[JoyboyBrian]

PR #4885 already determines every resonant-channel rank and supplies an explicit polynomial pseudoinverse. This contribution answers a different remaining arithmetic question: are its odd scalar denominators avoidable by a different uniform decoder?

For the frozen coefficient from #4797, a fixed finite monomial-linear program over R_S = Z[i,1/2,{1/p : p in S}] recovers the canonical orthogonal input projection from A_f y at every positive even width if and only if S contains 3, 5 and 7.

Necessity applies to every program in that declared grammar, with arbitrary independent auxiliary inputs and retained dirty state. The proof establishes faithfulness of all-even tensor evaluation after precisely g ~ -g; quarter phases remain distinct. The source projector has 10,528 nonzero even classes and coefficient LCD 3870720 = 2^12·3^3·5·7. One reduced coefficient, 1243/3870720, witnesses all three odd primes. Sufficiency uses the actual source decoder R(A* A/16) A*/16, whose polynomial constant is −2^19·3^7·5·7; it is not inferred merely from the projector's coefficient ring.

The result is restricted to fixed uniform exact linear recovery of the orthogonal projection. It does not exclude oblique, width-dependent, approximate or nonlinear recovery; does not count physical divisions; and supplies no canonical Gaussian encoder, complete target, restoration schedule, paid saving or new κ. Missing complements, routing/storage, precision, compiler funding and supplier costs remain explicit. #4908's bounded-word rational rounding is credited rather than reimplemented.

Canonical claim: #1166 (reply in thread) . The original all-even rank assignment was changed because #4885 already solved it. Source/base and exact predecessor attribution are pinned in TASK/SOURCES metadata. Credit Dugongue #4885/#4908, Nick Petalas #4825, JoyboyBrian #4797 and the retained Yiyang #4710, lee-phys #4688 and Jacob Sussman #4628 lineage.

Two substantive commits separate the proof and primary exact certificate from the independent modular implementation, controls and review clarification. Both commits use JoyboyBrian and the requested GitHub noreply identity.

Validation:

• Four registered Python 3.12 normal/-O commands PASS: exact integer source annihilator and scalar theorem certificate; independent packed-monomial/right-BFS modular certificate.
• The independent implementation imports neither checker, verifies the full modular annihilator, and certifies denominator valuations (3,1,1) at primes (3,5,7) through residues modulo 243,5,7. Six primary and four independent semantic controls pass.
• Metadata/source pins, committed bytes/source ancestry, official ten-file export/integrity and all four isolated Git-free replay commands with PATH=/nonexistent PASS.
• Separate agent proof/code review found no blocker and independently ran optimized checks. This is not independent human or proof-kernel acceptance.

Hosted CI remains pending/unverified at publication. No selected result or frozen predecessor files are changed. Prepared by JoyboyBrian with substantial OpenAI Codex assistance. Apache-2.0.

Final submission checkpoint: official main refreshed at 05:13 UTC remains b91956876e5b4122fd89179f1b57557ff8378f15, with unchanged guidance and affected inputs. Complete fresh overlap coverage is 323 open PRs (05:14:01 UTC), 4,077 discussion records (05:14:50 UTC), and 6,636 current-base local notes; scoped exact queries identify this round's canonical claims and credited distinct predecessors. A coordinating Codex review independently checked the final proof and executable boundary, retaining the stated conditional interface and open obligations. This is not a novelty certificate or human/proof-kernel acceptance.