Nonexistence of Williamson matrices of order 65

We prove by exhaustive computation that Williamson matrices of order $65$ do not exist: no four symmetric circulant sign matrices of that order have squares summing to $260$ times the identity. Compression to length $13$, common multipliers, conservative power spectral density bounds and Williamson's product identity reduce the search to matching pairs of full sequences. The CPU computation took about $271$ CPU-hours. Two checkers verify the compressed enumeration: one reuses the search's row generator, and one regenerates rows with separately written code. They share the compression model and per-representative enumeration records; neither repeats the decisive full-row lift search. Full-pipeline controls reproduce known smaller-order counts. The search is computational under stated arithmetic assumptions and is not formally verified. This work has not yet been reviewed by experts. The proofs, the search code, the enumeration checkers, the witness checkers and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction.

Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.