  We present Lattice QCD results on the neutron tensor charges including, for
the first time, a simultaneous extrapolation in the lattice spacing, volume,
and light quark masses to the physical point in the continuum limit. We find
that the "disconnected" contribution is smaller than the statistical error in
the "connected" contribution. Our estimates in the $\bar{\text{MS}}$ scheme at
$2$ GeV, including all systematics, are $g_T^{d-u}=1.020(76)$, $g_T^d =
0.774(66)$, $g_T^u = - 0.233(28)$, and $g_T^s = 0.008(9)$. The flavor diagonal
charges determine the size of the neutron electric dipole moment (EDM) induced
by quark EDMs that are generated in many new scenarios of CP-violation beyond
the Standard Model (BSM). We use our results to derive model-independent bounds
on the EDMs of light quarks and update the EDM phenomenology in split
Supersymmetry with gaugino mass unification, finding a stringent upper bound of
$d_n < 4 \times 10^{-28} \, e$ cm for the neutron EDM in this scenario.
