The Trajectory Quotient: Semantic Histories as Paths in the Rank-Fiber Graph

This note identifies the semantic histories of finite erasure processes with paths in a rank-fiber graph. Given a finite erased set and an antitone Galois connection, erasure orders are quotiented by equality of their semantic trajectories. The main result, Fiber-Path Exactness, proves that every source-to-sink path in the resulting layered rank-fiber graph is realized by an actual erasure order. Thus the trajectory quotient is exactly the path set of a finite graph computed from rank fibers alone. The proof uses the meet law of antitone Galois connections through a constructive Meet-Descent argument. The v0.3 frontier analysis isolates the more general mechanism: backward stability of the fiber partition is sufficient for exactness, Galois fibers are backward-stable, and hence the original theorem factors as meet law ⇒ backward stability ⇒ exactness. Both implications are strict. Forward stability independently suffices for exactness, while neither forward nor backward stability is necessary. Explicit finite witnesses separate these conditions and exhibit failure of exactness outside the sufficient classes. The paper therefore distinguishes three levels that should not be conflated: algebraic structure sufficient to force stability, local fiber stability sufficient to force global path realizability, and exactness itself. The exact characterization of fiber-path exactness beyond these sufficient conditions is left open. No probability, weighting, optimization, or scalar measure on trajectories is introduced.