Residue Structure of Legendre Intervals: From Punctured Systems to Higher-Order Phase Constraints

A unified survey consolidating Papers IX, X, XII, and XIII of the Titan Project into a single algebraic framework. For primes p = 2n−1, the interior of the Legendre interval [(n−1)², n²] consists of exactly p−1 consecutive integers, forming a punctured complete residue system modulo p missing the single class r ≡ 4⁻¹ (mod p). From this elementary observation, we derive a chain of exact results: (1) For any nontrivial character χ of order k | (p−1): Σ χ(x) = −χ(r) = −χ(2)ᵏ⁻², an exact character sum determined by the k-th power residue character of 2. (2) For k = 2 with p ≡ 3 (mod 4): the negation involution swaps QR ↔ NR, forcing zero quadratic skewness in each half-interval. (3) For k = 3 with p ≡ 1 (mod 6): the involution preserves cubic phases, yielding even phase counts in the right half and exactly one odd count (at the midpoint's phase) in the left half. Cubic zero-skew never occurs. (4) For k = 4 with p ≡ 5 (mod 8): the involution forces complementary pairing N(Φ₀) = N(Φ₂) and N(Φ₁) = N(Φ₃). The general criterion: the parity of (p−1)/k controls whether the involution preserves phases (even exponent → even counts) or swaps them (odd exponent → phase pairing/zero-skew). All results verified for primes up to n = 1000. This is the Phase I closing survey of the Titan Project.