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

The Park-Berry-Keating Conjecture introduces a critical formalization of the underlying functional analysis. We address the long-standing "non-normalizability" critique of the xp operator by defining a weighted Hilbert space HP (The Park Space) with a thermally regulated measure dmu_beta = e^-beta*x dx. We demonstrate that under the Park Limit (T -> 0), the eigenfunctions achieve a finite norm proportional to the exponential integral E1(beta), thereby satisfying the requirements for essential self-adjointness. This thermodynamic regularization ensures that the eigenvalues E are strictly real, providing a physical and mathematical framework for the Hilbert-Pólya program and the Riemann Hypothesis.