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This enabling theorem proves the sharp four-axis native monomial test window 9,10,11,12, improving the previous sufficient window 11,12,13,14. The exact residue thresholds are (12,9,10,11). Four tests are necessary, and no complete test set can have maximum width below 12. Complete kappa is unchanged; this reduces and settles an algebraic verification bound for fixed complete endpoints.
Proof, verifier, receipt, integer coset certificate, phase witnesses, source pins. Base: canonical main 6cf7a68, priority 7. The consumed proofs are Ali Romman's PR #5529 at 0fac48cff6252349cee0d5350914170d4acf9dfb, and Joseph Anglada's PR #5542 at dc2b1ff607c55e70786bd567b342f096fc0024f1.
The new integer GL(4,2) certificate vanishes on every pointwise-hyperplane stabilizer coset. Summing translations extends it to the affine group. A zero-phase code obstruction then gives four nonzero residuals invisible below their thresholds; a full-span tuple exposes diagonal entry 21 at each threshold. Translations are essential: the GL-only expression gives a false lower witness on a width-eight affine hyperplane.
Normal and optimized registered checks pass. They check all 65536 phase witnesses, two complete inventories of 20160 linear maps, all 37800 hyperplane cosets, all 730626 low-width multisets and five rejection controls. A separate C++17 enumeration of 30421755 multisets regenerates the exact decoded witness vector. Committed-byte, export-integrity and Git-free optimized replay checks pass.
Python 3.12 and the standard library suffice for verification; C++17 is needed only to regenerate witnesses. Expected markers:
PASS_FOUR_AXIS_SHARP_CUTOFFandPASS_REGISTERED_RESEARCH_PACKAGE.The grammar is fixed input-phase affine/quadratic Gaussian monomials, with fixed external coefficients and every independent dirty input and required return retained. Hadamards, width-dependent programs and undeclared aliases are excluded. This gives no efficient group-algebra expansion or physical supplier. No source caller, compiler, stock, precision, tape, outer admission, selected cold replay, full source-history ancestry or independent human/formal review was run. The checkout is shallow; hosted CI is separate.
Credit the linked three-axis theorem and higher-axis handoff to Joseph Anglada / fluorine, the grammar/Fourier method to Ali Romman's PR #5369, the representation-kernel distinction to Hiroki KUMAZAKI's PR #5522, and the earlier audit to Mihir Srivastava's PR #5521. Retain Jose Manuel, ywz65, Chris R and their native/phase/compiler lineage. Binary codes, Fourier projectors and group cosets are classical. Coordination. Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.