Prove one-column completion over composite cyclotomic alphabets

research/cyclotomic-column-completion/PROOF.md

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+# One-column cyclotomic completion
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+## Statement
+
+Write mu_m for the mth roots of unity. Let m,n>=2 and let A be an
+n-by-(n-1) matrix with entries in mu_m and A* A=nI. Then its unique
+orthogonal flat completion whose new column starts with 1 is
+
+```text
+w = (nI-AA*) e_1,       H = [A w].                         (1)
+```
+
+It satisfies H*H=nI and has entries in mu_lcm(2,m). Moreover:
+
+1. If m is even, every entry of w belongs to mu_m, for every n>=2.
+2. If m is an odd prime power and n>=3, every entry of w belongs to mu_m.
+
+Thus composite even alphabets and odd prime-power alphabets retain their
+original alphabet. Both the matrix order n and alphabet order m are arbitrary
+under these hypotheses; they need not agree. This extends the prime-order
+statement B097 of David Ganz's atlas, PR2849. That result and PR2855's
+prime pilot application are cited rather than reverified here.
+
+The n>=3 condition in the second conclusion is necessary. For any odd m,
+the valid two-row input A=(1,1)^T has normalized completion (1,-1)^T.
+Even allowing a global phase cannot put both entries in mu_m, since their
+ratio is -1. The doubled-alphabet conclusion is therefore sharp without
+a restriction on n. No sharpness assertion is made for odd composite m
+and n>=3.
+
+## The flat column and its integral entries
+
+Set C=nI-AA*. Since A*A=nI, C is Hermitian, C^2=nC, and rank C=1.
+Every diagonal entry of AA* is n-1, so C has diagonal 1. A rank-one
+Hermitian positive matrix with C_11=1 satisfies C=C e_1 e_1* C.
+Consequently C=ww*, |w_i|=1 and w_1=1. Also A*C=0, so A*w=0.
+The squared norm of w is n, proving H*H=nI. The orthogonal complement
+of A's columns has dimension one, proving normalized uniqueness.
+
+For i>1 the formula is explicitly
+
+```text
+w_i = -sum_(j=1)^(n-1) A_ij conjugate(A_1j).              (2)
+```
+
+Hence w_i belongs to Z[zeta_m] and is an algebraic integer. No division by
+n or a determinant denominator occurs. This is the useful integrality
+property behind the alphabet extension.
+
+## The classical cyclotomic facts
+
+We use cyclotomic irreducibility and [Q(zeta_r):Q]=phi(r). Here are the
+remaining arguments, to make their exact use explicit.
+
+If alpha in Z[zeta_m] satisfies |alpha|=1, then alpha conjugate(alpha)=1.
+All automorphisms of Q(zeta_m) commute with complex conjugation, since this
+Galois group is abelian. Applying an automorphism therefore shows that
+every algebraic conjugate of alpha has modulus one. Kronecker's theorem
+implies that alpha is a root of unity.
+
+For completeness, the needed Kronecker argument is finite. If alpha has
+degree d, each power alpha^k is an algebraic integer of degree at most d,
+and every conjugate still has modulus one. Coefficients of its monic
+minimal polynomial are integer elementary symmetric sums, bounded in
+absolute value by the corresponding binomial coefficients of degree at
+most d. Only finitely many such polynomials exist, and each has finitely
+many roots. Thus the sequence alpha,alpha^2,... has finitely many possible
+values. Two powers agree; alpha!=0 then implies a positive power equals 1.
+
+Put M=lcm(2,m). The roots of unity in Q(zeta_m)=Q(zeta_M) are exactly
+mu_M. Indeed, if a primitive tth root lies in this field, the compositum
+with Q(zeta_M) is Q(zeta_l), l=lcm(M,t). Thus phi(l)=phi(M).
+Because M is even and M divides l, the product formula for phi shows
+that increasing any existing prime exponent or adjoining any new odd
+prime strictly increases phi. It follows that l=M and t divides M.
+The reverse inclusion is immediate, using zeta_2m=-zeta_m^((m+1)/2)
+when m is odd. These are classical facts, not new number-theory claims.
+
+Applying them to every entry in (2) proves w_i in mu_M. For even m,
+M=m, proving the first conclusion.
+
+## Removing the sign for odd prime powers
+
+Let m=p^a for an odd prime p and a>=1. Evaluation at zeta_m=1 followed
+by reduction modulo p defines a ring map
+
+```text
+rho: Z[zeta_m] -> F_p,        rho(zeta_m)=1,
+```
+
+because Phi_(p^a)(1)=p. If n>=3, A has two distinct columns. Their
+orthogonality is a sum of n elements of mu_m equal to zero. Applying
+rho shows p divides n. Applying rho to (2) now gives
+
+```text
+rho(w_i) = -(n-1) = 1 in F_p.
+```
+
+Every element of mu_2m is either zeta_m^k or -zeta_m^k. Their images
+under rho are respectively 1 and -1, which differ because p is odd.
+Therefore each w_i lies in mu_m. The first entry already equals 1.
+This proves the second conclusion and completes the theorem.
+
+## Sources and scope
+
+David Ganz's atlas B082/B097 in PR2849 supplies the one-column and
+prime-order setting and explicitly leaves the composite alphabet outside
+its proof. The prime-phase pilot application in PR2855 supplies the exact
+recovery context. Their original files and credit remain unchanged.
+Kronecker's theorem, cyclotomic irreducibility, cyclotomic field degrees
+and the prime-power reduction are classical. Richard Stanley's
+[Algebraic Combinatorics, Theorem 12.14](https://math.mit.edu/~rstan/algcomb/algcomb.pdf)
+and Eric Moorhouse's
+[Cyclotomic Fields with Applications](https://ericmoorhouse.org/handouts/cyclotomic_fields.pdf)
+are accessible mathematical precedents for the integrality argument.
+The proof above supplies the actual implication used here.
+
+The contribution is the stated extension of this project's completion
+interface and its exact finite corroboration. Global literature priority,
+independent human and formal proof-kernel review are unclaimed. More than
+one missing column, approximate data, physical preparation, paid consumption,
+supplier admission and complete multiplication-theorem obligations remain
+outside scope. No new kappa or selected-frontier change follows.

research/cyclotomic-column-completion/README.md

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+# One-column completion without enlarging cyclotomic alphabets
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Claim type: scoped algebraic proof with exact finite corroboration. Let A
+be an n-by-(n-1) matrix over the mth roots of unity, with A* A=nI.
+Its normalized missing column is explicitly
+
+```text
+w = (nI-AA*) e_1,          w_1=1.
+```
+
+The completed matrix [A w] has orthogonal columns and retains the original
+alphabet whenever m is even, for every n>=2. It also retains the alphabet
+when m is an odd prime power and n>=3. For arbitrary odd m it always uses
+at most the 2mth roots. Matrix order and alphabet order need not agree.
+This extends the prime-order case of David Ganz's
+[atlas B097](../pure-math-atlas-400/volumes/02-geometry.md#b097), PR2849,
+and the prime pilot interface in his PR2855. Neither predecessor is audited
+or replayed here.
+
+[PROOF.md](PROOF.md) proves the universal statement. The missing entries
+are integral sums of roots and have modulus one. Classical Kronecker and
+cyclotomic field arguments put them in the original even alphabet. For
+an odd prime-power alphabet p^a, orthogonality implies p divides n;
+reduction at zeta=1 then excludes the extra minus sign.
+
+The n>=3 restriction for odd prime powers is necessary. For any odd m,
+A=(1,1)^T has missing column (1,-1)^T, even up to global phase. Thus
+alphabet doubling is unavoidable in the unrestricted odd-alphabet problem.
+For odd composite m and n>=3, this package gives the doubled bound and
+leaves original-alphabet completion open.
+
+## Exact finite evidence
+
+The standard-library verifier uses integer polynomial arithmetic modulo
+cyclotomic polynomials. It imports no predecessor executable and uses no
+floating-point comparisons. It reconstructs each missing column, checks
+literal column and row Gram identities, checks input preservation and
+checks same-alphabet membership where the theorem requires it.
+
+It checks all 314 dephased partial matrices in these ten domains:
+
+```text
+shape    alphabet orders m         numbers of valid inputs, in that order
+3x2      3, 6, 9, 15              2, 2, 2, 2
+4x3      2, 4, 6, 8, 10, 12       6, 24, 42, 60, 78, 96
+```
+
+Dephased means first row and first column have phase zero. The census
+enumerates 3,879 candidate nonconstant columns and 2,582 candidate input
+matrices. It does not identify gauge/permutation equivalents with each
+other. Each count above is directly enumerated.
+
+An additional 189 gauged tensor-Fourier erasures, at orders 4,6,8,9,12,15,
+compare the recovered column with the independently retained erased column.
+They include m=3,n=9 and m=9,n=9. Another 32 erasures of the 4-by-4
+one-parameter family over mu_8 test a changed non-Fourier phase. Four
+odd-alphabet two-row examples check the necessary sign boundary.
+In total the verifier completes 539 cases and recovers 3,325 entries;
+488 cases fall under the same-alphabet theorem. The other cases test the
+general doubled-alphabet conclusion without claiming a counterexample
+at n>=3. Nine malformed inputs reject. Three changed output entries fail
+the Gram check. These finite fixtures corroborate the proof rather than
+prove its unbounded quantifiers.
+
+## Reproduction
+
+From the repository root, using CPython 3.12:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/cyclotomic-column-completion/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/cyclotomic-column-completion/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cyclotomic-column-completion
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cyclotomic-column-completion --check-source-history
+```
+
+Another CPython 3.12 path may replace the local path. Both direct modes
+write nothing and emit `PASS_CYCLOTOMIC_COLUMN_COMPLETION`, the enumerated
+counts and a generated SHA256 receipt for the six package files. The
+registered commands each have a 60-second timeout. Source history,
+metadata and committed-byte checks are separate from mathematical review.
+
+The selected-baseline orientation passed with distinct complex, ordinary
+BIT, admitted BIT and outer budgets. Its scope is source-binding metadata,
+not a selected scientific replay. No selected cold replay was run.
+
+## Scope and credit
+
+This is an exact algebraic completion theorem and finite verifier. More
+than one missing column, approximate observations, physical preparation,
+paid consumption, dirty restoration, precision, tape costs and supplier
+admission remain unproved. No new kappa or selected-frontier change is
+claimed. Independent human and formal proof-kernel review, and global
+literature priority, are unclaimed. Lean and Docker were not run.
+
+Credit David Ganz's PR2849 for B082/B097 and the explicit composite-alphabet
+boundary, and his PR2855 for the exact pilot recovery application. Their
+proofs, implementation and original attribution remain unchanged.
+Kronecker's theorem, cyclotomic irreducibility and field degrees, prime-power
+reduction, Fourier tensors and the four-dimensional one-parameter Hadamard
+family are classical. The new contribution is the stated extension of
+this project's completion interface. Exact repository provenance is in
+[SOURCES.json](SOURCES.json); classical primary-source links are in the proof.
+
+The `formal-interface-lemma` card is explicitly changed to this algebraic
+side result. It removes a prime-only alphabet restriction, without
+constructing or pricing a multiplication supplier. The
+[advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19214393)
+and [TASK.json](TASK.json) record the exact scope and overlap limitations.
+
+## Open next questions
+
+1. For odd composite m and n>=3, must every such partial matrix complete
+   over mu_m? Start with m=15 and a dimension permitting mixed 3-cycle and
+   5-cycle vanishing sums. The sign exclusion used for a single prime power
+   has no direct replacement. A counterexample must be a complete partial
+   matrix, not just a sum of n-1 roots with the right modulus.
+2. Can this exact alphabet theorem support a robust completion method for
+   noisy composite-phase inputs? Start by bounding the perturbation of
+   (nI-AA*)e_1 and proving a separation radius for root rounding. Exact
+   orthogonality and integrality are used here, so no noise guarantee or
+   paid physical implementation follows.
+
+## Verifier bounds
+
+Only `verify.py` is executable. It has nonrecursive control flow and no
+subprocess, network, generated code or dynamic imports. Functions are below
+60 physical lines. Alphabets have order at most 15, cyclotomic conductors
+at most 30, and matrices at most 16 rows. The largest census scans at most
+12^3 candidate columns and at most 1,000^2 candidate pairs. Tensor fixtures
+have at most four factors, three gauges and sixteen erasure positions.
+All mathematical checks use explicit exceptions and remain active under -O.
+Six package reads are each bounded by 1 MiB. Warning-strict execution and
+syntax/function-size inspection are the available diagnostics; no external
+static analyzer is installed or claimed.
+
+Power of Ten rules 3 and 9 need managed-runtime deviations. Python integer,
+tuple, iterator and JSON operations allocate and use runtime dispatch.
+The fixed capacities above bound live data and call targets; this command
+has no steady service loop or external mutable resource. Both registered
+modes exercise these limits and controls. Reassess storage and validation
+before accepting external inputs or larger search domains.

research/cyclotomic-column-completion/SOURCES.json

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