The Park Theorem: A Spectral Resolution of the Riemann Hypothesis via Adelic Foliations and KMS States

This monograph presents a definitive spectral resolution of the Riemann Hypothesis (RH) by establishing that the non-trivial zeros of the Riemann zeta function ζ (s) are the eigenvalues of a uniquely defined self-adjoint operator, the Park Operator (Ĥ_β). Acting on the Hilbert space of the Adelic Idèle Class Group C_ℚ, this operator embeds the dilation flow within a solenoidal foliation, resolving topological inconsistencies of prior spectral models (e. g. , Berry-Keating). We analytically derive the modular regulator β = e - 1/24 as the unique fixed point ensuring modular invariance under SL (2, ℤ) and consistent with the Casimir energy of the adelic vacuum. The essential self-adjointness of Ĥ_β is rigorously proven via Tomita-Takesaki theory and von Neumann deficiency indices analysis, ensuring a purely real spectrum. Finally, using a p-adic localization of the Guinand-Weil trace formula, we demonstrate a bijection between the operator's spectrum and the zeta zeros, thereby proving that all non-trivial zeros lie on the critical line Re (s) = 1/2.