Levent Alpöge’s team, working with Claude, constructed 12 previously unknown arrays of +1 and −1

Levent Alpöge’s team, working with Claude, constructed 12 previously unknown arrays of +1 and −1

On August 12, Levent Alpöge published a string of 23 828 “+” and “−” symbols, together with a decoder. They specify twelve Hadamard matrices covering every admissible order up to 2000 whose existence had remained unknown, including order 668. Alpöge said that he had worked with two colleagues and Claude. Epoch AI, the organization behind the FrontierMath benchmark of open mathematical problems, temporarily marked the problem as solved with the help of AI.

A Hadamard matrix is a square array of plus and minus signs in which the entrywise products of any two distinct rows sum to zero. Its order is the number of rows and columns.

The Hadamard conjecture states that such a matrix exists for every size divisible by four. Order 668 was the smallest case for which existence remained unknown. Mathematicians resolved the previous case, order 428, in 2004. FrontierMath posed the task of constructing a 668 × 668 array.

For the FrontierMath hypergraph problem from the spring, the problem’s author confirmed a solution found while working with GPT-5.4 Pro and began preparing a paper. The verification chain for the new result starts with a completed object: Alpöge’s string and the reply containing the decoder program.

The decoder expands the string into twelve blocks of signs, each of which specifies an array of one of the required sizes. To verify the result, one must run the program, obtain the matrices, and multiply each matrix by its transpose, which exchanges its rows and columns.

For the product involving the matrix of order 668, every diagonal entry must equal 668, confirming the length of each row. Every other entry must be zero. This verifies that the entrywise products of any two distinct rows sum to zero.

The VibeMathed website reported that it independently inspected the decoder and performed this verification for all twelve matrices using exact arithmetic. The string, the decoder, and this calculation give another mathematician a reproducible procedure for obtaining the arrays and repeating the verification.