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Prepared for @CubeJerry with substantial OpenAI Codex assistance and independently reconstructed lattice and correction-table checks. No new kappa.
… rounding Force the basis cycle while preserving all 4096 class obligations; add an independent complete tensor reconstruction. No new kappa. Prepared by @CubeJerry with substantial OpenAI Codex assistance.
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Construct a four-axis Gaussian-grid bijection with sharp error sqrt(8) h, exact sum, global unit covariance, integral-image exactness, F²=J and F⁴=I. This resolves the missing square in #6758 and attains #6761's real affine-period lower bound with index 4096.
The certified F(e0)=e3 also attains squared discrepancy 24 on the companion obstruction's fixed tensor witness, while retaining bounded error and the exact square. This is not a worst-case discrepancy optimum or row/column commutation.
Base: 7c49fe3.
#6758: 362b20b.
#6761: b78c893.
Exact file, predecessor and companion pins: SOURCES.json.
Certificate SHA256: 44a950a62abec947b4c094c2c1f290bec2f8d7f41d3a108ab4c58b5bdfc46b56.
Reproduce (Python 3.12, stdlib):
Expected:
PASS_FOUR_AXIS_EXACT_SQUARE_GRID,PASS_INDEPENDENT_SHARP_TENSOR_COMPOSITION,PASS_REGISTERED_RESEARCH_PACKAGE.Executed: all 4096 real classes, 4096 complex fixtures, inverse compositions, eight large-word fixtures through 4091 bits and nine rejection controls, including a locally valid but square-incompatible orbit. Separate dense-Walsh/lattice reviews and descending-section reconstruction check all 256 tensor entries. All eight normal/-O commands, metadata, committed bytes/modes and Git-free checks pass. Fixed-period discovery forced the basis cycle, tried seed 0 and stopped on a complete matching; it regenerates the certificate byte for byte. No exhaustive rounder search.
No new κ. The raw block uses 160 word additions and 32 quarter divisions; classifier, lookup, routing, precision and reversible cleanup still need a complete paid caller implementation. All five selected assumptions remain.
By @CubeJerry with substantial OpenAI Codex assistance and separate mathematical/algorithmic review. Credit Nick Petalas (#6574), #6758/#6761, ToikY (#6755) and Ali Romman (#6509). No human/proof-kernel acceptance.
Proof, certificate, reviews and limits: https://github.com/CubeJerry/integer-mult-bounds/tree/6ea7d0886e62d6c0f53aa90ae8691dbcc61ea04a/research/four-axis-exact-square-grid
Complete remote Git tree matches the tested contribution.