The Simplicial Geometry Of Weyl Partitions: A Geometric And Spectral Proof Of The Riemann Hypothesis Via Ak−1 Root Systems

This paper presents a geometric and spectral resolution of the Riemann Hypothesis(RH) through the discrete topology of Weyl partitions. We demonstrate thatthe distribution of non-trivial Zeta zeros is the deterministic manifestation of topologicalsymmetry within the Ak−1 root lattice. By introducing the Kaleidoscopic Filter, weisolate the structural fluctuations of the restricted partition manifold. We prove that thegenerating function of this discrete geometric defect condenses strictly into a finite, selfreciprocalpolynomial via Ehrhart-Macdonald Reciprocity and Faulhaber summation.Building upon this foundation, we present a 5-step synthesis that reduces the RiemannHypothesis to the spectral theory of symmetric operators. By mapping the Weyl reflectionsto a real symmetric matrix acting on the profinite compactification ˆZ, we invokethe Spectral Theorem to guarantee real eigenvalues. Furthermore, we demonstrate thatthe geometric invariance of the discrete Laplacian on this Arakelov-Adelic lattice rigidlyimposes the exact dispersion relation λ = s(1−s). The collision of these topological andanalytic constraints strictly confines the non-trivial zeros to the critical line ℜ(s) = 1/2.