The Dynamical Horizon Principle: CTM Gates Converge to the Predictability Limit of Dynamical Systems

We present the Dynamical Horizon Principle (DHP): a Continuous Thought Machine (CTM) trained via gradient descent on multi-step prediction spontaneously allocates its temporal integration window to match the intrinsic predictability horizon of the underlying dynamical system.

Across three canonical regimes we confirm: (1) Markovian systems (mass-spring oscillator): τ* → 0 — no history needed; (2) Periodic systems (double pendulum): τ* → T, the natural period; (3) Chaotic systems (Lorenz attractor): τ* → τ_L ≈ 22 discrete steps, the Lyapunov time — the exact horizon past which future states become unpredictable.

A 150,432-parameter CTM trained solely on MSE prediction loss recovers τ_L to within 7% without any knowledge of Lyapunov exponents, delay embedding theory, or dynamical systems physics. The result is interpreted through Takens' theorem: the CTM learns to span the minimal embedding window required for topological reconstruction of the attractor. We further show that the learned window corresponds to the total embedding window T_W ≈ (m−1)τ, not the individual delay τ, clarifying the relationship to Fraser-Swinney mutual information criteria. The DHP suggests that information-optimal world models trained on sufficiently rich dynamics will spontaneously discover the predictability structure of their environment as a consequence of gradient descent on prediction loss alone.