  The $S=1$ Affleck-Kennedy-Lieb-Tasaki (AKLT) quantum spin chain was the first
rigorous example of an isotropic spin system in the Haldane phase. The
conjecture that the $S=3/2$ AKLT model on the hexagonal lattice is also in a
gapped phase has remained open, despite being a fundamental problem of ongoing
relevance to condensed-matter physics and quantum information theory. Here we
confirm this conjecture by demonstrating the size-independent lower bound
$\Delta >0.006$ on the spectral gap of the hexagonal model with periodic
boundary conditions in the thermodynamic limit. Our approach consists of two
steps combining mathematical physics and high-precision computational physics.
We first prove a mathematical finite-size criterion which gives an analytical,
size-independent bound on the spectral gap if the gap of a particular cut-out
subsystem of 36 spins exceeds a certain threshold value. Then we verify the
finite-size criterion numerically by performing state-of-the-art DMRG
calculations on the subsystem.
