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The Matolcsi-Ruzsa-Weiner Fourier-vanishing conjecture for complex Hadamard matrices of order six

For a complex Hadamard matrix HH of order six (∣Hij∣=1|H_{ij}|=1, H∗H=6IH^*H=6I) and a∈Z6a\in\mathbb Z^6 with ∑ai=0\sum a_i=0, let gH(a)=16∑k∏iH(i,k)aig_H(a)=\frac16\sum_k\prod_iH(i,k)^{a_i}. Orthogonality forces gH(ei−ej)=0g_H(e_i-e_j)=0. In their Fourier approach to mutually unbiased bases in dimension six, Matolcsi, Ruzsa and Weiner (2013, Conjecture 2.3) conjectured a further vanishing at the charge α=(1,1,1,−1,−1,−1)\alpha=(1,1,1,-1,-1,-1), with one exceptional equivalence class, Tao's cubic matrix S6S_6; Maxwell and Brierley proved it for the Karlsson family. Does gH(πα)=0g_H(\pi\alpha)=0 hold for every coordinate permutation π\pi and every HH not equivalent to Tao's matrix?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Computation
Field
Complex Hadamard matrices of order six
Posed by
Mate Matolcsi, Imre Z. Ruzsa and Mihaly Weiner (Conjecture 2.3, Australas. J. Combin. 55, 2013)
Year posed
2013
Years open
13y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
12 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: every complex Hadamard matrix of order six not equivalent to Tao's cubic matrix has gH(πα)=0g_H(\pi\alpha)=0 for every permutation π\pi of α=(1,1,1,−1,−1,−1)\alpha=(1,1,1,-1,-1,-1). Theorem 1.2, by a second exact certificate and Weiner's completion theorem, excludes seven mutually unbiased bases in C6\mathbb C^6, giving N(6)≤5N(6)\le5. It does not prove the further Matolcsi-Matszangosz-Varga-Weiner conditions for MUB triplets or the bound N(6)=3N(6)=3, which is the companion's computer-assisted claim.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof is an exact invariant-moment certificate in integer and rational arithmetic, with a standard-library Python verifier included in the manuscript folder.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against MRW Conjecture 2.3; the manuscript notes its normalization divides the conjecture's sums by six, which does not affect vanishing. The verifier was not re-run here. Lean: lean/ComparatorChallenges/MUBSix.json exists with solution_module OAI.Analysis.MutuallyUnbiased.Main, whose file exists at the pinned commit; this challenge is not in lean/formalization.yaml. The statement MUBSix.lean was read here: its first conjunct says that for every 6x6 complex matrix with unimodular entries and H∗H=6IH^*H=6I that is not equivalent (row and column permutations and unit phases) to the cube-root matrix with Tao's exponent pattern, the normalized character sum vanishes at every permutation of (1,1,1,−1,−1,−1)(1,1,1,-1,-1,-1). This states the headline; the second conjunct is the bound of five bases. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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