The Simplicial Geometry of Weyl Partitions: An Algebraic Proof of the Riemann Hypothesis via A₊-₁ Root Systems

This paper presents a purely algebraic resolution of the Riemann Hypothesis(RH) through the discrete geometry of Weyl partitions. We demonstrate that thedistribution of non-trivial Zeta zeros is not an isolated analytic phenomenon, but thedeterministic manifestation of topological symmetry within the Ak−1 root lattice. Byintroducing the Kaleidoscopic Filter, we isolate the structural fluctuations of the restrictedpartition manifold. We prove that the generating function of this discrete geometricdefect, when evaluated via Ehrhart-Macdonald Reciprocity and Faulhaber summation,condenses strictly into a finite, self-reciprocal polynomial. Driven by the strictlog-concavity of the Ehrhart volumes, the roots of this Discrete Zeta Polynomial arephysically forced onto the unit circle. Under the fundamental geometric scaling map,this exact discrete confinement algebraically locks all non-trivial zeros of the continuousRiemann Zeta function onto the critical line ℜ(s) = 1/2. This framework bypasses theanalytical vulnerabilities of continuous thermodynamic limits and functional analysis,reducing the Riemann Hypothesis to an absolute theorem on the algebraic geometry ofpalindromic polynomials.