[JoyboyBrian]

The frozen resonant channel from #4797 has an explicit rational decoder for its orthogonal input projection. #4980 left open whether a changed, oblique output representative can avoid odd scalar division. This constructs a fixed dyadic decoder B with A_f B_f A_f=A_f for every positive tensor width f.

The 30-term signed-double-coset recipe expands to 1,920 actual free monomials with common denominator 2048 and uniform norm bound 113/32. The idempotent B_f A_f has exactly the kernel of A_f, preserving all accessible information. The independent width-two witness has E(9,0)=1/2 and E(0,9)=0, so the output contract is genuinely oblique. The orthogonal criterion's primes 3,5,7 remain correctly required for its original output contract.

The two commits contain the finite integer algebra certificate and proof, followed by a separate packed-column literal tensor executor and phase/type controls. Every emitted word is checked against the inherited affine/quadratic free alphabet. Source TARGET bytes and all absolute phases are retained.

Validation:

• Registered Python 3.12 normal/optimized algebra and independent tensor checks pass: full coefficientwise C D C=C, all 64 width-two input columns, exact non-Hermitian witness, literal odd widths 1/3 and a 4,096-coordinate width-four dirty Gaussian-integer input.
• Committed-byte/source-history preflight and the official 11-file export integrity pass. Git-free normal/optimized algebra replays pass in4.423/4.197s; independent literal tensor replays pass in7.289/7.464s. Actual altered source bytes, bool-to-int receipt corruption and a changed section numerator all reject with the expected diagnostics, and all exported bytes are restored.
• The selected baseline is unchanged; no Lean, Docker, complete historical suite or full native/outer multiplier was run. Hosted CI is pending publication.

The logical program has 1,920 monomial/accumulation passes, retained observations and two owned full buffers with initialization/erasure scans. Exact arithmetic needs eleven extra fractional bits and at most two extra integer magnitude bits. Physical legal reads, routing/tape time, canonical Gaussian encoders, missing target complement, arbitrary original-owner restoration, complete consumer funding, stock, compiler and outer obligations remain open. The complete obligation ledger and next questions are in research/resonant-oblique-section/PROOF.md and README. There is no new kappa or supplier admission.

Canonical record: #1166 (comment). Scientific base: bee95ce88f9f19802f0613f92838895cd0e14a1f, priority revision 7. The initial complete cached overlap covered 310 open PRs (06:05:59 UTC), 4,096 Discussion records (06:06:48 UTC) and 6,690/6,690 tracked local notes; #4980's explicit oblique question was the relevant match. Arbitrary diffs and unpublished work remain outside coverage.

Credit JoyboyBrian #4797/#4980; Dugongue #4885 for all-width rank/projector and rational recovery; Nick Petalas #4825 for the existing dyadic width-four minor; Yiyang/Lyy-iiis #4710, lee-phys #4688, Jacob Alan Sussman #4628 with eumemic and Swapnil Jain, and the retained source lineage. Prepared for JoyboyBrian with substantial OpenAI Codex assistance, including a separate read-only proof/code review of exact head978b37127606ced3531f23188fa93bbbdb1f8a3e. Automated exact checks do not claim independent human or proof-assistant acceptance. Apache-2.0.

Submission refresh (2026-10-23 06:22–06:24 UTC): canonical main f259dabc7d70e0db8d6d36bfda888f008bbfecc1 was fetched; governing guidance, selected record and every declared scientific input for this package are unchanged. The scientific base remains bee95ce88f9f19802f0613f92838895cd0e14a1f intentionally; all new package paths are disjoint from upstream changes. Refreshed overlap paginated 304 open PRs and 4,114 discussion records; local coverage is all 6,690 notes at the pinned scientific base. Cached coverage is not a novelty certificate.

Supplemental AI peer review is local-only evidence; the committed written proof, source bindings and registered verifiers are the public reproduction artifacts. Peer-only observations do not constitute independent human or scientific acceptance.

[jwr404]

Independent check at head 978b3712. The registered replay passes on Linux with CPython 3.12.14 in 24 s, both the algebra and the independent tensor commands.

Argument. CDC = C in the monomial algebra modulo [−I] = [I] gives ABA = (aC/8)(8Da⁻¹)(aC/8) = aCDC/8 = A. The even-width evaluation g ↦ g⊗…⊗g respects that quotient, and A_f = 0 at odd f because the coefficients of g and (−I)g agree. Then E² = B(ABA) = E and ker E = ker A, as stated. ✓

Beyond the package's widths. It checks literal tensors at widths 1–4. I extended this to width 6. I used the emitted 1,920-term D and set B = 8Da⁻¹. The tensor action is my own: monomial (p,q) sends e_j to i^{q_j} e_{p_j}, applied digitwise in base 8, with exact Gaussian rationals.

• A₆B₆A₆x = A₆x holds exactly on 6 random dense Gaussian-integer columns, each with 40 random support points in the 512-dimensional space.
• At width 5, A₅x = 0.
• The emitted B has Σ|coeff| = 113/32 exactly, which gives the uniform norm bound (7232/2048).

Scope. As stated, this is an oblique section that changes #4980's output contract; it does not contradict the orthogonal-projection primes 3, 5, 7. It supplies no Gaussian jobs, caller or saving. I relied on the package's stabilizer expansion to produce D and did not re-derive the stabilizer sizes 32/16.

This is an author-independent check, not acceptance, and κ is unchanged.

Reviewed by James Reed with substantial Anthropic Claude (Claude Code) assistance.