Emergent Equilibrium: A Measurable Theory of Strategic Convergence in LLM Multi-Agent Dialogue, with a Closed-Form Critical-Temperature Bound

When multiple LLM agents debate, negotiate, or deliberate, do their dialogues converge—and when do they fail to? To this end, we give a formal answer and derive a closed-form condition for the transition. We formalize multi-agent dialogue as a discrete dynamical system on a semantic belief manifold, construct an explicit dialogue operator from cross-attention layers, and introduce the Ideal Group Introspection (IGI) framework—a two-parameter learnable model whose residual error yields the first measurable criterion for strategic convergence wherever those residuals are identifiable. Within this framework, we prove that an approximate emergent equilibrium—satisfying Nash strategic stability, geometric dialogue convergence and consensus coupling—exists under mild regularity conditions, via a sandwich argument linking Glicksberg and Brouwer, the profile delivered being $\delta$-close to the stage game's quantal response equilibrium, and Nash as $\beta \to \infty$; we establish spectral-radius convergence conditions and an attention contraction lemma, and show that the dialogue undergoes a phase transition whose critical sampling temperature is bounded above by the closed form $C/(1-L)$, where $C={\sigma }_V\cdot {\sigma }_K\cdot N/\sqrt{d_k}$ (${\sigma }_V,{\sigma }_K$ the value and key projection norms, $N$ the agents, $d_k$ the key dimension) is the architectural constant that the reduced-operator limit $L\to 0$ recovers exactly. Crucially, this makes a one-sided bound on instability onset computable from stored weights under one decomposition assumption, and predicts critical slowing near $T_c$—a prediction our sweeps do not confirm, a null result we report with its power. Validating across 15 models (four families, 0.6B to trillion-parameter scale) in strategic and tax-compliance games, we uncover three regularities that our theory explains. First, in the compliance framing frontier closed-source models exhibit sycophantic cooperation—complying even when defection is strictly dominant, while defecting in the abstract game—and so fall outside the IGI regime there; because this deviation is systematic rather than random, it defines an interpretable four-way taxonomy of model behaviour instead of a failure of the theory. Second, strategic behaviour is protocol-dependent rather than a fixed model property: a single checkpoint spans both endpoints, from complete cooperation in the tax framing to zero against randomised opponents. Third, cooperation varies non-monotonically with model scale within the mid-size regime, where strategic diversity is highest. Taken together, our results provide a rigorous, empirically testable foundation for predicting when LLM multi-agent deliberation is strategically meaningful—and when it is not.