[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. A whole-bank Gaussian-rational M-qubit Clifford satisfying Phi_n(U)=0 for odd n>1 necessarily has

phi(n)=2^a, 1<=a<=M
n = squarefree product of Fermat primes
rank_F2(S_U-I)=2a
native B-rank <= min(M,2a).

The degree-four native cap is sharp. An explicit four-qubit Clifford satisfies Phi5(W)=0 and has native B-rank four. This extends Ross Richardson's #4081 cubic response gate; his cubic result and Pauli character argument retain their credit. The cyclotomic extension and sharp degree-four example are the new steps. Classical cyclotomic irreducibility, totient arithmetic and Clifford/Bell gates are credited background. Global literature priority is unclaimed. No new kappa or selected-frontier change follows.

Evidence

Before, #4081 supplies the whole-bank cubic gate. This contribution adds the written all-odd-n theorem and the complete sharp degree-four construction without replaying a predecessor verifier.

• All 2,047 odd n in 3..4095 pass independent coprime-count versus factor-formula totient checks and the exact degree-condition equivalence. Eleven pass the necessary degree test; no existence census is claimed.
• Exact complete Clifford matrices check polynomial identities, unitarity, all Pauli generators, every fixed-Pauli sign, native and symplectic ranks and adjoint traces. The sharp example has trace -4, defect rank four and B-rank four.
• An order-five tensor control has defect rank eight and fails Phi5. Wrong-phase, primitive-power, nonunitary-companion and four invalid-domain controls pass.
• Local CPython 3.12.15 on macOS arm64: both registered warning-strict normal/optimized commands pass in under one second each, with identical generated six-file hash-bound receipts. The registered package runner, AST compilation, staged whitespace, exact committed bytes, executable modes and complete source-history checks pass at 91df334ccd2a4bc9959e311f118ac8aaf967ffc2.
• Official six-file export and declared-input integrity pass. Both isolated read-only Git-free replays produce identical receipts and preserve all seven export files, including its receipt.

From the repository root:

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

Any CPython 3.12 installation may replace that local interpreter path. Expected markers are PASS_ODD_CYCLOTOMIC_RESPONSE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Only the standard library is required. Reproduction and exact provenance are in research/odd-cyclotomic-response/{README.md,PROOF.md,SOURCES.json,VERIFY.json}.

Limitations and unrun obligations

The universal result is a written scoped proof, not a finite extrapolation or formal-kernel result. The necessary degree condition is not a sufficiency classification. Whole-bank primitive cyclotomic annihilation, Cliffordness and Q(i) entries are essential hypotheses. Order alone, non-Gaussian coefficients, several cyclotomic factors and restricted-input quotients are outside the theorem. A sparse reader relation of unbounded degree is outside the fixed-degree consequence.

No actual selected caller, complete paid/productive invoice, physical native implementation, dirty restoration, ownership, hosting, precision, stock, tape execution or supplier admission is supplied. All five selected framework assumptions remain. No selected scientific replay, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed.

Canonical main was refreshed before publication and remains a5bff3b, priority revision 8. The selected source-binding first command passed as metadata orientation only. TASK.json explicitly changes the joint-query-controller card to this algebraic response-interface side result. Refreshed overlap covers 310 open PR bodies and 5,911 Discussion records, main README searches including the general odd-n form, and targeted merged-PR searches. Sparse local-note coverage and unpublished work remain limitations.

Open next questions

• Determine which degree-permitted n admit whole-bank Gaussian-rational Clifford responses. Begin with an explicit primitive order-seventeen construction and its complete Pauli action; no general sufficiency construction is supplied here.
• Bound native rank under a fixed product of several odd cyclotomic factors. Begin with multiplicities and traces for Phi3*Phi5; trace cancellation can defeat the present nonzero-character argument.

A subsequent physical construction still needs its complete producer/consumer contract and paid bill before expansion.

Related open work

No open PR on the same odd-cyclotomic whole-bank Clifford response or its native-rank cap was found in the refreshed searches. Merged #4081 is the direct predecessor.

• Prove sharp 1/4 error for every real rank-seven P104 approximation (no κ) #6191 concerns unrestricted rank-seven approximation of the P104 Gram matrix, a different endpoint and rank quantity; no numerical comparison is asserted.
• Prove sharp boundary-rank tradeoff for cyclic reversal (no κ) #6193 studies ordinary rank(J-I) for reversing a regular cyclic coordinate stage. This PR studies the Pauli symplectic defect and native B rank of a primitive cyclotomic Clifford response; no common numerical quantity is computed.
• Prove coordinate charts suffice for coordinate-terminal cache trees (no κ) #6202 restricts intermediate Lagrangian charts in coordinate-terminal cache trees. Its rank-only edge costs differ from the symplectic defect of a whole-bank operator here; no common numerical result is compared.
• Prove exact rational cube-query costs for cubic reader repair (no κ) #6200 studies rational nonlinear cube-query repair, not a primitive cyclotomic identity of a linear Clifford operator; no numerical comparison is asserted.

The completion refresh read all ten other account PRs still open and created since the advance CLAIM at 2026-10-29T16:13:29Z. None computes the same odd-cyclotomic response quantity or supplies a conflicting same-instance number. No extra computations were run to check another contribution.

CI notes

Both hosted checks passed at exact head 91df334ccd2a4bc9959e311f118ac8aaf967ffc2 in run 40911294734: plan in 5m47s and replay (research/odd-cyclotomic-response) in 1m37s. The completed check list was read once; both entries report pass.

Hosted replay certifies the registered normal and optimized package checks at this head. The listed local exact, publication-byte, source-history and export checks also passed. Independent human and formal proof-kernel review remain unclaimed. No CI failure remains to report.

Advance CLAIM.

Merge Danger

Door: two-way. Six files are added in one isolated research package.

Blast Radius: package.