  Knabe's theorem lower bounds the spectral gap of a one dimensional
frustration-free local hamiltonian in terms of the local spectral gaps of
finite regions. It also provides a local spectral gap threshold for
hamiltonians that are gapless in the thermodynamic limit, showing that the
local spectral gap much scale inverse linearly with the length of the region
for such systems. Recent works have further improved upon this threshold,
tightening it in the one dimensional case and extending it to higher
dimensions. Here, we show a local spectral gap threshold for frustration-free
hamiltonians on a finite dimensional lattice, that is optimal up to a constant
factor that depends on the dimension of the lattice. Our proof is based on the
detectability lemma framework and uses the notion of coarse-grained hamiltonian
(introduced in [Phys. Rev. B 93, 205142]) as a link connecting the (global)
spectral gap and the local spectral gap.
