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 = R[ε]/(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.Declaration: AI useThe core geometric and number-theoretic approaches of this paper—such as the restriction of the so(4) spin connection and the introduction of dual numbers—were designed based on the author's visual intuition as an artist. In the process of translating and formulating this conceptual framework into a rigorous mathematical format, large language models (Gemini, Grok, and ChatGPT) were utilized as auxiliary tools for computation and drafting. The author's primary objective is to provide the visual and conceptual architecture of the theory; the author cannot assume responsibility for the mathematical calculations. The author declares no conflicts of interest.