Closure complexity of longest-edge bisection for triangular meshes

On triangular meshes, we analyze local mesh refinement based on longest-edge bisection equipped with the serial longest-edge propagation-path closure. Ties are resolved by terminal priority: if the incoming shared edge is a longest edge of the neighboring triangle, the pair is declared terminal and that edge is bisected. For every adaptive mesh sequence ${\mathcal{T}}_0,{\mathcal{T}}_1,\dots ,{\mathcal{T}}_L$ with marked subset sequence ${\mathcal{M}}_0,{\mathcal{M}}_1,\dots ,{\mathcal{M}}_{L-1}$, we prove the cumulative closure estimate $\mathrm{\#}{\mathcal{T}}_L-\mathrm{\#}{\mathcal{T}}_0\lesssim {\sum }_{\mathrm{\ell }=0}^{L-1}\mathrm{\#}{\mathcal{M}}_{\mathrm{\ell }}$. The proof has two ingredients. First, the finite-similarity-class theorem for planar longest-edge bisection turns the set of possible diameters into a finite union of $\sqrt{2}$-geometric lattices. Hence diameters grow by a uniform strict factor at every nonterminal step. This yields generation and spatial locality for all triangles created by a single mark. Second, a Binev–Dahmen–DeVore type charging argument converts this single-mark locality into the cumulative estimate.