AI-Assisted Construction Gives a Hadamard Matrix of Order 668

On August 12, 2026, Levent Alpöge, Philippe Voinov, and Saul Reynolds-Haertle announced a construction of a Hadamard matrix of order 668, crediting Claude as part of the collaboration. A Hadamard matrix is a square $\{\pm1\}$-matrix with pairwise orthogonal rows. Order 668 had been the smallest admissible order for which no example was known.

The announcement encodes sign data rather than presenting a conventional paper. VibeMathed reports decoding the author's follow-up script and independently checking in exact integer arithmetic that the resulting matrix satisfies $HH^{\mathsf T}=668I$. The same payload reportedly yields matrices for all twelve previously unresolved admissible orders below 2000. The public credit line names Claude but does not disclose its version or specify which mathematical, computational, or search steps it contributed.

This resolves the finite existence question for order 668, not the general Hadamard conjecture that matrices exist for every positive order divisible by four. No independent expert review or conventional proof write-up was linked at the time of the report, so the exact-arithmetic reproduction should not be read as broader community validation.