Active Finite Distinction Systems: A Formal Core for Boundary Maintenance under Finite Capacity

Paper also available at https://philarchive.org/rec/WUAFDP Official website: distinctiontheory.orgPublic portal for the start guide, papers, claim status, failure registry, prior-art boundary, and citation resources. Canonical GitHub repository:https://github.com/yiningwu-research/Distinction-Theory Formal Core Edition v1.0. This manuscript defines Active Finite Distinction Systems (FDS) as finite-capacity systems that maintain operational boundaries through state-dependent updates under representational, energetic, and temporal constraints. It extracts the formal FDS core from the broader Distinction Theory claim-space archive and presents it as a restricted mathematical framework rather than as an undifferentiated theory of everything. The central object is the twelve-component FDS tuple S = (X, E, B, M, Y, A, U, pi, ell, Phi, P, tau), where the components represent internal state, environment, boundary, memory, observation, action, update, finite projection, boundary-maintenance loss, resource budget, perturbation/pruning family, and update timescale. The manuscript develops the following formal and conditional components: the distinction primitive and its finite-system restriction; active and passive boundary criteria; internal representational capacity and finite distinction representation; capacity deficit as a task-relevant rate-distortion gap; conditional approximation proliferation under fixed coding families; a Landauer dissipation floor under explicit physical bridge assumptions; free-energy budgets and the prune-externalize-collapse trichotomy; invariant-supported persistence through quotient maps and perturbation families; normal-form Phase-A / Phase-B dynamics and failure modes; a protocol for quarantined domain applications. The document separates three layers: Formal FDS Core: definitions and conditional mathematical theorems about finite distinction systems. Physical Bridge: assumptions under which formal updates are realized by physical substrates coupled to thermal environments. Quarantined Applications: biology, cognition, artificial intelligence, civilization, geometry, black holes, and other domains, treated only through explicit bridge assumptions and operational tests. This manuscript does not claim that the distinction primitive alone derives thermodynamics, biology, cognition, artificial agency, spacetime geometry, black holes, or civilization. Those claims require additional finite-system restrictions, active-boundary hypotheses, physical bridge assumptions, and domain-specific mappings. Failure of a physical bridge or domain application does not propagate back to the formal FDS core unless a stated formal hypothesis is itself contradicted. This work is the formal FDS Core companion to: Wu, Y. Distinction Theory: A General Theory of Finite Systems. Zenodo, 2026. DOI: 10.5281/zenodo.20130174.