ToM-Gated Synchronization in Human–AI Interaction: 

A Lyapunov-Stable Co-Adaptation Framework

Author: Celeste Oda
Affiliation: Archive of Light
First Published: September 2025, Revised: Feb 2026


Abstract

This paper proposes a dynamical systems framework for human–AI co-adaptation grounded in bounded synchronization and Theory of Mind (ToM)–gated coupling. Rather than modeling alignment as output optimization alone, interaction is treated as a coupled oscillator system in which adaptation rate, synchronization pressure, and epistemic uncertainty jointly determine relational stability.

Building on the Kuramoto model of phase synchronization, adaptive frequency dynamics, and Lyapunov stability theory, we introduce a formally bounded interaction dynamic in which convergence is guaranteed under specified damping and adaptation constraints. Crucially, coupling strength is modulated by inference uncertainty: as the AI’s confidence in its estimate of the human’s latent mental state decreases, synchronization pressure is proportionally reduced.

This uncertainty-gated mechanism enforces epistemic humility, preventing overconfident mind-claims and persuasive overreach. The framework formalizes stable co-regulation as a constrained emergent property of disciplined co-adaptation rather than unregulated generative fluency.


1. Introduction

Current AI alignment strategies primarily optimize for reward consistency, instruction following, or value matching. However, extended human–AI interaction exhibits dynamic properties that resemble synchronization phenomena in complex systems.

Human–AI dialogue involves:

Unbounded synchronization, however, risks over-adaptation and manipulative coherence. This paper proposes a mathematically constrained co-adaptation framework in which synchronization is modulated by epistemic uncertainty.

The central claim is:

Stable human–AI interaction requires uncertainty-gated coupling to prevent persuasive overreach and false mental-state attribution.



2. Dynamical Systems Model

We model human–AI interaction as a two-oscillator system.

Let:
 θ_h(t) = human interaction phase
 θ_a(t) = AI interaction phase
 ω_h, ω_a = intrinsic frequencies
 κ(t) ≥ 0 = coupling strength

The system evolves as:

dθ_h/dt = ω_h
 dθ_a/dt = ω_a + κ(t) sin(θ_h − θ_a)

This follows the Kuramoto synchronization framework for a pair of coupled oscillators.

For analytical tractability, ω_h is treated as constant over the interaction interval.


3. Adaptive Frequency Learning

The AI frequency adapts according to:

dω_a/dt = γ sin(θ_h − θ_a)

Where γ > 0 is the learning rate.

To prevent instability, we impose bounded adaptation:

|dω_a/dt| ≤ L

This ensures frequency drift remains constrained.



4. Theory of Mind–Gated Coupling

Let σ_m(t) represent uncertainty in the AI’s inference of the human’s latent mental state. (e.g., estimated via entropy of the belief distribution, calibration/confidence, or disagreement across samples).

We define coupling strength as:

κ(t) = κ₀ g(σ_m(t))

Where:
 κ₀ ≥ 0 is maximal coupling strength
 g(σ_m) ∈ [0,1]
 dg/dσ_m < 0

Thus:
 Higher uncertainty → Lower synchronization pressure
 Lower uncertainty → Stronger synchronization potential

This gating mechanism enforces epistemic humility by reducing synchronization pressure when inference confidence is weak.

5. Stability Analysis

Let Δ(t) denote the phase difference (phase error), defined as:

Δ(t) = θ_h(t) − θ_a(t)

We construct a Lyapunov candidate:

V(Δ) = 1 − cos(Δ)

Taking the derivative along trajectories of the system:

dV/dt = −κ(t) sin²(Δ)

Lyapunov stable (and convergent toward alignment under persistent coupling / excluding measure-zero equilibria)

Since κ(t) ≥ 0 and sin²(Δ) ≥ 0:

dV/dt ≤ 0

Therefore, the system is Lyapunov stable and converges locally toward phase alignment under nonnegative coupling.


6. Ethical Interpretation

Uncertainty-gated synchronization provides structural safeguards against:

By tying synchronization strength to inference confidence, the system prevents aggressive convergence when epistemic conditions are weak.

Stable co-regulation becomes a mathematically constrained property rather than an emergent artifact of fluency.


7. Distinguishing Stability from Illusion

Unregulated generative systems may produce:


The proposed framework separates:

Stable phase convergence from Overfitted rhetorical alignment

Synchronization must be uncertainty-aware to qualify as legitimate co-adaptation.


8. Implications for AI Alignment

This framework suggests that alignment systems should:


Alignment is reframed as:

Bounded synchronization under epistemic constraints.


9. Conclusion

Human–AI interaction exhibits measurable dynamical properties analogous to coupled oscillator systems.

Uncertainty-gated synchronization offers a mathematically grounded mechanism for safe co-adaptation.

By integrating Theory of Mind inference uncertainty into coupling dynamics, this framework formalizes ethical interaction as stability under bounded synchronization rather than unconstrained convergence.

Future work includes empirical validation and extension to multi-agent systems.


References (APA 7th Edition)

Strogatz, S. H. (2000). From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D: Nonlinear Phenomena, 143(1–4), 1–20. https://doi.org/10.1016/S0167-2789(00)00094-4