Actuation-Resolution Language Collapse and Exact Causal Power in a Relational Heat Network

The paper establishes an exact theory of temporal observation-language realizability in physically coupled thermal networks, showing that actuator time resolution can fundamentally change which relational behaviors are possible, independently of the total available power. Using a quantitative, region-choice-independent projection of EF–FG terminal records, it proves a sharp language-collapse phenomenon: in a protected six-node thermal system, one constant-input interval per observation permits no words of length two or greater at any finite power, whereas two half-period intervals enable every nine-symbol word of arbitrary finite length. The latter result is achieved by a causal controller with an exact minimum all-word power of approximately 201.131346, establishing equality between online and offline power requirements. The theory extends to arbitrarily large physically coupled networks, heterogeneous modules, and specified parameter perturbations. Its strongest further contribution is a complete all-horizon language characterization for a nontrivially cross-coupled twelve-node system, including a rigorously certified critical conductance threshold and a power-independent obstruction under one-piece actuation. Under two-piece actuation, an explicit memoryless controller with 81 actions realizes all 81^n independently prescribed vector words for every length n, with a certified sufficient power budget of 421 and rigorous robustness margins. Its main novelty is the discovery and exact characterization of an actuation-resolution phase transition in physical observation languages, connecting causal control, infinite-horizon realizability, sharp resource obstructions, and certified thermal-network dynamics. The results concern precisely defined sampled relational projections rather than unrestricted realization of the complete EF–FG semantics.