Companion note : Mathematical consequences on the Millenium Problems

Part II explores the consequences of the q-dRIS structure for the foundational problems of mathematical physics. The bounded spectral domain, the holomorphic dependence on the frequency parameter, and the rigidity of the master operator impose analytic constraints that are incompatible with the pathologies underlying the classical formulations of the Millennium Problems. In this corrected framework, singularities, divergences, and combinatorial explosions arise not from intrinsic mathematical difficulty, but from the infinite-resolution assumption removed in Part I. We show that : — the Navier–Stokes equations admit global existence and smoothness under the bounded spectral structure induced by h0 ; — the Yang–Mills mass gap emerges from the doubled spectral geometry and the rigidity of the transfer operator; — the Hodge conjecture becomes a structural consequence of finite resolution and analytic closure; — theclassicaldistinctionbetweenP andNP collapsesintheq-dRIScomputationalstructure, where exponential combinatorics cannot be physically realized. Together,theseresultsdemonstratethattheq-dRISframeworkresolvestheanalyticandstructural inconsistencies at the root of the Millennium Problems, not by altering their statements, but by placing them in the only mathematically coherent setting for a universe with finite resolution and causal evolution.