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Odd maps commuting with the signed complement require affine-period index divisible by4096. The exact native square implies this symmetry in the componentwise real architecture. Includes exact integer witnesses, independent replay and adversarial proof review. No new kappa; no generic fourth-power impossibility or attaining construction. Prepared for @CubeJerry with substantial OpenAI Codex assistance and separate AI review. Apache-2.0.
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For A=W16/4, let S be the parity-sign diagonal and J complement the four coordinate bits. Prove that every full-rank affine period, with linear part A, of an odd integer map R commuting with K=SJ has index divisible by 4,096. No radius, sum or bijection premise is needed.
In the componentwise real native architecture F=-D R_C D, the exact square F^2=J implies the required commutation. Thus the 2,048-class phase-only period cannot support this additional identity. Existence at index 4,096 remains open. This does not exclude generic F^4=I.
New mechanism: close the declared period under K, then use two independent image classes against a rank-one quotient restriction. This proves divisibility without assuming a maximal or invariant declared period. It is a written obstruction, not a failed matching search.
Base: 7c49fe3.
#6574 head: 48e312f.
#6561 head: e8a55a6.
Exact source/blob/hash pins are in SOURCES.json. The phase-only construction is discovery context, not a runtime dependency:
https://github.com/CubeJerry/integer-mult-bounds/tree/362b20bc934500bc503e850059f0d34daa5dcae3/research/four-axis-phase-covariant-grid
Reproduce (CPython 3.12, stdlib):
Expected: PASS_INDEX4096_OBSTRUCTION_WITNESSES, PASS_INDEPENDENT_PERIOD_WITNESSES and PASS_REGISTERED_RESEARCH_PACKAGE.
Executed normal/-O: complete matrix identities, all 31 nonzero parity combinations, two exact image witnesses and three controls rejecting a malformed preimage, false image and missing sign phase. A separate Walsh-divisibility implementation constructs the second preimage. Adversarial proof review, committed bytes/modes, export integrity and four Git-free replays pass. Complete remote tree matches.
No new κ. No bound for nonperiodic maps, arbitrary coupled complex architectures, memory or physical tariffs. A larger invariant period and compatible correction cycles remain unconstructed. All five selected assumptions remain. No human/formal acceptance, hosted CI, selected cold replay or full-history preflight.
By @CubeJerry with substantial OpenAI Codex assistance and separate construction, witness and proof-review roles. Credit Nick Petalas (#6574/#6561), Ali Romman, David Ganz/Dugongue and retained native lineage. Apache-2.0.
Proof, reviewed hashes and limits:
https://github.com/CubeJerry/integer-mult-bounds/tree/b78c8938b6d04b7c340c06d4e958947aeecd9270/research/four-axis-exact-square-period-obstruction