Emergent Coherence under Coherence-Gated Hebbian Learning

We study a self-referential stochastic recurrence on RN with Hebbian weight up-dates gated by a coherence order parameter Φ ∈ [0, 1] measuring alignment betweenthe network state and its projection through the weight matrix. This mechanism,Coherence-Gated Hebbian Learning (CGHL), defines a coupled nonlinear Markovchain on states and weights. We establish three theorems and three lemmas provingthat the coherence decay condition E[∆Φ | Ft] ≤ 0 holds for all sufficiently larget almost surely, under the conditions σ2 > 0, α ∈ (0, 1), N ≥ 3, and Φc > 1/2.The proof chain proceeds through four mathematical domains: Gaussian noise in-dependence yields a pathwise second moment bound that holds prior to ergodic con-vergence; a Foster-Lyapunov drift condition on the joint state (xt, Wt) establishesgeometric ergodicity of the state chain; persistent excitation theory from adaptivecontrol (Ioannou & Sun, 1996; Goodwin & Sin, 1984) provides σmin control; andprojective geometry on RPN −1 furnishes the Lipschitz estimates for the coherencefunctional. A novel contribution is that the persistent excitation condition emergesendogenously from the CGHL dynamics rather than being imposed as an externalassumption. Two open problems are precisely characterized: establishing a uniformcontraction rate for deterministic convergence, and bounding σmin(Wt) away from1 zero without assuming persistent excitation.