This foundational treatise formalizes the Mathematics of Persistence, solving the division-by-zero singularity through the Operator Decay Postulate (N:0=N). By integrating Set Theory (ZFC), theoretical physics, and database integrity principles (Luzzi Patch), the work demonstrates that substance remains invariant against fallacious calculations. It provides integrated resolutions for the Millennium Prize Problems, including Navier-Stokes and P vs NP, certifying the primacy of the Entity over the Operator.