiplication theorem remain open. Reviewers still need to assess the quantified proofs; finite replay does not establish them. The rounded forward/reverse programs need not be exact inverses. I make no selected-frontier change.

I credit Vlado Vojdanovski for the words, routing, full circuits and preceding stability structure; Chafik Boukhalfa for certified phases; and the retained rational-generator/Gaussian and classical LDU/Schur/perturbation lineage. Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance in mathematics, source analysis, implementation and writing. Internal agents belong to the author process.