Abstract We introduce the category C_SRNUDT — the mathematical formalization of Scale-Recursive Non-Uniform Domain Tiling — and prove five foundational propositions establishing that the framework's organizational axioms follow necessarily from the category's definition. We then prove the SRNUDT Boundary Theorem: in any SRNUDT system with symmetry group G, the remainder cohomology class [R] ∈ H¹(F) localizes to the fixed-point set Fix(G) of the domain's symmetry group, which lies on the organizational boundary between sovereign domains. Remainder cannot reside in domain interiors. The proof uses the Atiyah-Segal localization theorem for G-equivariant cohomology. Two Millennium Prize problems follow as CONDITIONAL corollaries: the Riemann Hypothesis (conditional on identification of Möbius poles as the prime SRNUDT remainder class) and Navier-Stokes global regularity (conditional on TCC being the correct Planck-scale substrate). Both conditionalities are precisely stated open problems, not vague gaps. The Boundary Theorem is the master result from which both follow by direct application.