Mapping Multi-Power Local Obstructions in Higher-Degree Diophantine Equations via Orthogonal Cascades of the Body–Tail Sieve

We extend the multi-component "body–tail" residue framework to map and isolate localized modular obstructions within higher-degree Diophantine equations, specifically targeting Fermat's Last Theorem (a^n + b^n = c^n), Catalan's equation (x^p - y^q = 1), and Beal's Conjecture (A^x + B^y = C^z). By scaling the orthogonal sieve matrix up to the ultra-large ring perimeter 601,681,470,390 (which is the product of 90 and the primes 7, 11, 13, 17, 19, 23, 29, and 31), we analyze the interlocking trajectories of asymmetric power residue spaces over compounded finite fields. Experimental tracking via Chinese Remainder Theorem (CRT) deconstructions establishes that while trivial and quasi-trivial modular collisions persist due to invariant subgroup identity boundaries, the sieve forces strict geometric polarization. This local stabilization isolates valid modular intersections into ultra-sparse coordinate tracks, creating a comprehensive necessary perimeter for non-trivial integer solutions across generalized exponential spaces.