Finite Realism II: Sheaf-Theoretic and Categorical Formalization

We articulate a compact framework that turns minimal operational theses into a working metaphysics and epistemology. The primitives are records, contexts, channels, interventions; all further ontic commitments are earned by their power to compress records and enable counterfactual control under finite resources. Inquiry is organized as a site of contexts with effect algebras and admissible state spaces; channels compare by decision-theoretic sufficiency (Blackwell order); interventions are structural edits on causal mechanisms. Adequacy is measured by a compression–control functional: two-part MDL plus penalties for violating intervention invariants (with observational surrogates when interventions are unavailable). Spatiotemporal structure is reconstructed as invariants of response: order ⇒ a conformal class; a fastest-mode/clock/volume normalization fixes a representative metric when a single cone field dominates. We state five representation results under explicit operational assumptions: (T1) contextual obstruction forbids a global valuation; (T2) causal order determines a conformal light-cone; (T3) intersubjective merging of posteriors under truthful, capacity-limited exchange, with policy convergence under shared decision problems; (T4) compression-with-control bounds out-of-context regret with an explicit ∆shift; (T5) divergences diagnose window failure. A small end-to-end toy example (sensor-network latency geometry) illustrates the full stack.