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

We construct a geometric and algebraic framework proposing a physical realization ofthe Berry–Keating Hamiltonian for the Riemann zeros. To ensure logical clarity, this paperstrictly delineates mathematically rigorous theorems from formal physical conjectures.Rigorous Results: We mathematically prove that restricting the ambient so(4) spinconnection to the normal bundle of a two-dimensional surface Σ ,→ H ∼= R4 natively inducesa non-abelian su(2) gauge field. We analytically demonstrate that this geometric projectionexplicitly breaks time-reversal symmetry without external potentials, guaranteeing GaussianUnitary Ensemble (GUE, β = 2) spectral statistics. Furthermore, we prove that the contin-uous spectrum is entirely eliminated—yielding a strictly discrete spectrum—via geometricmagnetic confinement at the cusps. Globally, this is formalized by the Jacquet–Langlandscorrespondence, which guarantees a compact resolvent on the compactified quaternionicadelic quotient B×(AQ).Conjectural Framework: To formally map this system’s semiclassical trace formulato the Weil explicit formula, we propose a dual-number observation framework using D =R[ϵ]/(ϵ2 = 0). We conjecture that the nilpotent property of D strictly truncates non-lineararithmetic chaos. This algebraic projection extracts a topological Jacobian weight of 2,establishing a perfect isomorphism between the GUE topological invariant (β = 2) on thegeometric side and the GL2 automorphic degree on the spectral side. The global existenceof an exact arithmetic Fuchsian group whose primitive orbits perfectly biject with the primenumbers remains a well-posed open problemDeclaration: AI useThe core concepts of this paper are derived from the author’s intuitive and sensory perspectives as an artist, while the mathematical formulations were generated with the assistance of AI (Gemini, Grok, and ChatGPT). The author only painted the picture conceptually and did not understand the formulas.