The Exact Spectral Realization of the Riemann Zeta Function: A Geometric Proof of the Riemann Hypothesis via A₊-₁ Lattice Cohomology, Integer Partitions, and the Bonelli Operator

We present a rigorous proof of the Riemann Hypothesis derived from thegeometric and spectral analysis of the Ak−1 root lattice. We introduce the BonelliOperator Bk(n) an exact, completely closed-form structural identity derived via theEhrhart foliation of rational polytopes and partial fraction decomposition over cyclotomicfields. By resolving the periodic convolution tail via discrete Faulhaber summation, thisoperator strictly evaluates the discrete lattice volume in universal O(1) complexity withrespect to n, acting as a spectral sieve isomorphic to the M¨obius inversion. We provethat the Dirichlet polynomials generated by this exact operator satisfy an exact selfreciprocalfunctional equation and belong, in the thermodynamic scaling limit k ∼√n tothe Laguerre P´olya class of entire functions. Utilizing the spectral theory of self-adjointgeometric Laplacians and the theory of Mellin transforms, we demonstrate that thezeros of these operators are strictly confined to the critical line ℜ(s) = 1/2. By applyingFourier analysis and the Gutzwiller Trace Formula directly to the analytically closedBonelli Identity, we confirm that the constructive interference of the Weyl reflectionsgenerates a spectral density identical to the Riemann-von Mangoldt formula. Finally,we establish an explicit isomorphism between the geometric partition spectrum and theBosonic Fock Space, natively generating the Euler Product and verifying the functionalequation Ξ(s) = Ξ(1−s) via Ehrhart-Macdonald Reciprocity, definitively validating theHilbert-P´olya conjecture.