[aliromman]

The general sufficient native monomial tensor cutoff improves from 2n+4 to 2n+3 for n>=2. A separate exact three-axis calculation gives the sufficient four-width window 7,8,9,10, improving the retained window 10,11,12,13. For nine axes the general window becomes 21..24. The complete kappa calculation is unchanged.

Proof and scope, exact checks, source pins, registered replay. Base: canonical main 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision 7. Contribution head: 0fac48cff6252349cee0d5350914170d4acf9dfb.

The grammar is the predecessor input-phase affine/quadratic Gaussian monomial group, with fixed external coefficients and complete returned endpoints. An adaptive initial affine basis makes the residual binary Gram form nonalternating and saves one factor column. The three-axis phase-realization residue thresholds are exactly (8,9,10,7). Their representation consequence is sufficient; the least three-axis representation threshold is not claimed. The sharp two-axis cutoff six remains stronger in that dimension.

CPython 3.12.14 warning-strict registered normal and optimized checks pass: 1023 complete nonalternating binary forms through dimension four, 2624 constructed characters through 17 axes, 17408 literal Gaussian-phase checks and seven controls. All 65536 three-axis residue normal forms and an independent enumeration of 43758 multiplicity vectors agree on all 2048 exact full-span character minima. Both modes print PASS_SHORTER_NATIVE_PHASE_WITNESS; the launcher prints PASS_REGISTERED_RESEARCH_PACKAGE.

python3.12 -B scripts/research_package.py --package research/shorter-native-phase-witness

Python 3.12 and the standard library suffice. Committed-byte checks pass. No independent human or proof-kernel acceptance is claimed. This does not cover width-dependent all-width programs, Hadamard operators or physical supplier costs. Selected cold replay, physical emission, complete compiler, precision, stock and outer admission were not run. No predecessor executable is imported. Hosted status is unconfirmed at publication.

Credit Ali Romman (@aliromman) for PR #5369 and the pinned grammar, Fourier reduction and endpoint contract. Credit Hiroki KUMAZAKI (@kumagi) for PR #5522 and its pinned two-axis proof, including the representation-kernel distinction. PR #5521 audits one axis and the general proof; PR #5526 treats the full Clifford group. The source proofs retain the native implementation and finite-alphabet lineage. Binary factorization, Fourier inversion and affine bases are classical. The canonical claim keeps these scopes separate.

Next: determine the common group-algebra kernel of the nontrivial three-axis pointwise stabilizers at earlier widths. This is necessary before calling the sufficient cutoff sharp.

Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.