Thermodynamic Boundary Conditions for the xp Hamiltonian: A Zero-Entropy Approach to the Riemann Hypothesis

Description: This work formalizes the Park-Berry-Keating framework, providing a rigorous resolution to the historical non-normalizability paradox of the xp operator. By defining a weighted Hilbert space HP under the Park measure (dmu = exp (-beta * x) dx), we demonstrate that the eigenfunctions of the Berry-Keating Hamiltonian achieve a finite norm, establishing essential self-adjointness in the limit of vanishing entropy (T -> 0). Key Contributions: Symmetry Restoration: Introduction of the Park-Adjusted Operator (H-beta), which includes the "Park Term" (-ihbar * beta * x / 2) to compensate for the weighted measure and ensure strictly real eigenvalues. Numerical Validation: Computation of the convergence of the norm for various values of the inverse temperature (beta), proving spectral stabilization through the exponential integral E1 (beta). Thermodynamic Framework: A new perspective on the Hilbert-Pólya program, treating Riemann zeros as stable asymptotic states at absolute zero temperature. This framework builds upon and evolves the foundational models of Berry & Keating (1999) and the non-commutative geometry approach of Alain Connes (1999), providing a consistent physical environment for the distribution of non-trivial zeros on the critical line Re (s) = 1/2.