Claude Sweeps All Hadamard Matrices Below Order 2000: AI Starts Clearing the Math Open Problems List

Anthropic researcher and mathematician Levent Alpöge reported finding a solution for Hadamard matrices of order 668, crediting a team of three humans and Claude. The team actually produced 12 matrices, clearing all unresolved Hadamard matrix orders below 2000: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964. The Hadamard matrix problem involves constructing a square matrix with entries +1 and -1 such that any two rows are orthogonal. Before this, order 668 was the smallest unknown order since 2005, and it had stumped mathematicians for 30 years. The problem was included in the FrontierMath benchmark, a set of 50 open problems created by EpochAI with input from Fields Medalists and IMO coaches. This is the fourth open problem in FrontierMath solved by AI. The verification was done by reproducing the decoding process and checking the matrices with exact integer arithmetic. EpochAI has temporarily marked the problem as AI-solved but notes that it cannot determine whether the results come from an improved search strategy or a generalizable construction. The specific method and whether it can extend beyond order 2000 await the full technical report.