Intrinsic Phase Space Locking: A Physical Derivation of the Riemann Spectrum via Heisenberg Constraints

The Riemann Hypothesis remains one of the most significant open problems in mathematics,with deep connections to the spectral theory of quantum chaotic systems. TheBerry-Keating conjecture proposes that the Riemann zeros correspond to the eigenvalues ofa Hamiltonian H = xp, but this model suffers from inherent singularities in classical phasespace. In this paper, we demonstrate that these singularities are naturally regularized bythe Heisenberg Uncertainty Principle. We introduce the concept of “Intrinsic Phase SpaceLocking,” a mechanism where the quantum volume is constrained to Planck cells, forcing thecontinuous spectrum to discretize into the Riemann zeros. Unlike standard approaches thatassume deficiency indices of (1, 1), we demonstrate that the strict application of the Heisenberglimit imposes a (2, 2) deficiency index structure. By employing Wronskian boundaryanalysis and the Principle of Least Action, we derive an exact quantization condition thatreproduces the imaginary parts of the Riemann zeros without asymptotic error terms.