A counterexample to Sun's sign conjecture for cotangent permanents

For each odd prime p, Zhi-Wei Sun proved that the normalized cotangent permanent t′p is an integer and conjectured that χp(−1)t′p < 0, where χp denotes the Legendre symbol.For p = 29, we obtain t′29 = 84,643,985,981,440 > 0. Since χ29(−1) = 1, the cotangent inequality in Sun's Conjecture 4.7(ii) fails.We determine the value by specialization to finite fields and the Chinese remainder theorem, using an explicit Archimedean bound to ensure uniqueness. A separate calculation in Z[X]/(1 + X + ⋯ + X^28) verifies the corresponding cyclotomic identity coefficientwise.Preprint version v1.0.0, 2026-08-26. Author and correspondence: Yuhang Shi, [email protected] Codex was used for literature screening, development and checking of the exact verification programs, and language editing. Responsibility for the mathematical claims and the final manuscript rests with the author.