Thermodynamic Boundary Conditions for the Berry-Keating Hamiltonian: A Zero-Entropy Approach to the Riemann Hypothesis.

The Riemann Hypothesis relates the non-trivial zeros of the Riemann zeta function to the eigenvalues of a self-adjoint operator. The Berry-Keating Hamiltonian, H = xp, has long been the primary candidate for this operator. However, its quantization on the real line suffers from singularities and a lack of self-adjointness due to boundary terms at infinity. In this paper, we propose a novel physical regularization: the Park-Berry-Keating Conjecture. We posit that the system must be analyzed under a thermodynamic boundary condition of zero entropy (T -> 0). In this limit, the boundary terms that break Hermiticity vanish naturally without the need for artificial regularizers. We argue that the Riemann zeros represent the ground-state spectrum of the "xp" operator in a strictly zero-entropy regime, implying that all non-trivial zeros lie on the central critical line Re(s) = 1/2. Key Concepts: The Park Limit: The theoretical premise that the "xp" operator achieves essential self-adjointness at the limit of absolute zero temperature (Zero Kelvin). Thermodynamic Regularization: Using physical entropy constraints and the third law of thermodynamics to resolve mathematical boundary instabilities in the Hilbert space.