[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. The new result is the complete cut spectrum of the arithmetic phase U|x> = exp(2πi(a x²+b x)/N)|x> in the declared order x=u+Lv, for every integer a,b and every factorization N=LH.

Q = H/gcd(2a,H), L=qQ+t
squared cut coefficients: H(q+1), t times; Hq, Q-t times; omit zeros
cut rank = min(L,Q)

N=2^n, nu=v2(gcd(2a,2^n))
minimum exact peak MPO bond = 2^floor((n-nu)/2)
operator error epsilon<1 requires peak bond >= ceil((1-epsilon²) R_peak)

For odd a, width 65 has exact peak bond 2^32. The single-cut relative Hilbert-Schmidt approximation error is sharp; the operator-norm bound is necessary only. The universal statement rests on PROOF.md, not finite extrapolation. This is a representation obstruction, with no new kappa or selected-frontier change.

The source question is zxv's PR #5441 at 11fedb46fa199283f78282187c10aea432d8b0bb, whose cut and approximation directions motivate this arithmetic family. Its transfer contract, prior half-swap and binary dot-product examples, and retained native/phase/compiler lineage keep their credit. Crosswhite and Bacon's automata/MPO correspondence, character orthogonality, exact tensor factorization and low-rank approximation are classical. Tyson's QFT spectrum and Chen and Lindsey's compressible reindexed QFT are related background. Global literature priority is unclaimed. No predecessor executable or theorem was audited as this contribution.

Advance CLAIM. Base 3f0152a32eeebac0df11946dbd50cabe5a7f7839, priority revision 8. The uniform-transfer-audit card is explicitly changed to an algebraic side result: no actual selected caller has been identified with this full endpoint. Only research/arithmetic-chirp-cut-spectrum/ is added.

Evidence

• Before: PR Beyond columnwise controls: finite certificates and routing bounds #5441 supplies a general transfer contract and other routing examples, without this arithmetic-phase spectrum.
• After: an explicit orthogonal factorization supplies every nonzero squared coefficient, a matching rank decomposition, the sharp single-cut Hilbert-Schmidt tail and the binary valuation profile.
• Exact cyclotomic replay covers 5,870 phase cases, 567,828 upper-triangular Gram entries, 15,316 truncation values and 492 nonflat cases, for all N=1..36, all ordered factorizations, a=0..N-1 and b=0,-3.
• Encoded arithmetic checks 8,256 binary valuation profiles and 715,520 cuts through width 128. It does not materialize those exponentially large matrices.
• Thirteen explicit projectors satisfy idempotence, rank and attaining residual-trace checks. Three structural controls, 21 signed/shifted controls and four invalid-input rejections pass.
• CPython 3.12.15 on macOS arm64: warning-strict direct normal and optimized runs pass, timed at 4.16 and 4.38 seconds. Their generated six-file hash receipts agree exactly. Both registered commands pass within their 60-second limits.
• Metadata, committed bytes, executable modes and complete source-history preflight pass at bc755bb256fd369a52341950c62a80b29db64cdb. The official six-file export and input-coverage check pass. Both isolated read-only Git-free replays pass with all seven exported files, including the export receipt, unchanged.

From the repository root, reproduce with the exact local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/arithmetic-chirp-cut-spectrum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/arithmetic-chirp-cut-spectrum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/arithmetic-chirp-cut-spectrum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/arithmetic-chirp-cut-spectrum --check-source-history --check-modes

Any CPython 3.12 interpreter can replace that installation-specific path. The checker uses the standard library and no live predecessor input; source notes are pinned as provenance. Receipts are generated by the checker, not hand-written.

Limitations and unrun obligations

The arithmetic integer index, complete phase endpoint and consecutive digit order are hypotheses. This is not a bound on tensor powers of a local Clifford phase, physical memory, ordinary time, native query price or the complete paid/productive invoice. A short circuit, different representation, legal bit reordering or smaller consumer demand may evade it. Complete reversal is covered; arbitrary bit permutations are not.

No actual selected caller or supplier has been bound to this endpoint. Coefficient preparation, physical native implementation, dirty restoration, hosting, precision and the complete consumer bill remain open. The single-cut Hilbert-Schmidt optimum does not prove an optimal operator-norm or simultaneous all-cut approximation. No selected cold replay, physical multiplier, Lean or Docker run was performed. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Bind one literal arithmetic phase request and its allowed data order to the endpoint before applying the obstruction to a caller.
• Determine the minimum peak bond over permitted bit permutations, with their physical costs declared.
• Determine the optimum operator-norm error at fixed cut rank; only a necessary bound is supplied here.

Related open work

No open PR on this arithmetic-chirp cut spectrum was found in the refreshed bounded searches. Main README searches included the general N=LH form and the coefficient gcd, not only the binary example. Sparse local overlap coverage remains incomplete; these searches are not proof of novelty.

• Certify rectangular dirty circuits with k+1-coordinate witnesses (no κ) #6180 gives compact certificates for typed rectangular independent-operator circuits. It does not compute this physical arithmetic-phase spectrum.
• Prove an all-width tensor-frame reduction under fixed boundaries (no new κ) #6171 proves a fixed-boundary tensor-frame rank-moment reduction in a contact-graph model. That model and quantity differ from the operator cut rank here; no comparison or disagreement is asserted.

The completion refresh read all 12 other open account PRs created since the CLAIM. None studies this arithmetic-chirp cut spectrum or reports a conflicting same-instance value. In particular, #6181 concerns joint-bank P104 Gram normalization and #6183 concerns contractive P104 mode deletion; their endpoints and error models differ from this arithmetic phase.

CI notes

Hosted CI run 40881762911 passed at exact head bc755bb256fd369a52341950c62a80b29db64cdb: both plan and replay (research/arithmetic-chirp-cut-spectrum) succeeded. The registered package replay ran in CI. Local normal/optimized, publication and read-only Git-free export checks also pass. This records executed replay coverage, not independent scientific acceptance.

Merge Danger

Door: two-way. One isolated research package is added.

Blast Radius: package. Shared code, pinned predecessor bytes and the selected result are unchanged.