Phase Loop Dynamics: A Syntax of Drift, Repair, and Resonance

This work presents Phase Loop Dynamics (PLD), a unified theoretical and applied framework for modeling conversational and syntactic interactions as a dynamic topology of recurrent loops.PLD defines five primary loop types (𝓛₁–𝓛₅) and formalizes their operation using operator algebra, stability criteria, and gauge invariance principles. The framework incorporates Latent Phase Theory, distinguishing latent phases from silence and modeling their role in hesitation, repair, and re-entry phenomena.Cross-disciplinary mappings are provided to connect PLD constructs with psycholinguistics, HCI, AI dialogue systems, and interaction design. Applications include latency-aware interfaces, loop-augmented transformers, and engagement recovery in education.An implementation architecture and evaluation metrics are included, with diagrams illustrating operator hierarchy, phase space trajectories, and processing pipelines.This deposit contains the final 19-page PDF of the paper and the accompanying implementation diagram.