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The circulant Hadamard conjecture: real circulant Hadamard matrices exist only in orders 1 and 4

A real Hadamard matrix of order nn is an n×nn\times n matrix HH with entries ±1\pm1 and HHT=nInHH^T=nI_n; it is circulant if every row is the cyclic shift of the first. Order 4 has the example with first row (−1,1,1,1)(-1,1,1,1), and order 1 is trivial. The circulant Hadamard conjecture, traditionally attributed to Ryser (1963), asserts that there are no others. Turyn showed any further order is 4u24u^2 with uu odd and not a prime power; field descent and later methods of Schmidt and Leung-Schmidt, and the 2017 computation of Logan and Mossinghoff, excluded all but 4489 orders up to 4⋅10304\cdot10^{30}. Several claimed proofs have appeared. Is every real circulant Hadamard matrix of order 1 or 4?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Design theory; Hadamard matrices and difference sets
Posed by
Herbert J. Ryser (traditional attribution)
Year posed
1963
Years open
63y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: a real circulant Hadamard matrix of order n≥1n\ge1 exists if and only if n∈{1,4}n\in\{1,4\}. Section 2 reproves Turyn's restriction n=4u2n=4u^2 with uu odd, and the main argument excludes every odd u>1u>1 through an alternating product of character values over the primes dividing uu. Corollary 1.2 deduces the Barker-sequence classification. It says nothing about the (non-circulant) Hadamard conjecture on existence in every order divisible by 4, nor about complex or group-developed variants.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's 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 single manuscript (September 23, 2026) is the whole family; the Barker-sequence consequence is listed as a separate entry.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as stated in the manuscript and in Ryser's tradition; it is the full conjecture with no restriction on the order. The proof (group-ring and cyclotomic arguments) was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator CirculantHadamard, declaration OAI.CirculantHadamard.exists_iff_order_one_or_four). The comparator statement CirculantHadamard.lean was read here: for every n>0n>0, a real n×nn\times n matrix with ±1\pm1 entries, Hij=h(j−i)H_{ij}=h(j-i) with index arithmetic modulo nn, and HHT=nIHH^T=nI exists if and only if n=1n=1 or n=4n=4. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The manuscript lists five earlier claimed complete proofs (Oh-Hashi 2016, Orozco Lopez 2019, Morris 2023, Gallardo 2024, Manjhi-Kumar 2025) without assessing them.

Sources

Changelog1 change

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