Entropic Temporal-Interface Complexity on Classical Law Spaces with Quantum-Compatible Extensions

This preprint develops a two-component notion of entropic temporal interface complexity (TEC) for discrete-time law spaces describing stochastic dynamical systems observed through a finite interface. A “law” is formalised as a trajectory of probability measures on an interface alphabet, while implementations are arbitrary Markovian systems with hidden internal state. On each law space the paper assumes a general family of divergences DX (μ∥ν) DX (\|) DX (μ∥ν) satisfying positivity and mild lower–semicontinuity conditions, so that the construction is not tied to a specific choice such as relative entropy. From this setup the work defines an interface complexity rate CifaceC₈₅₀₂₄Ciface that quantifies per-step entropic reshaping of the observable interface distribution, and a temporal complexity rate KtimeKₓ₈₌₄Ktime that measures the average cost of deforming the law along a discrete law-time semigroup. These two components are combined into TEC, which is shown to be invariant under a broad class of implementation changes and stable under coarse-graining of the interface. The framework therefore separates what an external observer can see at the interface from how complex the internal temporal reshaping of laws must be. The paper then couples TEC to recent notions of semantic information based on viability. For a given choice of viability variable, it defines semantic information rates and semantic efficiencies that compare task-relevant information directly to interface and temporal complexity. Under explicit energetic comparability assumptions, temporal complexity is related to entropy-production-like functionals in finite-state reversible Markov jump models, yielding physical notions of semantic efficiency measured in “semantic information per unit dissipated energy”. Although the main development is classical, the same axioms extend to quantum law spaces by replacing probability measures with density operators and the divergence family with suitable quantum divergences. This shows that TEC is compatible with quantum channels and quantum instruments, and can serve as a bridge between classical interface descriptions and quantum-level implementations in self-organising or self-improving systems.