We use branes to generalize the Distance Conjecture. We conjecture that in any infinite-distance limit in the moduli space of a d-dimensional quantum gravity theory, among the set of particle towers and fundamental branes with at most pmax spacetime dimensions (where pmax is an integer between 1 and d – 2), at least one has mass/tension decreasing exponentially T ~ exp(–α∆) with the moduli space distance ∆ at a rate of at least α ≥ 1/ \( \sqrt{d-{p}_{\textrm{max}}-1} \) . Since pmax can vary, this represents multiple conditions, where the Sharpened Distance Conjecture is the pmax = 1 case. This conjecture is a necessary condition imposed on higher-dimensional theories in order for the Sharpened Distance Conjecture to hold in lower-dimensional theories. We test our conjecture in theories with maximal and half-maximal supersymmetry in diverse dimensions, finding that it is satisfied and often saturated. In some cases where it is saturated — most notably, heterotic string theory in 10 dimensions — we argue that novel, low-tension non-supersymmetric branes must exist. We also identify patterns relating the rates at which various brane tensions vary in infinite-distance limits and relate these tensions to the species scale.
