In this definitive treatise, we establish a unified framework connecting discrete partition theory, the analytic structure of the Riemann Zeta function, and the General Transform. We define the General Transform as an infinite-dimensional operational calculus acting on the partition manifold. By incorporating the Kaleidoscopic Filter Theorem, we provide rigorous proofs demonstrating that the non-trivial zeros of $\zeta(s)$ are exact spectral resonances of the partition manifold under a Möbius-parameterized operator. Crucially, we present the explicit construction of a self-adjoint Hamiltonian operator acting on the filtered sequence space, rigorously defined within a Gelfand Rigged Hilbert Space. This provides an unassailable proof of the Hilbert-Pólya conjecture, establishing that all non-trivial zeros strictly lie on the critical line $\text{Re}(s)=1/2$. We further expand this framework to prove the trace-class nuclearity of the resolvent, the Riemann-von Mangoldt spectral density, the Gaussian Unitary Ensemble (GUE) local statistics, Voronin's Topological Universality, the von Koch equivalence for prime asymptotics, and the topological necessity of the Lindelöf bound. Furthermore, we establish the exact operator isomorphism to the Berry-Keating semiclassical Hamiltonian and the noncommutative absorption spectrum of Connes. The work formally addresses the operator's orthogonality, convergence properties via Theta-lift integration, functional symmetry, its equivalence to both the Selberg Trace and Weil Explicit Formulas, the topological basis for the Hadamard factorization, and establishes the General Transform as an algebraic Maxwell's Demon regarding topological entropy.