Hydrodynamic Stability Hypothesis: Re-grounding Quantum Mechanics in Classical Measure Theory

This work identifies a “chronological fallacy” at the heart of quantum foundations: the axiomatization of quantum mechanics by von Neumann (1932) predated the rigorous formulation of classical probability by Kolmogorov (1933), creating an artificial schism between physics and statistics. To resolve the “spectral divergence” between the continuous ontology of hydrodynamic quantum models and the discrete phenomenology of spectroscopy, the hydrodynamic stability hypothesis is proposed. By synthesizing Madelung hydrodynamics, Radon-Nikodým measure theory, and dynamical systems theory, it is demonstrated that “quantization” is not an intrinsic property of the operator algebra, but an emergent dynamical stability phenomenon triggered by the measurement interaction. Eigenstates of the Hamiltonian act as basins of attraction for the probability fluid, making the “quantum jump” a continuous, deterministic relaxation process. This framework restores a Kolmogorovian probability structure to quantum mechanics, resolving the measurement problem without abandoning local realism or invoking discontinuous collapse.