Thermodynamics from Continuation Structure: A Structural Analogue of the Second Law in Temporal Rate Ontology

The second law of thermodynamics asserts that entropy increases in isolated systems, yet its foundations remain contested. Standard approaches ground the second law in probabilistic typicality arguments over microstates or in low-entropy boundary conditions (Albert 2000; Price 1996; Callender 2004). This paper proposes a structural analogue of the second law within Temporal Rate Ontology (TRO), in which physical systems are described in terms of admissible continuation structures. We define structural entropy as the logarithm of finite-horizon continuation multiplicity and prove a local growth condition: under the Principle of Maximal Freedom (PMF), structural entropy along PMF-selected trajectories satisfies S₋-₁ (sₓ+₁) >= SL (sₜ) - log|ℬ (sₜ) |, where |ℬ (sₜ) | is the local branching factor. The result is situated carefully: it is a structural property of PMF-selected trajectories, not a derivation of the thermodynamic second law, and three open problems (typicality, the W/ΩL correspondence, and the thermodynamic limit) are explicitly identified. The paper argues that the TRO structural entropy provides a non-probabilistic interpretive reframing of entropy increase as continuation-space expansion rather than as a brute thermodynamic fact or a consequence of probabilistic postulates.