Thermodynamic Manifold Framework (TMF): Physical Realization of Formal Relational Dynamics in Thermodynamic State Space

The Thermodynamic Manifold Framework (TMF) reformulates physical systems as evolving probability distributions mu (W, x, t) over a thermodynamic state space W = (T, p, rho, Yᵢ, phi), rather than as trajectories of mean-field variables. A composite operator Tₜheta = N ∘ Sₑps ∘ Eₑta ∘ Wₛigma ∘ Dₐlpha encodes dissipation, mixing, efficiency modulation, and survival threshold. TMF is identified as a physical realization of Formal Relational Dynamics (FRD; Tatai, 2026a; DOI: 10. 5281/zenodo. 19728746): the Wasserstein-2 distance on the space of thermodynamic distributions provides the relational metric, and the state-dependent dissipation functional kappa (mu) realizes the FRD contraction parameter. Key results: (1) Internal time tau (mu) = kappa (mu) ^-1 emerges from the distribution — entropy increase corresponds to internal time acceleration. (2) The Otto and Diesel thermodynamic cycles are analyzed stroke-by-stroke in distribution space: irreversibility appears as a geometric gap between forward and return manifold paths, invisible in P-V diagrams. The kappa (mu) profile of the Otto cycle reveals three distinct regimes; the isochoric heat addition is identified as the primary irreversibility event. (3) Classical continuum equations (Navier-Stokes, Fourier, Fick) are sketched as mean-field projections of the TMF evolution equation. (4) Five open problems are stated precisely, continuing the series from FRD. Four interactive HTML simulators accompany this paper as supplementary material. All run in any modern browser without installation. This is a working paper / preprint, not peer-reviewed. Results are marked as PROVEN, CONJECTURE, NUMERICAL, or OPEN throughout. This paper is the second in a series; the first is FRD (DOI: 10. 5281/zenodo. 19728746).