Lyapunov Stability of the Edge-Vertex Ratio in a Graph Dynamical System with Cohomological Feedback

We study a discrete-time dynamical system in which a finite graph G(t) = (V(t), E(t)) grows by two
competing mechanisms: bounded-rate edge addition and energy-threshold-gated vertex expansion. A
cellular sheaf over G(t) equips each vertex with a stalk Rd and each edge with linear restriction maps.
A spectral obstruction proxy derived from the near-zero spectrum of the sheaf Laplacian enters a
bounded pressure functional that governs whether vertex expansion fires on a given cycle.
Define the edge-vertex ratio r(t) = |E(t)|/|V(t)|. We construct a Lyapunov function V(r) = (r - r*)2 and
show that, given empirically verified monotone feedback and rate balance, V is decreasing along
trajectories outside a neighborhood of the equilibrium r*. Over the attractor regime, the equilibrium is
characterized by the stationary rate balance r* = mn/nn, where mn and nn are the time-averaged
edge and vertex creation rates. The attractor width scales as O(1/|V(t)|).
The stability of r implies, via the Euler characteristic, a bound on the first Betti number beta_1(t)
proportional to |V(t)|. We validate the result on a 500-cycle run (seed 42, ARPACK-corrected
eigensolver) starting from a 150-vertex seed and growing to 515 vertices. The ratio r(t) locks to 3.0 by
cycle 150 and remains there for 350 consecutive cycles. The measured stationary rate ratio mn/nn =
2.99 is consistent with the observed attractor to within rounding precision.