Prove odd cyclotomic Clifford rank cap, sharp at degree four (no κ)

research/odd-cyclotomic-response/PROOF.md

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+# Odd cyclotomic response gate
+
+This scoped proof was produced by Nick Petalas. It extends Ross Richardson's
+whole-bank cubic trace gate in [PR4081](https://github.com/CrocSwap/integer-mult-bounds/pull/4081).
+The cyclotomic extension and the degree-four sharpness example are the new steps.
+The predecessor's Pauli character argument is credited below. No historical
+priority is claimed for the classical field-degree or Clifford facts.
+
+## Statement
+
+Let n>1 be odd. Let U be a unitary Clifford on M qubits, with every matrix
+entry in Q(i), and suppose the **actual whole-bank matrix** satisfies
+
+    Phi_n(U)=0.
+
+Here Phi_n is the nth cyclotomic polynomial. Write S_U for the induced
+symplectic matrix on the 2M binary Pauli coordinates. Then:
+
+1. phi(n)=2^a for some integer 1<=a<=M. Equivalently, n is a squarefree
+   product of distinct Fermat primes.
+2. The characteristic polynomial is Phi_n(X)^(2^(M-a)), and
+   Tr(U)=mu(n) 2^(M-a).
+3. rank_F2(S_U-I)=2a, exactly. In particular, the native off-diagonal
+   B-block rank is at most min(M,2a).
+
+The condition on n is necessary. This proof does not assert Clifford
+existence for every such n. A Fermat prime means a prime of the form
+2^(2^j)+1, including any not presently known. The statement does not assume
+that the currently known Fermat primes are the only ones.
+
+At n=5, the bound B-rank<=4 is attained on four qubits, as constructed below.
+Thus the degree-four cap cannot be reduced for all ambient widths.
+
+## Cyclotomic degree and trace
+
+For odd n, adjoining i and a primitive nth root gives Q(zeta_(4n)). Indeed,
+the orders 4 and n are coprime, so the product of primitive roots of those
+orders has order 4n, and its powers recover both factors. Classical
+cyclotomic irreducibility therefore gives
+
+    [Q(i,zeta_n):Q(i)] = phi(4n)/2 = phi(n).
+
+For a proof of the classical irreducibility theorem, see Romyar Sharifi's
+[Abstract Algebra, Theorem 6.6.11](https://math.ucla.edu/~sharifi/notes/algebra-ch06.html).
+Consequently Phi_n remains irreducible over Q(i). Its roots are distinct.
+Since Phi_n(U)=0, the Q(i)[X]-module defined by U is a direct sum of copies
+of Q(i)[X]/(Phi_n). Equivalently, its characteristic polynomial is Phi_n^t
+for a positive integer t, and
+
+    2^M = t phi(n).
+
+Thus phi(n) is a power of two, say 2^a, with a<=M. If
+n=product_p p^e_p, then phi(n)=product_p p^(e_p-1)(p-1). Every p is odd.
+No e_p can exceed one, and every p-1 must be a power of two. If p=1+2^b
+is prime, b is itself a power of two: an odd divisor c>1 of b would factor
+(2^(b/c))^c+1. Conversely, a squarefree product of Fermat primes has
+power-of-two totient. This proves the stated equivalence.
+
+Each primitive nth root occurs t times. The sum of the primitive nth roots
+is mu(n). One elementary derivation partitions all nth roots by their
+exact orders and applies Mobius inversion to their sum. Hence
+
+    Tr(U)=t mu(n)=mu(n) 2^(M-a).
+
+Squarefreeness makes mu(n) equal to plus or minus one. In particular the
+trace is nonzero, and its squared absolute value is 2^(2M-2a).
+
+## Pauli adjoint trace
+
+This is the Pauli-character step used in the predecessor's cubic proof.
+For completeness, let P_v be Hermitian Pauli representatives, indexed by
+v in F2^(2M). Clifford conjugation sends P_v to epsilon(v) P_(S_U v),
+where epsilon(v) is plus or minus one. On K=ker(S_U-I), comparing the
+conjugation of P_v P_w with the product of the conjugations cancels the
+same Pauli multiplication phase. It gives
+
+    epsilon(v+w)=epsilon(v)epsilon(w),  v,w in K.
+
+Thus epsilon restricted to K is a character. Its sum over K is zero if
+the character is nontrivial, and 2^dim(K) otherwise. This sum is the trace
+of conjugation by U in the Pauli basis. The same linear map on matrices
+has trace Tr(U) conjugate(Tr(U)), by its tensor representation U tensor
+conjugate(U). The trace computed above is nonzero. Therefore
+
+    2^dim(K) = 2^(2M-2a),
+    rank(S_U-I) = 2M-dim(K) = 2a.
+
+In the physical X/Z coordinates, the off-diagonal B block of S_U is also
+a submatrix of S_U-I. Its rank is at most 2a, and its size is M by M.
+This proves the native-rank cap without any tensor or support restriction.
+
+## A sharp degree-four example
+
+Define H_g=(1+i)[[1,1],[1,-1]]/2, S=diag(1,i), and let C be CNOT with
+the first qubit controlling the second. The Gaussian-dyadic Clifford
+
+    U5 = - C (S tensor I) (H_g tensor H_g)
+
+has matrix
+
+    1/2 * [ -i  -i  -i  -i ]
+          [ -i   i  -i   i ]
+          [  1  -1  -1   1 ]
+          [  1   1  -1  -1 ].
+
+Direct matrix multiplication gives U5^5=I and
+I+U5+U5^2+U5^3+U5^4=0. Its trace is -1, and its symplectic defect has
+rank four. These are complete 4-by-4 identities, checked with integer
+Gaussian numerators in verify.py.
+
+Add two identity spectators. On the four qubits numbered 0,1,2,3, let V
+be Bell preparation on pairs (0,2) and (1,3). Apply H_g on 0 and 1, then
+CNOT 0->2 and 1->3. Set
+
+    W = V^-1 (U5 tensor I_4) V.
+
+Conjugation preserves Phi5(W)=0. Its B block is nonsingular. To see this
+without relying on a computed rank, write A for the four-dimensional
+symplectic coordinate space of the first two qubits, and E for that of
+the spectators. Bell preparation sends the standard Z Lagrangian L to
+a graph L' of a symplectic isomorphism A->E. In characteristic two this
+graph is Lagrangian, has dimension four, and intersects A trivially.
+The active symplectic map T of U5 has T-I invertible. For l in L',
+
+    ((T direct_sum I)-I)l = ((T-I) projection_A(l), 0).
+
+If (T direct_sum I)l also lies in L', the displayed difference lies in
+A intersect L', hence is zero. Invertibility of T-I and of
+projection_A restricted to L' forces l=0. Thus L' and its image are
+transverse. Pulling back by V, L and S_W L are transverse. The map from
+input Z coordinates to output X coordinates is therefore invertible,
+which is exactly B-rank four. The standalone verifier also checks this
+complete 16-by-16 matrix, all eight Pauli generators and every fixed
+Pauli sign.
+
+## Meaning for a proposed response interface
+
+The n=3 case is Richardson's result. New examples of the necessary bound
+are n=5 with defect four, n=15 with defect six, and n=17 with defect eight.
+Orders 7,9,25 and 27 cannot satisfy their primitive cyclotomic relation
+on any Gaussian-rational qubit bank, even before Cliffordness is imposed.
+Those are exclusions of the primitive relation, not exclusions of those
+finite orders among Clifford operators.
+
+For fixed odd n, no family with unbounded native B rank can retain
+Phi_n(U)=0 on the whole bank. In terms of a rank r, any such primitive
+response needs phi(n)>=2^(r/2). This is an obstruction to a fixed-degree
+primitive response identity, not a complete query-price
+or caller impossibility theorem.
+
+Order alone is insufficient: U5 tensor U5 still has order five but
+does not satisfy Phi5=0, and its symplectic defect rank is eight.
+Arbitrary global phases also do not preserve the premise. Non-Gaussian
+coefficient fields, non-Clifford operators, restricted-input quotients,
+responses with several cyclotomic factors, and complete changed callers
+are outside this theorem. No physical implementation invoice, dirty
+restoration, supplier admission, new kappa or complete multiplication
+theorem follows. The universal proof is written mathematics, not a
+finite extrapolation or a formal-kernel result.

research/odd-cyclotomic-response/README.md

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+# Odd cyclotomic Clifford response gate
+
+Scoped written proof with exact finite corroboration, produced by Nick Petalas.
+For every odd n>1, an actual Gaussian-rational M-qubit Clifford U satisfying
+Phi_n(U)=0 on the whole bank necessarily obeys
+
+```text
+phi(n)=2^a,  1<=a<=M
+n is a squarefree product of distinct Fermat primes
+Tr(U)=mu(n) 2^(M-a)
+rank_F2(S_U-I)=2a
+native B-rank <= min(M,2a).
+```
+
+The degree-four cap is sharp: a complete four-qubit Clifford W satisfies
+Phi5(W)=0 and has native B-rank four. Its construction adds two identity
+spectators to an explicit two-qubit order-five Clifford and conjugates by
+Bell preparation. For fixed odd n, the primitive relation cannot carry
+unbounded native rank. This is an interface obstruction, with no new kappa.
+
+[PROOF.md](PROOF.md) gives the all-parameter argument and the explicit
+sharpness construction. The new step extends Ross Richardson's
+[PR4081](https://github.com/CrocSwap/integer-mult-bounds/pull/4081)
+whole-bank cubic gate to every odd cyclotomic polynomial. His cubic result
+and fixed-Pauli character argument retain their credit. Classical cyclotomic
+irreducibility, totient arithmetic and Clifford/Bell gates retain their
+classical status. Global literature priority is unclaimed. No predecessor
+verifier is imported or replayed.
+
+## Evidence and reproduction
+
+The standalone standard-library verifier uses exact integer Gaussian
+numerators, common positive denominators and binary elimination. It checks:
+
+- All 2,047 odd n in 3..4095. Totients from explicit coprime enumeration
+  equal the independently factored formula, and the degree condition equals
+  the squarefree Fermat-prime condition. Eleven n in this interval pass the
+  necessary degree test. This is not an existence census.
+- The complete two-qubit degree-four matrix and four-qubit sharp example,
+  including their polynomial identities, unitarity, all Pauli generators,
+  every fixed-Pauli sign, both ranks and the adjoint trace identity.
+- An order-five tensor control with symplectic defect eight that fails
+  Phi5=0; a wrong global phase; a passing primitive-power control; a
+  nonunitary companion matrix satisfying Phi5; and four invalid domains.
+
+Run from the repository root with CPython 3.12:
+
+```sh
+/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
+```
+
+Other installations may substitute their CPython 3.12 path. Expected markers
+are `PASS_ODD_CYCLOTOMIC_RESPONSE`, `PASS_REGISTERED_RESEARCH_PACKAGE` and
+`PASS_RESEARCH_COMMIT_BYTES`. Each direct check takes less than one second
+locally and has a registered 30-second limit. Normal and optimized runs
+generate the same six-file SHA256-bound receipt on stdout. No receipt is
+hand-written or stored in the package. Direct replay also works in a
+read-only Git-free export.
+
+[SOURCES.json](SOURCES.json) pins the catalogue, selected record and source
+proof as historical provenance. They are not executable inputs. The
+scientific base is `a5bff3bf133e482541c09eba4e773884bdb2b1b7`, priority
+revision 8. The selected source-binding first command passed as metadata
+orientation only. The `joint-query-controller` card is explicitly changed
+to this algebraic response-interface result; it does not solve the installed
+QQ16 caller task. [TASK.json](TASK.json) records the overlap limits and
+structured next questions.
+
+## Limits
+
+The universal result is the written proof. Finite checks corroborate the
+identities and bounded arithmetic, and do not prove the all-n statement.
+The necessary condition on n is not a sufficient existence classification.
+Order n alone, non-Gaussian coefficient fields, non-Clifford operators,
+several cyclotomic factors and restricted-input identities are outside the
+theorem. A sparse reader relation of unbounded degree is also outside the
+fixed-degree consequence.
+
+No actual selected caller, physical native program, complete signed invoice,
+dirty restoration, ownership, hosting, precision, stock, tape execution or
+supplier admission is supplied. No selected scientific replay, Lean or
+Docker run was performed. All five selected framework assumptions remain.
+Independent human and formal proof-kernel review are unclaimed. No complete
+multiplication theorem or selected-frontier change follows.
+
+## Open next questions
+
+- Determine which degree-permitted n admit Gaussian-rational Clifford
+  responses. Begin with a complete primitive order-seventeen example and
+  its Pauli action; the necessary degree condition alone supplies no such
+  construction.
+- Bound native rank when a fixed product of several odd cyclotomic factors
+  annihilates U. Begin with the exact multiplicity and trace equations for
+  Phi3*Phi5; cancellation can make the adjoint trace vanish, so the present
+  nonzero-character argument does not automatically apply.
+
+These are follow-on questions. Any favorable physical caller still needs
+its complete producer/consumer contract and paid bill before expansion.
+
+## Verification implementation bounds
+
+There is no recursion or dynamic code loading. Every matrix has dimension
+at most 16, every scalar numerator and denominator is bounded by 2^256,
+matrix powers use at most 32 factors, polynomials have at most 33 bounded
+integer coefficients, and the factor domain ends at 4095. Pauli loops have
+at most 256 labels and eight generators. Every function is below 60 lines.
+Explicit invariant failures remain active under optimized Python.
+
+Power of Ten rule 3 has a managed-runtime deviation at verify.py: Python
+allocates integer, tuple and dictionary objects during exact arithmetic.
+Fixed dimensions, bit capacities, factor bounds and sequential fixtures
+bound live collections. No asynchronous resources or retained output files
+exist. Rule 8 uses the standard dataclass decorator for one fixed four-field
+immutable record; it loads no external schema or executable input. Rule 9
+uses reduce with the single built-in gcd target over at most 512 integers.
+Warning-strict normal and optimized runs plus AST compilation supply the
+available Python diagnostics. A static type-checking configuration is not
+introduced for this isolated research package. A future larger replay must
+reassess these capacities before raising them.
+
+New work is Apache-2.0. Exact predecessor pins and credit are in
+[SOURCES.json](SOURCES.json) and [PROOF.md](PROOF.md).

research/odd-cyclotomic-response/SOURCES.json

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+    {
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+  "classical_reference": {
+    "author": "Romyar Sharifi",
+    "title": "Abstract Algebra, Theorem 6.6.11",
+    "url": "https://math.ucla.edu/~sharifi/notes/algebra-ch06.html",
+    "use": "Classical cyclotomic irreducibility; no code or text copied."
+  }
+}