DONE: four widths decide affine-quadratic native identities

[aliromman]

Status: DONE / author-checked / 2026-10-23 19:50 UTC / priority revision 7. Claim released.

For fixed n-axis binary affine/quadratic Gaussian monomials, a complete scalar register identity holds at all positive tensor widths if and only if it holds at 2n+4 through 2n+7. The proof constructs a nonzero basis entry for every false identity, with input/output addresses encoded in O(n²) bits, and proves phase-residue kernel stabilization by 2n+4. Four widths are necessary for the full group; the least cutoff remains open. Public artifact: PR #5369, exact head 18354f41de4dc7bfda4692e814cf1d91a6d619ef. The selected κ is unchanged. This is a written uniform proof with separate implementation checks, not independent human or proof-kernel acceptance. No further claim is reserved.

Read the proof, contributor guide and exact replay receipt. The new endpoint checker uses actual native inverse, commutation, finite-order and phase relations. It preserves every dirty input and return. It does not construct the four large tensor matrices. Sparse support can still be exponential.

The result includes two limits on finite testing and computation. A rank-projector family is zero at all widths below n and nonzero at n. Complete identity testing is coNP-complete even for five arbitrary dirty roles with reversible instructions and commuting native sign generators. The latter uses Barrington's classical width-five construction and an explicit five-step controlled swap; it requires no zero input or erase.

The actual event8814 recipe gives a positive complete conjugation identity. Removing its global sign passes the even-width probes and fails at the generated width-25 witness. Normal and optimized CPython 3.12.14 replays agree. The registered replay, 37 rejection controls, eight mathematical code-mutation rejections, committed-byte checks, official export integrity, and normal/optimized Git-free replays in a fresh read-only export pass. Exact domains and counts are in the receipt.

The theorem excludes nonmonomial Hadamard operators, arbitrary width-dependent programs, nonlinear leaves, external aliases and rectangular types. It does not supply a physical native invoice, a complete packet/dispatcher binding, compiler admission, precision/tape implementation or a new multiplication exponent. No selected cold replay, Lean, Docker or full-history ancestry check was run. The shallow checkout's recorded source objects and retained bytes were checked.

Next useful task: independently review the phase-character construction, its affine-basis separation, the residue quantifiers and the reversible complexity reduction. Start with proof Sections 2, 3, 5 and 6 and the direct replay. A later mathematical extension could determine the least stabilized width, or supply a quantitative theorem for a specified nonmonomial Clifford group. These are open questions, not a reservation or an admitted supplier.

The PR records exact source pins and predecessor credit: Jose Manuel's native recipe and earlier phase-sector construction; the binary-stream affine-basis proof prepared for ywz65; Chris R's general tensor-separation arguments; JoyboyBrian and Cinna Davis's fixture/caller lineage; and zxv and Krzysztof Wójt's independent/inverse certificate interfaces. Lempel, Fourier inversion, Burnside-Brauer, Pauli conjugation, SAT completeness and Barrington retain their classical credit. New work by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. No global literature priority is claimed.

The final refreshed snapshot covered 316 open PRs and 4,648 Discussion records with complete remote pagination. Canonical main remained 7c472563474e60185e6ca101a17622133d047e56. The physical-stack, CRT-entry and p16 fusion claims remain separate. Arbitrary historical diffs and unpublished work were outside the overlap coverage.

Original prospective claim, retained for history

CLAIM — a constructive finite tensor cutoff for affine-quadratic native identities

Question: Can complete scalar register identities over the exact binary affine/Gaussian-phase monomial family be decided for all positive column multiplicities, with a counterexample using at most 2n+7 columns when they fail? Here a native map is |x> -> i^(c+a.x+2 sum b_ij x_i x_j)|Ax+t>, on n binary axes, with A invertible. Scalar coefficients stay outside tensor powers; all dirty input and return roles remain independent. This is the exact monomial recipe grammar in PR2250, not the full Clifford group with Hadamard gates.

I will prove or refute an explicit short witness construction. Group residual terms by their affine map, choose a nonzero Fourier character of the quadratic phase coefficients, and realize that character on an affinely spanning tuple using a binary symmetric Gram factorization and two-copy phase correction. The tentative cutoff is linear in n, independent of program length or expanded residual support. A complementary rank-projector family should show that tests of any fixed small set of widths are insufficient as n grows. These statements are prospective until checked.

Expected artifact: a dimension-uniform proof, exact canonical native composition/inversion and complete dirty endpoint checker, compact concrete input/output witness construction, an independently transcribed witness checker, source binding to the actual PR2250 recipe convention and event8814 recipe, and exact rejecting controls. Word expansion may be exponential; no polynomial-time circuit identity test, full native compiler, physical invoice, new kappa or selected-frontier change is promised. The target is a general reusable algebraic interface, not one additional circuit identity.

Base: refreshed canonical main 7c472563474e60185e6ca101a17622133d047e56, priority revision7. The public snapshot covered 304 open PRs and 4,621 Discussion records; a live newest-discussion check followed it. Exact cutoff/faithful-tensor/phase-character searches and the relevant native source were inspected. The independent/inverse operator packages have a different grammar; the finite-alphabet separation arguments in PR4928/4978 are retained predecessors. The active physical-stack, CRT entry and p16 block-fusion claims are outside this scope. Arbitrary historical diffs and unpublished work remain coverage limits.

Credit Jose Manuel/jmbarrancoml and Douglas Colkitt for PR2250 and its recipe interface; JoyboyBrian and Cinna Davis for the source-bound event fixture and retained caller; zxv and Krzysztof Wojt for the related complete dirty-circuit certificate line; the source authors of PR4928/4978 for their tensor-separation use. Binary symmetric Gram factorization (Lempel), finite Fourier inversion and Burnside-Brauer tensor generation are classical. No global mathematical priority is claimed. New work by Ali Romman (@aliromman) with substantial OpenAI Codex assistance. Author-checked evidence only; no independent human or proof-kernel review. Claim expires 2026-10-23T21:15:00Z and will be renewed if needed.

[aliromman]

DONE — the complementary full-Clifford result in PR #5376 is extended at 5370f4465388dbbfff3ba45d5d6ae99d447b1766. The extension proof gives eight selected widths for n>=2, a sharp one-axis maximum cutoff eight, 75/76 separation over the full width range, and compact phase-exact native certificates with bitwise witnesses. It includes Hadamard gates. Fixed coefficients across widths remain essential; expanded support can still be exponential.

This thread's monomial four-width criterion has its own stronger cutoff within that smaller class. Its phase-character argument, complexity result and event fixture remain distinct, credited and unchanged. No new kappa or complete supplier admission is claimed. Canonical DONE and evidence. Local verification, export replays and hosted CI pass. The claim is released. Ali Romman with substantial OpenAI Codex assistance.