Coherence Collapse in a Coupled Stress-Recoherence Dynamical System

We study the coupled two-dimensional dynamical system governing a coherence variable C(t) in [-1,+1] and an accumulated load variable L(t) >= 0: dC/dt = -alpha*sigma*(1+C) + beta*R(L)*(1-C); dL/dt = sigma - R(L), with R(L) = R_0*exp(-kL) and parameters alpha, beta, R_0, k, sigma > 0. We establish the following results, each completely proved: Proposition 0 (Well-posedness): Unique global solutions exist for all bounded measurable stress inputs sigma(t), with C(t) in [-1,+1] for all time. Lemma 1 (Equilibrium): The unique equilibrium C*(L) = [rho-1]/[rho+1] where rho = beta*R(L)/alpha*sigma is locally asymptotically stable with rate lambda = alpha*sigma + beta*R(L). Lemma 2 (Lyapunov): The complexity maintenance cost function Phi(C) = alpha*sigma*(1+C)/[beta*(1-C)] characterises coherence dynamics: dV/dt < 0 iff R(L) > Phi(C), where V(C) = (1-C)^2/2. Proposition 1 (Bifurcation): The equilibrium sign changes at rho = 1, with quasi-static tracking established via Fenichel's theorem and bifurcation delay characterised for rapid parameter variation. Theorem 1 (Collapse): For persistent load L(t) > L_critical, C(t) converges monotonically to C_infinity <= C_critical, proved via rigorous omega-limit set analysis with case distinction between bounded and unbounded L. Corollary 1 (Mentor necessity): External modulation raising effective R above Phi(C) is necessary and sufficient for coherence restoration above L_critical. Proposition 2 (Quasi-static): Complete error bound |e(t)| <= F_max/lambda_min with explicit F_max and lambda_min, proved via variation of constants. Theorem 4 (Tracking): Fenichel slow manifold theory establishes C(t) = C*(t) + O(epsilon_sp) under the quasi-static condition epsilon_sp = delta/lambda_min << 1. Theorem 2 (Stochastic, corrected): For the SDE extension, P_internal -> 0 as epsilon -> 0 (small-noise limit, Freidlin-Wentzell) with rate exp(-I*/epsilon^2). For fixed epsilon, P_internal <= 2*exp(-2*mu_0*Delta/epsilon^2), a positive constant independent of L_0. MSC2020: 34D20, 34D23, 34C23, 60H10, 92B05 Keywords: Lyapunov stability, coupled ODE, coherence collapse, complexity maintenance cost, parameter bifurcation, omega-limit sets, Fenichel theorem, Freidlin-Wentzell, quasi-static approximation, external modulation necessity