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Construct and certify a minimal-period odd lattice map. Bind its exact inverse to retained two-/three-call ports and save two fractional bits at four axes. Preserve explicit fourth-power counterexample and complete scoped cost limits. Prepared for @CubeJerry with substantial OpenAI Codex assistance.
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Complete the phase-covariance obligation left by #6574 while retaining its sharp Euclidean error sqrt(8)h, exact complex sum, zero and literal inverse. An explicit 2,048-class real quotient and 1,008 negation-orbit witnesses give F(ix)=iF(x). Prove the period index is minimal for odd affine-period real maps; the native minimum is 2^22, including coupled corrections. These are quotient indices, not memory lower bounds.
Bind the new map to #6509's actual retained two-/three-call ports. At f=4, certified coordinate errors allow g=q+1 instead of g=q+3: two fractional bits and two retained word bits saved, with unchanged external requests and exact dirty-bank return. The direct bill includes transforms, classification, table access, wrappers and scratch work. #6755's concurrent review identifies these obligations but does not endorse this result.
Base: 7c49fe3.
#6574: 48e312f.
#6509: a4173e8.
All source/blob/hash pins are retained.
Reproduce (CPython 3.12, stdlib):
Expected: PASS_PHASE_COVARIANT_GRID, PASS_FOUR_AXIS_RETAINED_PORTS and PASS_REGISTERED_RESEARCH_PACKAGE.
Executed normal/-O: all 2,048 source and target classes; independent rational lattice normal forms; 4,096 native-coefficient fixtures; 11 core controls; 16 port-direction cases, 40 actual cuts, four reduced-grid boundary cases and five caller controls. False radius, target collisions, wrong inverse/axis and overflow reject. Separate proof review, committed bytes/modes, export integrity and Git-free replays pass. Complete remote tree matches.
No new κ. The certificate explicitly fails F^4=I; its exact witness is retained. Exact group order, graph consistency, physical/tape/compiler pricing and supplier admission remain open; all five selected assumptions remain. No human/proof-kernel acceptance, hosted CI or selected cold replay; full-history preflight refused the shallow checkout.
By @CubeJerry with substantial OpenAI Codex assistance and separate construction/review roles. Credit Nick Petalas (#6574), Ali Romman (#6509/#6534/#6536), JoyboyBrian (#5485), David Ganz/Dugongue (#5815), #4743 and retained native lineage. Apache-2.0.
Proof, exact pins, caller bill and open questions:
https://github.com/CubeJerry/integer-mult-bounds/tree/362b20bc934500bc503e850059f0d34daa5dcae3/research/four-axis-phase-covariant-grid