Hook Counterexamples to Sign-Row Extremality in Symmetric-Group Character Tables, and an Exact Phase Representation of the Sign Bias

We investigate a question posed by S. Hopkins (OPAC-038) on whether the sign representation maximizes the number of negative entries in the character table of the symmetric group S_n for all sufficiently large n. We present certified exact counterexamples to this extremality conjecture across fifteen values of n, beginning at n = 112 and extending through the complete interval 127 ≤ n ≤ 134. These counterexamples isolate a narrow, structured regime of proportional hooks (n - a, 1^a) with a/n ≈ 0.65. For n ≡ 1 (mod 3), the counterexamples coincide exactly with a shifted hook family η_m = (m + 2, 1²ᵐ⁻¹) ⊢ 3m + 1. To analyze this family, we prove a series of exact structural identities:(1) Even parts of the conjugacy class act as finite differences on a positive subset-sum profile.(2) On odd classes, cycle-deletion reduces the character to an explicit cubic polynomial in small multiplicities, isolating a resonance family that guarantees a large lower bound of negative entries.(3) The character sign can be mapped to an exact phase representation where the signed row bias corresponds to the odd-harmonic content of a canonical complex number. An alignment identity shows that for η_m, the third harmonic of the phase is invariant under variations in both the sign and the number of parts of the partition. Finally, we provide high-precision uniform sampling data up to n = 601 to track the decay rate of the zero density versus the third-harmonic bias, highlighting the role of phase rotation in determining whether these violations persist indefinitely or vanish asymptotically.