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

Paper also available at https://philarchive.org/rec/WUTAIB 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-O2 develops an operational finite-record model of time for bounded physical observers. Building on FDS-O1, where an observer is defined as a finite distinction-register and measurement as stable record formation, this paper asks what temporal order means for such a finite register. The central claim is that finite systems access temporal order only through register time: causal dependencies, clock labels, logs, residues, update traces, synchronization records, and finite memories carried by physical records. Coordinate time, proper time, and clock readings remain valid in their standard domains; FDS-O2 studies the finite record structure through which bounded observers use temporal order. The 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. It makes causal dependency prior to temporal labeling: a record is operationally later than another when the update that forms it depends on the earlier record through an accessible finite update chain. When finite memory forces overwrite, compression, projection, or garbage collection, the update map is many-to-one. The paper quantifies this non-injectivity by LU = H(X|Z), the residual set of compatible past states given the current record. This loss defines an operational arrow of register time: future records may contain traces of past records, but erased distinctions cannot be reconstructed from the current finite record without external memory. This version adds synchronization bottlenecks between finite observers, latency-induced apparent causal inversion, finite clock coarse-graining, dissipative projection, compatible-history interpretation, an operational time-resolution / energy-throughput tradeoff, and long-context temporal collapse in AI-like finite-window systems. The accompanying replication package includes deterministic synthetic simulations, figures, CSV outputs, LaTeX source, and Python code. The simulations 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. These are illustrative model demonstrations, not empirical fits to detector data, clock data, cosmological observations, AI-system benchmarks, or human time perception. Scope and boundary. This paper does not claim that microscopic laws are fundamentally nonunitary, that coordinate time in equations is invalid, that relativity requires a preferred foliation, or that cosmology no longer requires boundary conditions. It does not derive general relativity, quantum measurement, or the thermodynamic Second Law in full. Its contribution is narrower: it converts finite record formation into a concrete operational model of time as causal, irreversible, finite-register update.