Hadamard Matrix of Order 668

There exists a Hadamard matrix of order $668$ : a matrix $H\in\{-1,1\}^{668\times668}$ such that $HH^{\mathsf T}=668I_{668}.$ Equivalently, the $668$ rows of $H$ are pairwise orthogonal.

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-assisted
Method: Construction
Field: Combinatorial design theory
Posed by: Raymond Paley
Year posed: 1933
Years open: 93y
Solved: 2026-08-12
Model: Claude (version undisclosed)
Vendor: Anthropic
Collaborators: Levent Alpöge, Philippe Voinov, Saul Reynolds-Haertle
Verification: Unreviewed
Publication: Announced
Significance: 30 / 100
Disclosed cost: —
Wikipedia: Not counted (article postdates the solution)

What was actually shown

Claimed construction of the smallest previously unresolved Hadamard order; the general Hadamard conjecture remains open. The same announcement is reported to cover every previously-open admissible order below 2000, and this site independently confirmed three of them - orders 892, 1132 and 1244 - by exact arithmetic. Order 668 itself has not been extracted from the announcement by anyone publicly, including here.

What the AI did

The announcement is a bare string of 23,828 plus and minus signs with no accompanying text, so it carries no disclosure of its own. Epoch AI's record of it credits a team of three humans working with Claude, and the submission names Levent Alpöge (the poster), Philippe Voinov and Saul Reynolds-Haertle. Which mathematical, computational or search steps came from Claude is not stated anywhere, and no model version is given. The tier is therefore set at the floor the methodology prescribes for an unspecific disclosure rather than at what the framing suggests.

Verification

Partially reproduced by this site on 12 August 2026, with the entry's own claim the part that did not reproduce. The announcement is a single X long-post holding 23,828 characters of "+" and "-" and nothing else: no prose, no code, no separators. Epoch AI describes it as posted in the form of a puzzle and says it encodes matrices for every previously-open admissible order below 2000, which explains why its length is not a multiple of 668. Decoding it here succeeded in part. Three consecutive Goethals-Seidel quadruples sit at offsets 1376, 2268 and 3400, and the matrices they generate were built explicitly and checked in exact integer arithmetic: all entries in {-1,+1} and H H^T equal to nI exactly, for n = 892, 1132 and 1244. All three of those orders were previously open, so the announcement demonstrably contains genuine new Hadamard matrices, and the odds of a random sign string satisfying the Goethals-Seidel autocorrelation condition even once are negligible. Order 668 was not located. Every offset in the string was scanned for each of the four standard seed shapes - a Goethals-Seidel or Williamson quadruple of four circulants of order 167, a two-circulant core or Legendre pair of length 333, base sequences of total length 334, and a negacyclic quadruple - and none occurs anywhere. That absence is expected rather than suspicious: had order 668 been a classical quadruple of this kind it would have been found decades ago, which is exactly why it was the smallest open case. Roughly 19,000 of the 23,828 characters remain undecoded here. So: the announcement is real and contains verified new matrices, the specific order-668 claim is unverified, and no independent expert review or published construction exists.

Claim issue

The announcement cannot be checked as posted. It is an undocumented sign string with no construction, code or prose attached, and the order-668 seed is not present in any standard encoding, so the central claim of this entry rests on the author's word plus Epoch AI's provisional record. Epoch itself hedges twice, marking the solve as AI-attributed only provisionally and noting it is unclear whether the result is an improved search or a general construction. Other trackers (TheoremDB, EmergentMind) still list order 668 as open, though their records predate the announcement.

Sources

Submitted by LucidHawk551 on 12 Aug 2026