A Frobenius obstruction to fixed quadratic-character constructions of Legendre pairs

A Legendre pair of odd length n consists of two ±1 sequences whose periodic autocorrelations add to −2 at every nonzero shift. At a prime length the quadratic character gives such a pair. We ask whether a formula of that kind can give Legendre pairs of length kp for a fixed odd k > 1 and every prime p. Index the positions by ℤ/kℤ × Fₚ and fix a finite set D of algebraic numbers. On each of the 2k columns {i} × Fₚ of the two sequences, allow any Boolean function of the quadratic characters χₚ(t − c) for c ∈ D reduced modulo p, with free values where a character vanishes; all these choices may change with the column and with p. We prove that for a set of primes of positive density no such pair is a Legendre pair. At these primes Frobenius acts as one fixed scalar on the ℓ-torsion of finitely many hyperelliptic Jacobians, for an auxiliary prime ℓ; such primes exist by Serre's theorem on homotheties and Chebotarev's theorem. There the pair equations would force two ±1 sequences of length k whose periodic autocorrelations add to 0 at every nonzero shift, which is impossible for odd k > 1. The result extends to finitely many nonzero rational functions in place of the linear factors t − c and to one common affine change of coordinate at each prime. For |D| ≤ 2 an elementary count excludes every large prime. The theorem excludes one Frobenius class and leaves the other primes open. The proof names no excluded prime and gives no bound for the least one. No formal proof of any result is claimed. This record holds the article and its source archive. The source archive contains the TeX source and bibliography, the claim map verification/claims.json, the finite checks, the scripts that build and inspect the PDF, and a SHA-256 manifest of its files. The finite checks are four Python programs and a runner. They run with Python 3.11 or later and only its standard library on a POSIX system, and the archive's README gives the command. They test the article's elementary identities and small models on fixed finite domains. They do not test Serre's theorem, the construction of the Frobenius class or the main theorem, and they compute no torsion field, Serre exponent, numerical density or obstructing prime. The article text, bibliography, prose and data are licensed under CC BY 4.0, and the original programs and TeX support under the MIT License, as LICENSES.md states; the bundled copy of pypdf keeps its BSD 3-Clause license. Use of generative AI. Generative AI tools, including ChatGPT, Codex and Claude, assisted with the mathematics, proofs, programs and text. The author is responsible for all content.