We study a discrete-time dynamical system in which a finite graph G(t) = (V(t), E(t)) grows by twocompeting mechanisms: bounded-rate edge addition and energy-threshold-gated vertex expansion. Acellular 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 abounded 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 andshow that, given empirically verified monotone feedback and rate balance, V is decreasing alongtrajectories outside a neighborhood of the equilibrium r*. Over the attractor regime, the equilibrium ischaracterized by the stationary rate balance r* = mn/nn, where mn and nn are the time-averagededge 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-correctedeigensolver) starting from a 150-vertex seed and growing to 515 vertices. The ratio r(t) locks to 3.0 bycycle 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.