Coarse-Grained Anti-Recurrence and Informational Hysteresis in Finite Memory Systems: Lost Preimages, Side Records, and Capacity-Recovery Asymmetry

FDS-P4 develops a finite-memory account of coarse-grained anti-recurrence and informational hysteresis. It studies what happens when a finite effective representation applies a non-injective truncation: once preimage information is absent from the effective record, later capacity recovery does not reconstruct the discarded distinctions. The central claim is representation-relative, not metaphysically irreversible. P4 does not deny microscopic reversibility, unitary dynamics, Poincaré recurrence, reversible computation, or fluctuation theorems under their own assumptions. It asks a narrower operational question: if exact preimage recovery is later possible, where was the inverse information stored, and what accounting boundary carries it? The paper formalizes this question using preimage entropy, a Bayes-optimal exact recovery bound, residual inverse uncertainty, side-record recurrence criteria, capacity-recovery asymmetry, and an informational hysteresis observable. It also replaces purely metric closure error with information-theoretic Jensen-Shannon/KL Markov-closure errors and relates projected hidden-state burden to Mori-Zwanzig memory-kernel structure. The main result is that capacity recovery is not distinction recovery. A system may regain storage capacity after a bottleneck, but it does not thereby regain inverse information erased during the bottleneck unless that information was preserved in side records, reversible garbage registers, environmental traces, protected invariants, external ledgers, or an enlarged accounting boundary. P4 further distinguishes benign from malignant informational hysteresis. Benign hysteresis occurs when erased distinctions are task-irrelevant or replaced by cheaper invariant variables without increasing task loss. Malignant hysteresis occurs when erased distinctions later become task-relevant and residual inverse uncertainty increases expected loss. The paper includes deterministic synthetic demonstrations of recovery bounds, capacity hysteresis, externalization break-even, Mori-Zwanzig memory-kernel burden, Jensen-Shannon Markov closure error, benign versus malignant hysteresis, low-cost Phase-B survivor selection, and the transition from ordinary P4 truncation to P7 invariant/topological protection. This work is part of the Finite Distinction Systems / Distinction Theory physical-operational sequence. It provides the representation-side bridge between capacity overflow, finite-memory record turnover, entropy/resource ledgers, and invariant-supported persistence.