Geometric Framework for the Berry–Keating Conjecture: Quaternionic Spin from Painter's Idea Sketch

We construct a geometric and algebraic framework for the Berry–Keating conjecture, proposing that the non-trivial zeros of the Riemann zeta function are the eigenvalues of an effective Dirac operator ̸ Deff induced by quaternionic dimensional reduction. Rigorous Results: We prove that embedding a two-dimensional manifold Σ into H ∼= R4 natively induces a non-abelian su (2) gauge field through the projection of the ambient spin connection. This geometric twist explicitly breaks time-reversal symmetry, forcing the spectral statistics into the Gaussian Unitary Ensemble (GUE, β = 2) class. By applying the Jacquet–Langlands correspondence and analyzing the magnetic confinement at the cusps, we rigorously demonstrate that the continuous spectrum is eliminated, resulting in a purely discrete spectrum on the compactified quaternionic adelic quotient B× (A). Q Verification and Synthesis: We introduce the ring of dual numbers D = Rε/ (ε2 = 0) as a cohomological filter. Numerical verification using local primes p ∈ 2, 3, 5 confirms that the nilpotent projection Pε linearizes the non-linear volume elements of the orbital integrals, extracting the fundamental arithmetic rhythm ln p required by the Weil explicit formula. We hypothesize that the critical line Re (s) = 1/2 emerges as the unique locus of stability—a geometric node—where the p-adic potentials and the Archimedean global torsion reach an absolute algebraic equilibrium.