Finite-Bath Memory, Markovianization, and Environmental Forgetting in Finite Distinction Systems: Side Records, Memory Kernels, and the Loss of Recoverable Distinctions

Abstract

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 FDS-P3 develops the environmental-forgetting paper in the physical bridge sequence of Finite Distinction Systems (FDS). It studies how finite systems depend on environments, baths, instruments, logs, niches, databases, and external carriers whose own memory is finite, partially accessible, and costly to recover. FDS-P4 studied internal coarse-grained anti-recurrence: once a non-injective truncation removes preimage information from an effective representation, later capacity recovery does not reconstruct the erased distinction unless inverse information is preserved elsewhere. FDS-P3 studies that “elsewhere.” It asks whether environmental side records remain accessible, recoverable, affordable, and timely for the finite system that needs them. The relevant object in P3 is not the full bath state, but the accessible environmental readout REt = gt(Et), induced by the observation channel, finite resolution, latency, cost, readout protocol, and accounting boundary. Environmental forgetting means loss of operational recovery relative to an accounting boundary, an observation channel, an update window, and a task tolerance; it is not a claim that microscopic information is annihilated. The paper defines environmental side records, accessible and hidden bath memory, accessible inverse information, residual-entropy recovery, Bayes recovery probability, distortion-based recovery, Markov closure error, memory-kernel burden, finite-bath saturation, environmental resource ledgers, and operational Markovianization. The main claims include environmental side-record recovery criteria, an operational Markovianization criterion, a finite-bath non-monotonicity and recurrence caveat, and a finite-bath saturation / record-collision theorem. FDS-P3 connects the FDS physical bridge sequence: P4 describes internal preimage loss, P3 describes environmental side-record loss, O3 accounts for finite record reuse and entropy/resource ledgers, P6 asks whether recovery can occur within the required update window, and P7 describes protected exceptions through invariant quotients and topological side-ledgers. The paper relates FDS-P3 to open-system physics, non-Markovianity, Mori–Zwanzig and Nakajima–Zwanzig projection methods, Markov-chain lumpability, hidden-state memory, information theory, stochastic thermodynamics, and Poincaré recurrence. It treats these bodies of theory as implementation bridges rather than replacements. Deterministic normal-form simulations are included. They illustrate finite-bath memory decay, Markov closure error, memory-kernel burden, side-record recovery probability, finite-bath record saturation, operational Markovianization regimes, environmental ledger costs, a robot external-memory worked example, and P3/P4/P7 regime structure. The figures are normal-form demonstrations, not empirical phase diagrams or fitted physical data. This release includes the paper PDF, LaTeX source, reproducibility code, generated figures, and CSV / JSON outputs.