Time as Irreversible Distinction Update: Finite Records, Causal Ordering, and Register-Time Collapse

Time is often treated either as a continuous coordinate in physical equations or as a global ordering whose arrow is explained by statistical mechanics, cosmology, or boundary conditions. FDS-O2 gives a narrower operational account: for a finite physical observer, usable time is the ordered structure of irreversible distinction updates in finite records. Building on FDS-O1, where an observer is defined as a finite distinction-register and measurement as stable record formation, this paper defines a temporal register as a physical system that records, orders, timestamps, updates, synchronizes, and reuses distinctions under finite memory, finite clock precision, finite buffering, finite communication bandwidth, finite latency, and finite thermodynamic budgets. The central claim is not that coordi- nate time is unreal or that microscopic dynamics must be nonunitary. It is that finite systems access temporal order only through register time: causal dependencies, clock labels, logs, residues, and up- date traces carried by finite records. We make causal dependency prior to temporal labeling: record zj is operationally after zi when the update that forms zj depends on zi through an accessible finite update chain. When finite memory forces overwrite, compression, projection, or garbage collection, the update map is many-to-one. Its non-injectivity can be quantified by LU = H(X|Z), the residual set of compatible past states given the current record. This loss defines an operational arrow: future records may contain traces of past records, but erased distinctions cannot be reconstructed from the current finite record without external memory. The paper adds synchronization bottlenecks between finite observers, latency-induced apparent causal inversion, finite clock coarse-graining, dissipative projection, and long-context temporal collapse in AI-like finite-window systems. Deterministic sim- ulations illustrate memory-fill-driven non-injectivity, fixed-memory overwrite versus append-only logging, latency-based order inversion, clock-tick collisions, synchronization ambiguity, projection- semigroup loss, and context-window temporal collapse. The paper does not solve the cosmological arrow of time, the Past Hypothesis, the full Second Law, general relativity, or quantum measure- ment. Its contribution is narrower: it converts finite record formation into a concrete operational model of time as causal, irreversible, finite-register update.