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.
Benedek Blackthorn· Zenodo (CERN European Organi...· 0 citations
A Legendre pair consists of two cyclic sign sequences whose periodic autocorrelations sum to −2 at every nonzero shift. An explicit pair of length 117 is constructed whose sums over equal residues modulo 13 are 1 at residue zero and respectively 3χ(r) and −3χ(r) elsewhere, where χ is the quadratic character modulo 13. This verifies one case of Kotsireas’s prescribed lifting conjecture, without establishing the general conjecture or classifying all lifts. Both sequences are fixed by an involution that reflects one Chinese remainder coordinate. A short correlation table proves the pair identities and gives a Hadamard matrix of order 236. The source archive contains the literal rows, manuscript and Lean sources, and parent and lift-search programs. 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 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.
Benedek Blackthorn· Zenodo (CERN European Organi...· 0 citations
This record accompanies A prime-square multiplier obstruction for Legendre pairs, version 1.3.0. A Legendre pair consists of two cyclic sign sequences whose periodic autocorrelations sum to −2 at every nonzero shift. For odd length n = p²q, where p is an odd prime and q is a positive integer, neither row of a Legendre pair can be invariant under i ↦ (1 + pq)i + c, for any offset c modulo n. No symmetry is required of the companion row. A short odd-cycle argument gives individually sharp autocorrelation bounds, including for sequences of sum 1. The general direct obstruction was previously stated publicly under the GitHub handle byclaude. This note supplies a self-contained proof with sharp bounds, derives the affine consequence, and provides formal verification. Applications include lengths 117 and 333. The result excludes specified symmetries; it neither decides existence of Legendre pairs at length 333 nor refutes the prescribed-compression conjecture. Version 1.3.0 revises the exposition, attribution, and explanation of scope. The mathematical statements and Lean proofs are unchanged from version 1.2.0. Files include the article PDF, reader package, formal-verification package, arXiv source archive, and build record. The accompanying Lean 4 development formally verifies every original mathematical result stated in the article. Literature priority and the cited subgroup-table transcription remain outside that verification. The article discloses generative-AI use. Licensing is file-class specific: article and documentation under CC BY 4.0; original executable, build, and verification material under MIT. Factual and third-party material is not relicensed. See the packages’ license map.
Benedek Blackthorn· Zenodo (CERN European Organi...· 0 citations
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