[apoursh]

Change and mathematical scope

Answers the order-17 existence question in Nick Petalas's #6192 with an explicit four-qubit Gaussian-dyadic Clifford:
U17 = i S_3 CZ_03 CZ_12 CZ_23 H_g^(tensor 4), where H_g=(1+i)[[1,1],[1,-1]]/2.

The actual matrix satisfies Phi17(U17)=0, has order 17, trace −1, symplectic defect rank eight and native B rank four. Applying #6192's Bell-spectator method on four additional qubits gives B=[[I,A+I],[I,I]] of rank eight. This attains the predecessor's cap for every ambient width M>=8. The new contribution is this explicit instance and its exact certificate; the general cap and Bell method retain predecessor credit.

Scoped construction plus written proof and complete finite checks. No new κ or selected-frontier change. Source/consumer matching, native chronology, ownership, dirty restoration, the complete paid/target invoice and supplier admission remain open.

Task and overlap

Canonical main was refreshed first and before submission: a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8. Task joint-query-controller, scope key 58d8c9133832c5149bdc, scope changed to the released algebraic existence subproblem; the original full supplier acceptance gate remains unmet.

Advance claim and canonical investigation. Exact predecessor #6192 head: 91df334ccd2a4bc9959e311f118ac8aaf967ffc2 (still pending). Its two cited documents are retained verbatim with checked SHA256 pins; no predecessor executable or unavailable Git-ancestor dependency is imported.

The local overlap helper covered 30/8623 notes in the sparse checkout; live checks covered the newest 15 handoffs and all-state searches for Clifford/seventeen, Phi17 and order-17, refreshed before submission. No competing explicit instance appeared in those bounded results; no exhaustive novelty claim.

Validation

Publication head: 4e3f55105e436dfa364029f9032c90f6a239fbe0. CPython 3.12.14, macOS arm64, standard library only.

python3.12 -B scripts/research_package.py --package research/order-seventeen-clifford
python3.12 -B scripts/check_research_commit.py --package research/order-seventeen-clifford --check-source-history --check-modes

Both registered checkers pass normally and under optimized Python. Primary coverage: complete 16×16 unitarity, order and polynomial identities; all sixteen nonidentity powers; eight signed Pauli-generator images; the full binary Bell conjugation; seven hypothesis controls. The separately written checker uses unreduced Gaussian integers and imports no primary code. The 256-dimensional polynomial identity is derived by tensor/conjugation in the proof, not claimed as a dense replay.

Committed bytes/modes and declared source history pass. The official export passes integrity checking, and all four direct replays pass from a read-only, Git-free export with empty PATH; all 11 files including its receipt retain their bytes and modes. Corrupting a retained source document is rejected under optimized Python. A separate AI mathematical audit found no blockers.

Not run: selected scientific cold replay, physical/native supplier replay, Lean/formal-kernel proof or independent human review. Prior binding-only orientation was reused after confirming all 1227 bound inputs were unchanged; no fresh scientific replay is implied. Hosted CI is pending at submission.

Credit

Prepared by apoursh with substantial OpenAI Codex assistance for construction, independently written checkers, mathematical audit, proof and packaging. Nick Petalas retains #6192's theorem and Bell method; Ross Richardson retains the earlier cubic/Pauli-character lineage. Classical cyclic Clifford existence is credited to the literature, including Kern–Ranade–Seyfarth, arXiv:0912.4661; no historical priority is claimed. Apache-2.0, with predecessor attribution retained.

Open next questions

README and structured task identify the next gate: find one actual caller requiring this exact response, establish its restoration contract, then price the displayed circuit before exploring a cheaper equivalent implementation. Without that contract, native rank attainment is not a paid-work saving.