Anthropic researcher and Claude build every open Hadamard matrix below size 2000

A 668-order Hadamard matrix is a square array of 668 rows and 668 columns, every entry either +1 or −1, every pair of distinct rows meeting at a right angle. The smallest open case in that family, named after French mathematician Jacques Hadamard in 1893, has been unsolved since 2005. Anthropic researcher Levent Alpöge and two human collaborators, with Claude as a co-author, have now built it. The same run cleared all eleven other open orders below 2000.

The team is Levent Alpöge, Philippe Voinov, Saul Reynolds-Haertle, and Claude, listed in that order on Alpöge's public writeup of the run. Twelve matrices came out of the project, one for every open size in the family under 2000: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964. Each one fills an open case in a problem set that has been a target of both human mathematicians and every deep-reasoning AI model tested so far. The 668 case had been the smallest open order for two decades; it took that position in 2005 when a separate construction at order 428 closed.

Epoch AI has provisionally marked the 668-order case as solved by AI on the FrontierMath open-problems page, the fourth of fifty benchmark items cleared. The benchmark itself was built by more than sixty contributors, including Fields medalists and International Mathematical Olympiad coaches, and described in the 2024 paper at arXiv:2411.04872. Epoch AI's caveat is explicit: the construction method is not yet public, and the agency cannot yet say whether the approach generalizes beyond the orders Alpöge's run covered, or whether it amounts to a faster search rather than a general construction.

Alpöge's X post carried 23,828 plus and minus characters, the rows of the matrix in flattened form, and a small obfuscated Shell script that turned them back into structured output. One Claude variant, Fable, refused to interpret the script. A different model family, GPT-5.6 Sol, was used to read the puzzle. An independent math-results database called VibeMathed has since reproduced the decoding and verified the underlying object: every entry is plus or minus one, the inner product of any two distinct rows is zero, and the diagonal entries are all 668, the correct shape for a true Hadamard matrix.

The general Hadamard conjecture, that every multiple of 4 admits a Hadamard matrix, remains open. The 668 case is one specific finite order that had been stuck, not a fall of the 150-year-old existence question. A separate researcher produced a 668-order matrix modulo 64 last year, which is a related but weaker object; a true integer Hadamard matrix at that order was the open problem the team closed.

Epoch AI's provisional mark could move back to open if the technical report does not check out, or harden into the permanent record once Alpöge and co-authors publish the construction. The deeper question is whether the method Alpöge used, whatever it is, scales above 2000, or whether the orders below 2000 were the easier ones and the next band of open cases requires something the team has not yet found.