Short Communication: The 720-Corridor Invariant within the Exceptional Triad of the Body–Tail Sieve

We report the discovery of a rigid algebraic invariant emerging from the multi-component "body–tail" residue framework under orthogonal prime field scaling. When evaluating the exceptional residue codes 49, 77, and 97 modulo 100, which are structurally linked to the properties of seven, their expansion from the baseline modular ring Z_90 to the comprehensive super-modulo ring Z_90090 yields a strict cardinality identity. For any composite target N congruent to one of these exceptional codes modulo 90, with gcd(N,30)=1, the framework structurally preserves exactly 720 valid coordinate tunnels, matching the order of the symmetric permutation group S_6. This brief note provides the formal proof using the Chinese Remainder Theorem, multiplicative group decomposition, and Euler's totient function, showing that the number of coprime extensions is phi(1001)=720. The invariant holds for each exceptional code, confirming that these codes act as perfectly uniform algebraic channels. The result enables deterministic distribution of 720 parallel threads in hardware vectorization without collision, offering practical implications for massive parallel factorization systems.