Phangara's conjecture: every order-six S-Hadamard matrix is equivalent to Tao's cubic matrix
Lisonek (2019) called a complex Hadamard matrix an S-Hadamard matrix if its entrywise square is also Hadamard. Tao's cubic matrix of order six, with cube-root-of-unity entries, is an example. Phangara (MSc thesis, Simon Fraser University, 2026, Conjecture 3.1.6) conjectured that it is the only one in order six. Is every S-Hadamard matrix of order six equivalent, under row and column permutations and unit phase multiplications, to Tao's cubic matrix?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Complex Hadamard matrices; S-Hadamard matrices
- Posed by
- Jasleen Phangara (Conjecture 3.1.6, MSc thesis, Simon Fraser University, 2026)
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 5 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Proposition 3.1: if and its entrywise square are both complex Hadamard matrices of order six, then is equivalent to Tao's cubic matrix; in particular all its dephased entries are cube roots of unity. The manuscript notes that uniqueness of the cube-root class was already known from Butson classifications (Lampio-Ostergard-Szollosi; Liang-Chen-Long-Qiu); the new step derives cube-root entries from the two Hadamard conditions for arbitrary phases. It says nothing about other orders.
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 result is Proposition 3.1 of the companion Fourier-certificate manuscript, used there as a lemma.
Verification
No independent mathematician has checked this yet. Checked here: Proposition 3.1 of the manuscript was read against Phangara's conjecture as the manuscript quotes it; it states that if and are order-six Hadamard matrices then is equivalent to Tao's matrix. The proof is a short Newton-identity argument on row ratios. The Lean challenge HadamardCubeFiber formalizes only a supporting cancellation lemma (lean/docs/266.md says so), not this classification, so the entry is Unreviewed. The thesis itself was not opened here.