A prime-square multiplier obstruction for Legendre pairs
Abstract
A Legendre pair of odd length n consists of two cyclic sign sequences whose periodic autocorrelations sum to −2 at every nonzero shift; it yields a Hadamard matrix of order 2n + 2. Let p be an odd prime and q a positive integer, with n = p²q odd. For every offset c modulo n, neither sequence can be invariant under the affine map i ↦ (1 + pq)i + c. An odd-cycle argument gives autocorrelation bounds that are individually sharp even for sequences of sum 1. The result excludes specified symmetries in searches for Legendre pairs, including at lengths 117 and 333; it does not decide existence at length 333. The general direct obstruction was previously stated publicly under the GitHub handle byclaude. This article gives a self-contained proof with sharp bounds and derives the affine consequence. The source archive contains the manuscript, Lean formalization, claim map and verification instructions. The article and explanatory text are licensed under CC BY 4.0; original code and Lean sources are licensed under MIT. Third-party terms are retained as described in LICENSES.md. 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 preparing the text. The author is responsible for the mathematical statements, computations and references.
