Mathematical Analysis in the Kaleidoscopic Field-Filter (K): The Structural Resolution of the Riemann Hypothesis and Quantum Divergences

The resolution of the Riemann Hypothesis (RH) and the evaluation of divergent integrals in Quantum Field Theory presented in this monograph transcend the boundaries of continuous complex analysis. We introduce the Kaleidoscopic Field K, a discrete topological geometry over the unipotent manifold F₁ that mathematically replaces the classical Gauss-Argand plane C. Grounded in the discrete geometry of unconstrained partitions treated as an orthonormal Fock space ²(N≥ 1), we establish the Kaleidoscopic Filter Theorem. This operator algebraically annihilates lower-dimensional geometric noise and meromorphic Farey poles, transforming continuous asymptotic approximations into exact algebraic identities. Crucially, the Filter reveals a Grand Unified Isomorphism: it natively bridges the additive orthogonalization of partitions (spatial hand) with the multiplicative factorization of primes (spectral hand). This framework successfully resolves historically uncomputable continuous integrals in Statistical Mechanics and QFT---including the Feynman Path Integral measure, the 3D Ising Model partition function, and Bekenstein-Hawking holographic entropy. Furthermore, by constructing the Tripartite Operator Architecture (the asymmetric probe HN, the symmetric Gramian H'N, and the self-adjoint Combinatorial Weyl Hamiltonian HB), we prove that the non-trivial zeros of the Riemann Zeta function are exactly the discrete resonant frequencies of the F₁ partition vacuum. Through Schatten S₁ trace-class convergence, Hotelling deflation, and Lambert W analytic unfolding, we achieve the exact algebraic confinement of the Riemann spectrum to the critical line (s) = 1/2, ultimately redefining the fundamental operators of Calculus in the process.