  The Distance Conjecture holds that any infinite-distance limit in the scalar
field moduli space of a consistent theory of quantum gravity must be
accompanied by a tower of light particles whose masses scale exponentially with
proper field distance $\Vert\phi\Vert$ as $m \sim \exp(- \lambda
\Vert\phi\Vert)$, where $\lambda$ is order-one in Planck units. While the
evidence for this conjecture is formidable, there is at present no consensus on
which values of $\lambda$ are allowed. In this paper, we propose a sharp lower
bound for the lightest tower in a given infinite-distance limit in $d$
dimensions: $\lambda \geq 1/\sqrt{d-2}$. In support of this proposal, we show
that (1) it is exactly preserved under dimensional reduction, (2) it is
saturated in many examples of string/M-theory compactifications, including
maximal supergravity in $d= \text{4 - 10}$ dimensions, and (3) it is saturated
in many examples of minimal supergravity in $d= \text{4 - 10}$ dimensions,
assuming appropriate versions of the Weak Gravity Conjecture. We argue that
towers with $\lambda < 1/\sqrt{d-2}$ discussed previously in the literature are
always accompanied by even lighter towers with $\lambda \geq 1/\sqrt{d-2}$,
thereby satisfying our proposed bound. We discuss connections with and
implications for the Emergent String Conjecture, the Scalar Weak Gravity
Conjecture, the Repulsive Force Conjecture, large-field inflation, and scalar
field potentials in quantum gravity. In particular, we argue that if our
proposed bound applies beyond massless moduli spaces to scalar fields with
potentials, then accelerated cosmological expansion cannot occur in asymptotic
regimes of scalar field space in quantum gravity.
