Grounded Coordination in LLM Dialogue: A Repeated-Measures Test for the Shared Frame Subspace

Opponent-shaping methods align agents' behavior without requiring an explicit representation of what a partner believes or intends beyond the reward function, leaving them brittle exactly where representational mismatch matters most: in novel situations requiring multi-agent coordination. We propose a representational account of grounding: a single activation space $W$ factors into a shared frame subspace $F$, situational structure common to both agents, and agent-indexed self/other subspaces $I$ and $O$, with a grounding registry $G$ tracking which regions of $O$ have survived communicative testing. We identify $F$ as the subspace on which an orthogonal-Procrustes alignment residual between two agents' activations stays small. Across twelve instruction-tuned models, pooled activations show significance across episodes, but a within-episode-centered analysis, is significant in exactly one model, Qwen2.5-32B-Instruct. Treating turn as a repeated-measures factor for this model, the systematic turn-to-turn representational shift falls outside the identified frame, consistent with grounding-relevant change concentrating in the agent-indexed complement. A turn-stratified decoding probes the dialogue's disambiguating content across turns. We report these initial findings on a single controlled task and outline how this instrument extends toward certifying grounded coordination across populations of interacting AI agents.