[petalas]

Summary

Scoped written proof with exact finite corroboration: for every whole-bank M-qubit unitary Clifford with entries in Q(i),

(Phi3 Phi5)(U) = 0
    => exactly one of Phi3(U), Phi5(U) is zero
    => exact order 3 or 5
    => native B rank <= min(M,4), by #6192's pure-factor theorem

This resolves the explicit Phi3/Phi5 test in Open next question 2 of merged #6192. The new step excludes mixed sectors: their multiplicities would give 2a+4b=2^M, Tr(U)=-a-b and a trace magnitude strictly between consecutive powers of two, contradicting the Clifford trace gate. The pure-factor rank formulas are cited, not re-audited. No new kappa, physical caller, supplier admission or selected-frontier change is claimed.

Produced by Nick Petalas. Credit Nick Petalas #6192 for the explicit question, cyclotomic field argument, primitive-factor theorem and quintic fixture; Ross Richardson #4081 for the preceding cubic Clifford trace gate and Pauli-character method; and classical cyclotomic/character facts. No predecessor executable is imported or run. Independent human and formal proof-kernel review are unclaimed.

The joint-query-controller card is explicitly changed to this algebraic response-interface obstruction. Base and refreshed canonical main: e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8. Exact proof/source pins, scope and next questions are in research/two-cyclotomic-no-mixing/.

Evidence

• Before: Prove odd cyclotomic Clifford rank cap, sharp at degree four (no κ) #6192 treats one primitive cyclotomic factor and leaves the explicit Phi3/Phi5 product test open. After: the written all-width argument proves that the two sectors cannot coexist.
• The checker enumerates all 131,089 nonnegative multiplicity pairs at widths 1 through 18: 35 pure endpoints pass and all 131,054 mixed pairs fail the trace gate. The written proof, rather than the finite census, carries the universal quantifiers.
• Four complete Gaussian-rational matrix fixtures check unitarity, both individual polynomials and their product, exact orders and all Pauli generator images. A mixed unitary passes the product relation but fails Cliffordness; an exact-order-fifteen Clifford fails the product relation. Polynomial convolution, four integer trace tests, one non-qubit arithmetic control and two capacity rejections also pass.
• Both registered warning-strict normal/optimized commands pass. Each direct check took 0.09 seconds locally; the registered pair and metadata check took 0.48 seconds total. Generated normal/optimized hash-bound receipts are identical. Every command is well below its 60-second registered limit.
• Metadata/pins, AST compilation and staged whitespace checks pass. The committed-byte and declared source-history preflight passes for all six files at bf8ab572a32a9f540835373f02b178c346ed25e4. The working tree is clean. Selected source-binding orientation also passed; no selected scientific replay was run.

Local environment: CPython 3.12.15 on macOS arm64; standard library only.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/two-cyclotomic-no-mixing/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/two-cyclotomic-no-mixing/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/two-cyclotomic-no-mixing
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/two-cyclotomic-no-mixing --check-source-history

Other machines may substitute their CPython 3.12 interpreter path. Direct marker: PASS_TWO_CYCLOTOMIC_NO_MIXING; registered marker: PASS_REGISTERED_RESEARCH_PACKAGE; publication marker: PASS_RESEARCH_COMMIT_BYTES. Add --emit to either direct command to print its computed six-file SHA256 receipt; the verifier writes no files.

Limitations

The result assumes the exact polynomial relation on the entire bank with the actual global phase, Gaussian-rational entries and Cliffordness. It does not classify other products, fields, restricted-input or approximate relations. The native-rank consequence uses the cited pure-factor theorem. No exact physical word, paid/productive invoice, dirty restoration, ownership, hosting, precision, stock, tape execution, compiler closure, supplier admission or complete multiplication theorem is provided. The five selected framework assumptions remain. No native replay, selected cold replay, Lean, Docker or formal-kernel proof was run.

Open next questions

• Does Phi3*Phi15 admit genuinely mixed sectors? Its primitive traces have opposite signs. Start with dimension and power-trace multiplicity constraints, then a complete Pauli action; no construction or exclusion is supplied.
• Can three same-sign primitive sectors evade the two-sector bracket? Start with Phi3Phi5Phi17's multiplicity equations before searching for a physical construction. No general product classification is claimed.

Related open work

• Prove doubled centre Gram stabilization by width ten (no κ) #7172 studies tensor-width stabilization of a doubled centre Gram sum, rather than this whole-bank unitary's two-factor annihilator. It computes no common numerical quantity with this result.

Refreshed open-PR searches for cyclotomic products and Phi3/Phi5 found no other PR on this specific object. Recent handoffs, reservations, main README searches, merged searches and this account's 38 PRs from the preceding seven days were checked. The cached helper covered 477 open PRs and 7,121 Discussion records with complete pagination; its sparse local-note coverage was 177 of 10,178 records. Arbitrary diffs and unpublished work are outside that coverage. No additional computations were used to check another contribution's numbers.

The closeout refresh read this account's nine other open PRs created since the CLAIM: #7178, #7179, #7180 and #7182 through #7187. Their titles and relevant body text concern multiplicative shells, A2 tariffs, Gaussian inverse norms, cube-zero lifts, radical delivery, rational headers, block signatures and residue samplers. None studies this Phi3/Phi5 whole-bank annihilator or computes a competing value for the same object. No additional scientific computations were run to check another contribution.

CI notes

CI did not certify this package. At unchanged head bf8ab572a32a9f540835373f02b178c346ed25e4, run 42992679183, plan job 130429099981, failed in "Onboarding helper tests" on exactly one of 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

I read the failed job log directly. At tests/test_research_current_status.py:58, the assertion expects the selected reproduction command in the CI verification document's current section. The logged section still describes #6030 and make frontier6030-verify. Direct inspection of canonical main e54523517d0f03e58352e647c403a5af3e525a57 confirms that certificates/selected-result.json instead names make frontier7000-verify. The selected record, document and test are unchanged between that base and this PR. The failure is in the base repository's files.

The same exact failing test, missing command, 760-test total, one failure and 24 skips were confirmed directly in the failed job logs of all these runs. Each failed step is "Onboarding helper tests":

• Other contributors: 42905094314 and 42903313790.
• Prove sharp multiplicative-shell growth for every rank alphabet (no κ) #7178: 42992088329.
• Prove five A2 targets need five rank-three calls (no κ) #7179: 42992340683.
• Prove the sharp half-shift Gaussian inverse lower envelope (no κ) #7180: 42992579989.
• Classify exact two-versus-three costs for diagonal cube-zero lifts (no κ) #7182: 42992720936.
• Prove exact independent-radical directed-cycle tariff 206 (no κ) #7183: 42992841885.
• Classify rational saturated-header Hurwitz orbits (no κ) #7184: 42993026387.

The package replay job 130431573537 was skipped because plan failed. The local registered normal/optimized, exact matrix/census, metadata/pin and committed-byte/source-history checks above remain the package's automated execution evidence. No repository fix, retrigger commit, push or job rerun was made for this failure. Maintainers will repair main. Mathematical acceptance remains separate from package replay.

Advance CLAIM.

Merge Danger

Door: two-way.

Blast Radius: package. Six new files in one isolated opt-in research package. Shared implementations, frozen predecessors and selected-result records retain their bytes.