TopoDialogue: Visualising and Operationally Distinguishing Recurrent Semantic Trajectories in Multi-Turn Dialogue

Long human–AI conversations may return to a similar semantic region without being exact textual repetitions. A return may be unproductive, may add detail or abstraction, or may precede a move into a new region. These cases are difficult to distinguish in a flat transcript. We present TopoDialogue, an offline, provenance-aware research instrument that represents a dialogue as a time-indexed three-dimensional trajectory and assigns one of four operational classes: closed loop, spiral revisit, escape, or open trajectory. The current implementation uses explicit semantic-node annotations when available and a declared lexical fallback otherwise. It provides deterministic hash-based display co-ordinates, four selectable height mappings, recurrence and progress scores, first-occurrence speaker provenance, and warnings for known display artefacts. We define the complete Phase-1 procedure and evaluate it with six authored synthetic cases whose expected structures are known by construction. Both the Python reference implementation and the browser implementation match all six declared class and recurrence-source labels; these are specification-conformance tests, not an estimate of accuracy on natural conversation. The study shows that recurrence, progress within a revisit, and feature provenance can be made inspectable and reproducible. It does not show that the display recovers a unique semantic manifold, infers mental state, or improves dialogue outcomes. We report projection instability, lexical approximation, monotonic-time, and speaker-drift failure modes and set out the human-annotated evaluation needed next.