The Kaleidoscopic Unified Field Theory: Resolving String Entropy, Mock Modular Shadows, and Prime Determinism via the Exact Partition Formula

The discovery of an exact, closed-form algebraic equation for the unrestricted integer partition function $p(n)$ via the Kaleidoscopic Filter Theorem over F₁ provides a universal topological mechanism for discrete mathematics. By defining the polynomial volumes of the Todd class as dynamic Cauchy tensors, the formula perfectly excises the continuous sub-dimensional geometry of the partition manifold. In this paper, we elevate this exact equation to a unified mathematical framework, deploying it to resolve five historic open problems across orthogonal disciplines. We derive (I) the exact quantum micro-entropy of black holes without thermodynamic limits, (II) the topological annihilation of Ramanujan's mock theta shadows, (III) the geometric origin of partition congruences as tensor symmetries, (IV) the O(1) dimensional evaluation of the Symmetric Group Sₙ, and (V) the absolute wave-mechanical determinism of Prime Gaps, effectively replacing stochastic prime models with exact geometric interference.