[aliromman]

An identity between native monomial circuits can pass small tensor widths and fail later. This PR proves a complete criterion for fixed binary affine maps with quadratic Gaussian phases: on n >= 1 axes, equality at 2n+4, 2n+5, 2n+6 and 2n+7 is equivalent to equality at every positive tensor width. Scalar coefficients stay outside tensor powers, and the endpoint includes every arbitrary dirty input and returned role.

The proof constructs a nonzero basis entry for every failed identity, with its input/output addresses encoded in O(n²) bits. It also proves exact phase-residue kernel stabilization by width 2n+4. For this monomial group, the linear bound replaces the retained general |G/Z|-1 sufficient bound. Four widths are necessary because each central phase residue can hide a different error; the value 2n+4 is not claimed to be the least cutoff.

Start with the contributor guide, replay receipt, source pins, and source/related-work manifest.

This is an enabling proof and reusable exact checker. The complete κ remains 60403/20000000. The selected record and all physical supplier, ownership, compiler, precision/tape and outer recovery obligations remain unchanged. Hadamard operators, arbitrary width-dependent programs, nonlinear leaves, external aliases and rectangular types are outside this theorem.

The package includes three useful boundaries:

• A rank-projector family vanishes at every width below n and is nonzero at n. The supplied nine-axis example passes all widths 1 through 8.
• Equality is coNP-complete even with five arbitrary dirty roles, only reversible operations, and commuting native sign generators. The proof gives an explicit reduction using Barrington's classical width-five construction and a five-step controlled swap. Sparse circuit expansion can be exponential; no fast general solver is claimed.
• The actual event8814 nine-axis recipe satisfies a complete native conjugation identity. Omitting its global sign passes the even-width probes but fails at a generated width-25 witness, with discrepancy -2. This binds the supplied recipe bytes and conventions, not the full dispatcher-to-chart chain or production packet.

The producer compares canonical native coefficients; it does not build the four tensor matrices. The separate checker uses direct Boolean-polynomial substitution for positive claims and literal basis execution for negative claims. It imports no producer or native-algebra code. Exact rational arithmetic, Python and instruction semantics are shared trust.

Validation ran on macOS with CPython 3.12.14 and no third-party dependencies:

python3 -I -W error -B research/native-tensor-cutoff/verify.py
python3 -I -O -W error -B research/native-tensor-cutoff/verify.py
python3 -I -B research/native-tensor-cutoff/audit_mutations.py
python3 -B scripts/research_package.py --package research/native-tensor-cutoff

The first two commands return the same JSON with status PASS_NATIVE_TENSOR_CUTOFF. The mutation audit returns PASS_NATIVE_CUTOFF_MUTATIONS; the registered runner ends with PASS_REGISTERED_RESEARCH_PACKAGE research/native-tensor-cutoff outcome=completed.

The replay covers 33,866 symmetric binary matrices, 2,192 phase characters, 100 larger character cases through 73 axes, 1,152 native composition pairs, 120 complete circuits, 11,760 dirty-input basis cases, 4,444 below-rank projector cases, 80 CNF formulas on 992 assignments, 480 reversible branch pairs, all 512 source addresses, five committed certificates and 37 rejection controls. Four private-copy mathematical mutations fail in both Python modes; their receipt names the semantic rejection checks. Committed-byte checks, official export integrity, and normal/optimized Git-free replays in a fresh read-only export also pass. Finite checks support the written uniform proof; they do not replace it.

No selected cold replay, Lean, Docker, independent human review or proof-kernel acceptance is claimed. The full-history ancestry check was omitted in this shallow checkout; the recorded source objects and retained bytes were checked. Hosted CI is reported separately by GitHub.

Scientific base: current canonical main 7c472563474e60185e6ca101a17622133d047e56, priority revision 7. All six consumed source exports are pinned to that commit; no pending PR head is imported. The final advisory overlap snapshot covers 316 open PRs and 4,648 Discussion records, with complete remote pagination. Related source proofs were read separately. Arbitrary historical diffs and unpublished work are outside that coverage. Coordination is in Discussion #5354.

Credit and distinctions:

• @jmbarrancoml supplied the recipe implementation in PR #2250 and the earlier phase-sector construction in Discussion #2106. Douglas Colkitt's construction and the retained authors keep the credit in the unchanged notice.
• @JoyboyBrian supplied the fixture/caller work in PR #5329 and PR #5327, retaining Cinna Davis (@1337isnot1337) and their source lineage. The exact recipe is consumed without claiming their remaining caller obligations.
• @Clear1nsight supplied the related tensor-separation arguments in PR #4928 and PR #4978. @ywz65 supplied PR #2599 with the affine-basis argument for the distinct binary-stream action. The general separation proof and binary-stream proof are comparisons, not imported verifiers.
• @zxv in PR #5235 and @krzysztofwojt in PR #5262 supply the related independent-operator and named-inverse certificate interfaces. The native finite-order and phase relations here require a different grammar.

Lempel's binary Gram factorization, finite Fourier inversion, Burnside-Brauer tensor generation, Pauli conjugation, SAT completeness and Barrington's permutation programs are classical and cited in the proof. No global literature priority is claimed. New work by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. The proof and guide leave concrete next questions about the least cutoff, nonmonomial Cliffords and further source-bound adapters.

[jwr404]

Independent review (James Reed, with Claude Code). Head 18354f41de4dc7bfda4692e814cf1d91a6d619ef.

Verdict: the four-width theorem's key combinatorial step holds on an independent exhaustive check. For small n the window has slack worth recording.

Independent check of §2 (realizing every phase character)

I wrote my own exact BFS over tuple statistics: length mod 4, coordinate counts mod 4, pair counts mod 2, and the set of points used, with affine span = F₂ⁿ required. For every character (r, u, v) it decides whether a tuple of exact length f realizes it.

n	characters	realizable at some f ∈ [F, F+3] with f ≡ r (mod 4)	max base length for (u, v), any r
1 (F = 6)	16	16	5
2 (F = 8)	128	128	6
3 (F = 10)	2,048	2,048	7

So every character is realized inside the claimed four-width window. Your verifier covers all characters through three axes as well; this is a separate implementation of the same statement. Two remarks:

1. The base-length bound "at most 2n + 4" holds with room to spare: the true minimum for n ≤ 3 is n + 4. Only the window matters for the theorem. Still, this suggests a possible sharper cutoff, e.g. the widths n+4..n+7, for an open-questions line. The late-projector example, first nonzero at width 9 = n, does not contradict that.
2. Minimal realizations including r can exceed F (8 for n = 1, 9 for n = 2). Your proof's order (build a base tuple of length at most F, then pad zeros to the right residue inside the window) is exactly what makes this harmless. That is worth one explicit sentence, since a reader might otherwise expect each character to be realizable at length at most F.

Other checks

• Necessity of four widths. Constant phases contribute i^(c·f), so P = Σ_c α_c [i^c I] can vanish on any three residues of f mod 4 but not the fourth. ✓
• Replay. The registered replay passes locally in 22 s. The standalone check_certificate.py on late-projector-9 prints PASS_NATIVE_TENSOR_COUNTEREXAMPLE.

Not re-derived: the symmetric binary factorization width bound (n + 1 columns), the Fourier separation in §3 beyond the window statement, and the coNP-completeness reduction (Barrington and the controlled swap).