The Riemann Hypothesis as a Self-Adjointness Condition

This note is a programmatic synthesis of two companion working papers. It contains no new theorems; its purpose is to state, in standard language, the structural reformulation of the Riemann hypothesis that both papers approach from different directions, and to exhibit the convergence of three independent frameworks onto a single conjectural object. The reformulation is the classical Hilbert–Pólya statement — that the non-trivial zeros are the spectrum of a self-adjoint operator — reached via a specific route: the functional-equation involution s ↦ 1−s is read as a two-pole symmetry whose balance locus is Re(s) = ½, and the requirement that the generating arithmetic operator preserve this balance is identified with self-adjointness. The three convergent frameworks are the Bost–Connes system over ℚ(ζ₄₂₀), a chiral interference model on a logarithmic spiral, and a quaternionic realisation via the SU(2)×SU(2) splitting of SO(4) and its Shimura-curve geometry. A local–global reading recasts the per-component refinement as p-adic completion and the no-drift condition on the adelic assembly as the adelic form of the hypothesis. The note is explicit that this is a reformulation and a convergence, not a proof: the single open problem common to all frameworks is the spectral identification itself.