V.T137 — No UV divergences in tau

Every spectral sum over boundary characters in H_partial[omega] converges. For any sector X and depth N: sum ||chi_X(alpha_n)||^2 <= kappa(X)^2 N. Physical observables are obtained via profinite completion, not continuous integration. The limit exists because boundary characters are bounded. No regularization, renormalization, or cutoff is needed.