TopoDialogue: A Reproducible Instrument for Distinguishing Recurrent, Progressive, and Escaping Semantic Trajectories

Long human-AI dialogues can return to a similar semantic region without repeating the same words. A return may simply close a loop, add features and abstraction, or precede a move into a new region. We present TopoDialogue, a provenance-aware research instrument for making these distinctions inspectable and reproducible. The Phase-1 implementation represents turns as a time-indexed three-dimensional display trajectory, computes recurrence with a declared feature-set rule, records whether a feature first appeared in a user or assistant turn, and assigns one of four operational classes: closed loop, spiral revisit, escape, or open trajectory. It also reports the input hash, algorithm version, projection seed, axis definitions, and warnings for known display artefacts. We evaluate the method on six authored synthetic cases with known structures and run the same manifest through Python and JavaScript implementations. All twelve declared class and recurrence-source outputs agree. This is specification conformance, not accuracy on natural conversation. The result is a reproducible measurement instrument and a set of testable boundaries, not a psychological, clinical, or topological claim.