slug: cuntz2009
bibkey: cuntz2009integralmodulardata
doi: 10.1007/s10801-008-0139-y
url: https://arxiv.org/abs/math/0611233v2
triage: theorem
motivation_gids:
  - D5/S3/Fourier/CuntzZeroFreeColumnRefutation.result

Cuntz’s integral Fourier zero-free column question

Problem

Michael Cuntz, Integral modular data and congruences, Section 7, Question (5), arXiv:math/0611233v2 p. 21, journal p. 380, asks:

Does an integral Fourier matrix $s$ exist with a column $k$ such that $\forall i$ $s_{ik}\ne 0$ and $\exists i$ $s_{ik}\ne \pm 1$?

The answer is negative for the exact existential assertion. Preregistration: #22614. The source note is Cuntz.

Statement echo

Definition 4.1 names integral Fourier matrices before adding congruence conditions (5)–(6); neither these conditions nor a $T$-matrix is required. The matrix equation fixes row norms. The reversed indices in Definition 3.1’s explanatory norm sum are not an additional hypothesis. No extra binder or conjunct weakens the source question.

Motivation

A zero-free column is necessarily a sign vector. The obstruction follows from orthogonality, normalization and integrality, independently of the source’s further structure.

Gap

The source presents Question (5) as open. The literature screen recorded in #22614 located no full settlement within its searched scope. Its completeness is ASSUMED-UNVERIFIED.

Literature status

The preregistration records a Semantic Scholar list of eight citing works, seven screened and one inaccessible publisher PDF, against eleven citations reported by OpenAlex. OpenAlex citing-work and author queries returned HTTP 429. Cuntz’s arXiv author list supplied 51 records; later relevant papers 0704.2384 and 0809.5186 were read in relevant parts. The matrix-specific searches included Singh’s 1610.05353 and 1610.05763. Broad arXiv result sets and later papers unrelated by title and abstract were not exhaustively read. Google returned interstitial pages, which supply no absence evidence. These are attributed preregistration readings, not a new exhaustive literature search.

Route

Let $A$ be the real cast of $s$ and $D=\mathrm{diag}(d)$, with $d_i>0$. Since $A{A}^T=D$,

$$A(A^T{D}^{-1})=I,(A^T{D}^{-1})A=I.$$

The second identity follows from the first for square matrices. Its diagonal entries give

$$\sum _i\frac{s_{ik}^2}{d_i}=1.$$

The first column is one, hence ${\sum }_{i}1/d_i=1$. In a zero-free integer column, every square is at least one. Equality of the two positive weighted sums forces every square to be one. The private helpers weighted_column_squares, positive_weight_integer_square_rigidity and zero_free_column_units implement this argument and are consumed by result.

Falsifier

The refutation would fail if a square integer matrix satisfying the stated axioms had a zero-free nonunit column. The kernel-checked result excludes precisely this existential assertion. Rational entries, an unnormalized first column or a rectangular matrix lie outside the surviving rigidity statement.

Evidence

Settling node: D5/S3/Fourier/CuntzZeroFreeColumnRefutation.result. The claim has type Prop and the result has type Not claim. The axiom closure of every public declaration is contained in $\{\mathrm{propext},Classical.choice,Quot.sound\}$.

The module has utility: none: its arbitrary-size structural proof is neither bounded enumeration, a checker, a numeric reduction nor a certified finite instance. Utility basis: not-applicable(kind=none). The escape audit for its only public theorem, result, is unfinished in #22721; the three proof helpers are private. No four-slot registration is claimed.

Experiment entry

Pinned experiment directory, script check.py, command python3 check.py, exit 0. Script SHA-256: 55dd994bb876901a2ee5d0f4be187ac928c6006a9aadfdd394c838b2b4bbb1ea. Final reading: ALL_CHECKS_OK. The script uses exact integer and rational arithmetic. The readings below concern the pinned script and its stated finite scope.

Triage

What the settlement shows

1. Proved, kernel-checked — exact negative answer. result refutes the unchanged positive-size existential assertion in Section 7(5).

2. Proved, kernel-checked — surviving general statement. Private zero_free_column_units treats arbitrary square integer matrices whose first column is one and whose row Gram matrix is diagonal with positive entries. It does not require a determinant premise, symmetrization, fusion integrality or $T$.

3. Proved in prose — mechanism. The kernel-checked private theorem weighted_column_squares gives ${\sum }_i{s}_{ik}^2/d_i=1$. At the first column, $s_{i0}=1$, so the same identity gives ${\sum }_{i}1/d_i=1$. For a zero-free integer column, $s_{ik}^2\ge 1$ and every weight $1/d_i$ is positive. Equality of the sums therefore forces each $s_{ik}^2=1$, hence $s_{ik}=1$ or $s_{ik}=-1$. The private theorem positive_weight_integer_square_rigidity states the corresponding sign conclusion from these weighted-sum hypotheses.

4. Computed — source example and finite enumeration. The experiment entry checks the source’s 13-dimensional example, all $13^3=2197$ structure constants, Fourier equations (1)–(4) and congruence conditions (5)–(6). It enumerates square integer matrices of sizes $1,2,3,4$ with entries in $[-2,2]$, first column one and mutually orthogonal rows, up to row permutation: respectively $1,1,8,26$ matrices, $36$ total. Every matrix passes the weighted Gram, reciprocal-norm, balance and zero-free-column checks.

5. Computed — necessary structural boundaries. Exact checks verify orthogonality and zero-free nonunit columns for rational rows $(1,7/5),(1,-5/7)$, unnormalized integer rows $(1,2),(2,-1)$, and rectangular integer rows $(1,1,2),(1,1,-1)$. These show why each named condition of the general helper matters; they are not Fourier-matrix counterexamples.

6. Computed — zero/excess balance. Throughout the enumeration window and the source example, the script checks

   $$\sum _{i:s_{ik}=0}\frac{1}{d_i}=\sum _{i:|s_{ik}|\ge 2}\frac{s_{ik}^2-1}{d_i}.$$

   A universal balance theorem is not formalized here. The identity is explained by separating zero, unit and larger-square terms in the weighted sum; the asserted computational coverage is only the finite window and source example.

7. Open — classifications and further source questions. Equal-norm classification, classification of all Fourier matrices and existence of a $T$-matrix are not settled. The argument imposes no change on the source’s proved results and supplies no classification or $T$ construction. A question asking for an example of the excluded zero-free nonunit type has no witness; other questions require their own arguments.

8. Open — exhaustive literature absence. No full settlement was located within the preregistration’s searched scope. Unread or inaccessible sources and later unindexed results remain outside that claim.

ASSUMED-UNVERIFIED

Completeness of the external literature screen is ASSUMED-UNVERIFIED. Literature readings are attributed to the preregistration producers. The finite checker does not prove an unbounded enumeration, the universal balance identity, either classification, or existence of $T$.