Higher-Order Residue Deficits in Legendre Intervals

We generalize the Spin Asymmetry Theorem (Paper IX, Zenodo: 18706876) from quadratic residues to arbitrary k-th power residues in Legendre intervals. For p = 2n−1 prime and k | (p−1), the interior of (n−1) ², n² exhibits a universal phase deficit: the k-th power residue class containing r ≡ 4⁻¹ (mod p) has exactly one fewer representative than each other class. Since r ≡ 4⁻¹, the deficient phase is determined by the k-th power character of 2: - k = 2 (quadratic): deficit always in Φ₀, recovering the Spin Asymmetry Theorem. - k = 3 (cubic): deficient phase = χ₃ (2), determined by the representation p = a² + 27b². - k = 4 (quartic): deficit in Φ₀ when p ≡ 1 (mod 8), in Φ₂ when p ≡ 5 (mod 8) ; the phases Φ₁, Φ₃ are never deficient. For any nontrivial character χ of order k: Σ χ (x) = −χ (2) ^k−2 over the interior, an exact character sum evaluation. Verified for k = 2 (549/549), k = 3 (148/148), k = 4 (146/146), k = 5 (73/73), all primes up to n = 1000. Source code: https: //github. com/Ruqing1963/higher-order-residue-deficits This is Paper XII of the Titan Project.