Exclude mixed cubic and quintic Clifford response sectors (no κ)

research/two-cyclotomic-no-mixing/PROOF.md

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+# Cubic and quintic sectors cannot mix
+
+This proof was produced by Nick Petalas. Apache-2.0.
+
+## Statement and credit
+
+Let M be a positive integer. Let U be a unitary Clifford on M qubits,
+with every entry in Q(i), and suppose the actual matrix on the entire bank
+satisfies
+
+    (U² + U + I)(U⁴ + U³ + U² + U + I) = 0.
+
+Then exactly one of the two factors is zero. Consequently U has exact order
+three or five, and its binary Pauli symplectic matrix S satisfies
+
+    rank(S-I) = 2  in the cubic case,
+    rank(S-I) = 4  in the quintic case.
+
+Its native B-block rank is at most min(M,4), and at most min(M,2) in
+the cubic case. The quintic case requires M >= 2.
+
+The new implication is the exclusion of mixed sectors under the product
+relation. This settles the explicit Phi3/Phi5 test in Open next question 2 of
+[Nick Petalas's #6192](https://github.com/CrocSwap/integer-mult-bounds/pull/6192).
+The pure-factor defect formulas above are cited consequences of that package,
+not new formulas or an audit. Ross Richardson's
+[#4081](https://github.com/CrocSwap/integer-mult-bounds/pull/4081)
+supplies the earlier cubic Clifford trace gate and its Pauli-character method.
+Cyclotomic irreducibility and the character calculation are classical tools;
+no literature-wide priority claim is made for them.
+
+## Spectral multiplicities
+
+For odd n, the fields Q(i,zeta_n) and Q(zeta_(4n)) coincide: primitive roots
+of coprime orders 4 and n generate the same compositum. Classical cyclotomic
+irreducibility gives
+
+    [Q(i,zeta_n):Q(i)] = phi(4n)/2 = phi(n).
+
+See [Romyar Sharifi, Abstract Algebra, Theorem 6.6.11](https://www.math.ucla.edu/~sharifi/notes/algebra-ch06.html)
+for the irreducibility theorem over Q. The displayed degree argument gives
+the extension to Q(i); it is also the field argument retained in #6192.
+In particular Phi3 and Phi5 remain irreducible over Q(i), with degrees
+two and four.
+
+Their product is squarefree. The assumed annihilation therefore decomposes
+the Q(i)[X]-module into copies of the two simple modules. Equivalently, there
+are nonnegative integers a,b, not both zero, such that
+
+    characteristic_polynomial(U) = Phi3^a Phi5^b,
+    2a + 4b = 2^M,
+    Tr(U) = -a-b.
+
+The trace follows because the sum of the primitive roots is -1 for each
+of the primes three and five. In particular the trace cannot vanish.
+This is a spectral statement about the actual whole-bank matrix and its
+actual global phase, rather than a relation modulo phase or on a subspace.
+
+## Clifford trace restriction
+
+Here is the short Pauli argument from Richardson's trace gate. Label
+Hermitian Paulis by v in F2^(2M). Clifford conjugation has the form
+
+    U P_v U* = epsilon(v) P_(Sv),  epsilon(v) in {+1,-1}.
+
+On K=ker(S-I), the sign epsilon is a character. To see this, conjugate
+P_v P_w for v,w in K. Its Pauli multiplication phase is the same before
+and after conjugation, so epsilon(v+w)=epsilon(v)epsilon(w). Thus the trace
+of conjugation on the matrix space is either zero or |K|. The same map is
+represented by U tensor conjugate(U), with trace |Tr(U)|². Hence
+
+    |Tr(U)|² is zero or a power of two.
+
+Since a+b is a positive integer, (a+b)² being a power of two forces a+b
+itself to be a power of two. Unique prime factorization proves this last
+step, including that the exponent of the square is even.
+
+## The new no-mixing step
+
+Suppose a>0 and b>0. From 2a+4b=2^M one obtains
+
+    2^(M-2) < a+b < 2^(M-1).
+
+The left strict inequality is equivalent to 2a>0; the right one is
+equivalent to 2b>0. There is no power of two strictly between these two
+consecutive powers. This contradicts the Clifford trace restriction.
+Thus ab=0. If b=0, the minimal polynomial is Phi3, so Phi3(U)=0;
+if a=0, it is Phi5, so Phi5(U)=0. The two alternatives cannot both hold
+because the polynomials are coprime and the bank has positive dimension.
+This proves the new implication at every M without finite extrapolation.
+
+The rank consequences now follow directly from #6192's pure-factor theorem:
+its exponent a in phi(n)=2^a is one at n=3 and two at n=5. This use of the
+letter a in the cited theorem is separate from the cubic multiplicity above.
+The native B block is a submatrix of S-I, so its rank has the stated cap.
+No mixed-product rank or native implementation theorem is assumed.
+
+## Boundary examples and exact checks
+
+The checker uses the cubic Clifford
+
+    C3 = (-I + iX + iY + iZ)/2
+
+and the four-by-four quintic matrix displayed in #6192's proof, denoted C5.
+The matrices serve as positive fixtures, not as new constructions. It checks
+unitarity, the polynomial relations, exact orders and all Pauli generator
+images using Gaussian rational arithmetic. No predecessor verifier is run.
+
+Two eight-dimensional examples explain the scope of the implication:
+
+* diag(C3,C3,C5) is Gaussian rational and unitary, satisfies the product
+  relation, and satisfies neither individual relation. Its multiplicities
+  are (a,b)=(2,1), trace -3 and squared trace 9. It fails the Pauli-normalizer
+  test, so unitarity and the coefficient field alone do not suffice.
+* C3 tensor C5 is a Gaussian-rational Clifford of exact order fifteen.
+  It fails the product relation. Thus order dividing fifteen alone cannot
+  replace the stated annihilator hypothesis.
+
+An arithmetic control at dimension twelve, a=b=2, passes the trace test
+despite having two sectors. This is only a control for the power-of-two
+dimension argument; no twelve-dimensional Clifford construction is claimed.
+
+The finite census enumerates every nonnegative solution of 2a+4b=2^M for
+1 <= M <= 18. It tests the trace square as an integer and checks that
+exactly the pure endpoints pass. The written proof supplies the unbounded
+quantifiers. The matrix examples do not classify other cyclotomic products.
+
+## Limitations
+
+There is no emitted physical caller, productive-target credit, full paid bill,
+dirty-helper contract, precision/tape realization, supplier admission or new
+kappa here. The selected result and its five inherited assumptions are
+unchanged. Several factors, other degree pairs, restricted-input relations,
+different coefficient fields and approximate identities are not covered.
+Independent human and formal proof-kernel review are unclaimed.

research/two-cyclotomic-no-mixing/README.md

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+# Cubic and quintic response sectors cannot mix
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+Claim type: scoped written proof with exact finite corroboration.
+Independent human and formal proof-kernel review are unclaimed.
+
+Let U be a unitary Clifford on M qubits with entries in Q(i). If the actual
+whole-bank matrix satisfies `(Phi3 Phi5)(U)=0`, then exactly one of Phi3(U)
+and Phi5(U) is zero. The operator has exact order three or five. By the cited
+pure-factor theorem of [#6192](https://github.com/CrocSwap/integer-mult-bounds/pull/6192),
+its Pauli symplectic defect has rank two or four, respectively, and its
+native B rank is at most min(M,4). This product relation cannot create an
+unbounded-native-rank response family. No new kappa is claimed.
+
+The new step answers the explicit Phi3/Phi5 test in #6192's Open next
+question 2. It does not recheck the primitive-factor theorem. The
+[proof](PROOF.md) derives `2a+4b=2^M` and `Tr(U)=-a-b`. A nonzero integer
+Clifford trace has magnitude a power of two. If both sectors occur,
+`2^(M-2)<a+b<2^(M-1)`, a contradiction.
+
+Ross Richardson's [#4081](https://github.com/CrocSwap/integer-mult-bounds/pull/4081)
+supplies the earlier Clifford trace gate; #6192 supplies the field and
+pure-factor background. Classical cyclotomic and Pauli-character facts retain
+their classical status. No literature-wide priority claim is made.
+
+## Reproduction and evidence
+
+From the repository root, using CPython 3.12 and its standard library:
+
+```sh
+/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
+```
+
+Any CPython 3.12 installation may replace the local interpreter path.
+The direct marker is `PASS_TWO_CYCLOTOMIC_NO_MIXING`; the registered runner
+also prints `PASS_REGISTERED_RESEARCH_PACKAGE`. Both registered commands
+cover the same complete finite domain, with checks active under `-O`.
+The verifier writes no files. `--emit` prints a computed receipt binding
+all six package files by SHA256 and recording the interpreter version;
+redirect it outside the checkout.
+
+All 131,089 nonnegative spectral multiplicity pairs at widths 1 through 18
+are enumerated. Exactly 35 pure endpoints pass the trace gate; all 131,054
+mixed pairs fail. The written proof supplies the unbounded quantifiers.
+
+Four complete Gaussian-rational matrix fixtures check unitarity, the two
+polynomials and their product, exact order and every Pauli generator image.
+The cubic and quintic matrices are positive fixtures. An eight-by-eight
+unitary with both sectors satisfies the product but fails Cliffordness.
+A Clifford of exact order fifteen fails the product relation. These controls
+prevent replacing the hypotheses by unitarity or finite order alone.
+Polynomial convolution, four integer trace tests, a non-qubit arithmetic
+control and two capacity rejections supply additional checks. The receipt
+records every matrix's outcomes. The quintic fixture is credited to #6192;
+no predecessor executable is imported or run.
+
+## Scope and provenance
+
+The `joint-query-controller` card is explicitly changed to this algebraic
+response-interface obstruction. A bounded-degree response is a proposed way
+to avoid growing native cost; this result closes one named product-polynomial
+escape. It supplies no completed producer/consumer or favorable paid invoice.
+The selected frontier, kappa and five framework assumptions retain their status.
+
+[SOURCES.json](SOURCES.json) pins the cited statements and assignment
+provenance to main `e54523517d0f03e58352e647c403a5af3e525a57`, revision 8,
+selected #7000. All predecessor pins are provenance, not executable inputs.
+The selected metadata baseline passed after its sparse source closure was
+materialized; that check was orientation, not a scientific replay.
+
+Overlap checks covered live open PRs, this account's 38 PRs created within
+seven days, recent handoffs and active Discussions, main README searches,
+merged cyclotomic/Phi3/Phi5 searches and local reservations. The cached
+helper reported 477 open PRs and 7,121 Discussion records with complete
+pagination, but only 177 of 10,178 local note records. Both the general
+mixed-factor form and the literal polynomial were searched. Arbitrary diffs
+and unpublished work remain coverage limits.
+
+The exact whole-bank relation and actual global phase are essential to the
+statement. Other products, coefficient fields, restricted-input relations
+and approximate identities are not classified. Physical caller realization,
+dirty restoration, ownership, hosting, precision, stock, tape execution,
+compiler closure and supplier admission remain outside this contribution.
+No selected cold replay, native replay, Lean or Docker build was run.
+
+## Verification implementation limits
+
+The fixed-domain checker uses at most eight-by-eight matrices, three qubits
+for matrix fixtures, 128 signed Paulis, polynomial degree at most fifteen,
+and fifteen multiplications for an order test. The multiplicity loop has
+eighteen widths and at most 65,537 pairs at one width. Functions fit within
+sixty physical lines. Explicit invariant failures remain active under
+optimization. There is no recursion, dynamic code loading, subprocess or
+external dependency. Matrices and coefficients are fixed in the source.
+The only file reads are the six package files for receipt hashes, each
+capped at 100,000 bytes after reading.
+
+Power of Ten rule 3 has a managed-runtime deviation: Python allocates rational
+numbers and collections dynamically. Fixed matrices, finite Pauli tables,
+a streamed multiplicity census and the six-file receipt bound live data.
+Rule 9 uses built-in `len` as the sole `map` callback over at most four
+blocks. Warning-strict normal/optimized runs and AST compilation provide the
+available diagnostics; no external static analyzer is claimed.
+
+## Open next questions
+
+1. Can Phi3*Phi15 support genuinely mixed sectors in a whole-bank
+   Gaussian-rational Clifford? Its primitive traces have opposite signs,
+   so the present bracket does not apply. First solve multiplicity and
+   power-trace constraints, then seek a full Pauli action. No construction
+   or exclusion is supplied here.
+2. Can three same-sign primitive sectors evade this two-sector argument?
+   First enumerate dimension and trace constraints for Phi3*Phi5*Phi17.
+   The present proof gives no general product classification or physical
+   caller.
+
+Structured relevance, first steps and blockers are in [TASK.json](TASK.json).

research/two-cyclotomic-no-mixing/SOURCES.json

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