Manifold Trajectories in Next-Token Prediction: From Replicator Dynamics to Softmax Equilibrium

The paper asserts that for the replicator ODE ṗ_i = (1/T) p_i (s_i − s̄) with fixed scores s, the free energy F(p)=⟨p,s⟩+T H(p) is a Lyapunov function and that trajectories converge to the softmax π(s) (Theorem 4.1 and Proposition 3.7). However, under this ODE the explicit solution p_i(t)=p_i(0) e^{(s_i/T)t}/∑_j p_j(0) e^{(s_j/T)t} implies convergence to the argmax face, not to π(s); moreover, dF/dt is not sign-definite and can decrease (e.g., V=2, s=(0,1), T=1, p(0)=(1/2,1/2)). The paper conflates the natural-gradient of ⟨p,s⟩ (which yields the replicator field) with the natural-gradient of F (which yields the entropic/natural-gradient replicator). These claims in the paper are therefore false, while the model’s counterexample and corrected flow are correct. See the paper’s explicit claims in Theorem 4.1 and Proposition 3.7 for contrast .