Hilbert’s Sixth Problem Revisited Through Viscous Time Theory: A Kinetic–Spectral Framework Based on Admissibility, Coherence, and Recoverability

Hilbert’s Sixth Problem challenges us to rigorously axiomatize physics, particularly the bridge between microscopic dynamics and macroscopic laws. Yet, a conceptual gap remains: probability is usually treated as a fundamental assumption rather than a derived consequence of physical evolution. To address this, we introduce a Viscous Time Theory (VTT) framework governing evolution through admissibility, coherence, and recoverability. Applying an informational action principle, probability naturally emerges as an induced statistical measure over bundles of admissible trajectories. We validate this approach by analyzing a viscous-time kinetic transport operator, mapping out its contraction semigroup structure, spectral gap, and hypocoercive convergence. We further extend the model to nonlinear interaction kernels and evaluate its hydrodynamic scaling limit. Our analysis proves this diffusion-driven operator achieves strict spectral stability, exponential entropy decay, and global nonlinear stability. Furthermore, the macroscopic scaling limit rigorously yields nonlinear diffusion dynamics for coherence density. Ultimately, this provides an analytically tractable layer connecting microscopic evolution to macroscopic behavior. It demonstrates that probability, irreversibility, and transport laws can cohesively emerge from informational geometry, advancing the structural program envisioned by Hilbert.