Legendre pairs of lengths 169, 195 and 205

We construct cyclic Legendre pairs of lengths 169, 195 and 205, and hence Hadamard matrices of orders 340, 392 and 412. In each pair the negative supports of the two sequences are unions of orbits of a group of multipliers, so a short table of difference counts proves the Legendre equations. At lengths 169 and 205 the second sequence is the first decimated by a square root of −1. At length 195 the sequences are chosen independently, and in no equivalent pair is one a decimation of the other. Exactly 32 normalized ordered pairs of length 195 have both sequences fixed by the multiplier group of order 12 used in the construction, and they are all equivalent. At length 205 we classify the pairs equivalent to a pair with a16i = ai and bi = aui, where u2 ≡ −1 (mod 205): there are exactly six equivalence classes. The six resulting Hadamard matrices are pairwise inequivalent, even up to transposition. Each has an automorphism group of order 4100, and the six remain pairwise inequivalent after any number of Sylvester doublings. The census, the classification and the matrix results rest on four exhaustive computations, each carried out by two separately written programs. The proofs are verified in Lean, with these four computations as the only hypotheses. The source archive contains the pairs and the six order-412 matrices, the outputs of both programs for each of the four computations, the programs that check them, the Lean development and the LaTeX source. Its README gives a command for each check; verification/check_witnesses.py checks every pair in the article and its Hadamard matrix in about a second with Python alone. The article and data are licensed under CC BY 4.0; the programs and the Lean development are licensed under MIT, as LICENSES.md in the archive states. Use of generative AI. OpenAI's ChatGPT and Codex, together with Anthropic's Claude, were used in developing the mathematics, writing and checking the programs and the Lean proofs, and preparing the text. The author is responsible for the mathematical statements, computations and references. Version 1.0.1. Corrects an out-of-date sentence on order 668 in the introduction, gives the kernel step of Theorem 7.5 the argument checked in Lean, and describes the archive contents exactly. The results are unchanged.