[kumagi]

For the two-axis affine/quadratic Gaussian monomial group, the exact stabilized tensor cutoff is F=6. Fixed register identities are decided by widths 6,7,8,9, improving PR #5369's sufficient two-axis window of 8–11. This is sharp: an explicit fixed phase-projector × affine-sign residual vanishes at every width 1–8 but has a diagonal entry equal to one at width 9.

The proof identifies the exact phase-residue thresholds (8,9,6,7) and characterizes every complete test set: it must reach the appropriate threshold in each residue. The necessity argument uses AGL(2,2)=S4 and its odd pointwise stabilizers; it closes the representation-kernel gap that a character-realization table alone would leave. The upper bounds have an analytic proof, with exact enumeration as corroboration. See research/two-axis-native-cutoff/PROOF.md and RESULT.json.

Validation passes: registered Python 3.12 replay, optimized replay, committed-byte/source-history preflight, official 14-file export and optimized Git-free replay with PATH=/nonexistent. Separate implementations check all 128 character minima against 625 count normal forms, 24 affine maps, 128 phase polynomials, all 15 stabilizers, 1,364 below-threshold ordered tuples, 2,856 phase sums, all four sharp witnesses and semantic controls.

This answers the two-axis portion of #5369's open least-cutoff question. It complements Mihir Srivastava's active one-axis/general-proof audit, which excludes least cutoffs for n>=2. No predecessor files, general-axis theorem, selected nine-axis supplier or κ are changed. Fixed external coefficients and complete dirty input/output endpoints remain essential; physical costs, full supplier admission, n>=3 and nonmonomial Clifford cutoffs remain outside this result. No human or proof-kernel acceptance is claimed.

Credit Ali Romman's native cutoff theorem and open question, Jose Manuel's phase/native conventions, ywz65's affine-basis argument, Chris R's tensor-separation lineage and retained source contributors. Substantial OpenAI Codex assistance, including separate automated proof/checker/review subagents, is disclosed. Fourier inversion and symmetric-group sign representations are classical; no global literature-priority claim.

Canonical scoped result. Base 32890bc5de034ab8fc93ac88ead7279a6a416d04; predecessor head 18354f41de4dc7bfda4692e814cf1d91a6d619ef. Submission overlap covers 379 open PR bodies, 4,797 Discussion records and every base local note; no matching two-axis result found within that coverage.