Geometric Orbit Cancellation and Asymptotic Residual Compression in Multigrade Diophantine Equations

This work develops an analytic framework for reorganizing the multigrade moment structure of the Prouhet--Tarry--Escott (PTE) equations.  By decomposing the classical power-sum constraints into a centralized moment system and a structured quadratic orbit substitution,  we obtain a composite analytic method capable of expressing integer configurations through controlled exponential-type parametrizations.  This approach suppresses the dominant polynomial divergences that typically obstruct high-degree Diophantine analysis and provides  a rigorous pathway for evaluating integer moment balances without relying on exhaustive search.  The resulting structure offers a general analytic template for verifying and supporting integer identities in symmetric Diophantine systems.