Hubble-Coupled Entropy and the Emergence of Thermodynamic Time: A Proposed KMS Framework for Cosmic Modular Flow

We present a framework in which time and temperature emerge as successive deriva- tives of entropy defined over probability space, with the proportionality constant determined by the Hubble parameter H. Replacing the static Boltzmann constant with the dimensionless ratio H/H0 dynamically couples entropy to cosmic expansion, transforming it from a static scalar into a generative quantity. The transition from decelerating to accelerating expansion — occurring at redshift z ≈0.5 — marks the boundary condition for positive entropy production and the emer- gence of thermodynamic time. Prior to this transition, the mathematical structure of physical law existed without the derivative relationships that constitute temporal flow. We further show that the cosmic microwave background defines a universal KMS state, and that the Hubble parameter serves as the modular generator breaking the time-reversal symmetry of that state. This suggests that time is not fundamental but emergent from the curvature of probability space under specific cosmological boundary conditions, with H/H0 serving as the fundamental coupling between cosmic dynamics and local thermodynamics.