Algebraic Rigidity and Quadratic Residue Asymmetry in Legendre Intervals

We prove that when the length p = 2n−1 of a Legendre interval [(n−1)², n²] is prime, the p−1 interior integers form a punctured complete residue system modulo p, missing exactly one quadratic residue class. Consequently, quadratic non-residues always exceed quadratic residues by exactly 1 — an unconditional algebraic identity requiring no unproven conjectures. The theorem is computationally verified for all 549 qualifying intervals with n ≤ 2000. We complement this algebraic result with empirical measurements of the discrete Wronskian compression ratio for maximal prime-free subsequences near n², observing stabilization near q_W ≈ 1.20 across six orders of magnitude. Source code, verification data, and scripts are available at: https://github.com/Ruqing1963/legendre-spin-asymmetry This is Paper IX of the Titan Project, a programme investigating geometric and algebraic structures underlying classical prime number conjectures.