  Establishing the origin of the water D/H ratio in the Solar System is central
to our understanding of the chemical trail of water during the star and planet
formation process. Recent modeling suggests that comparisons of the D$_2$O/HDO
and HDO/H$_2$O ratios are a powerful way to trace the chemical evolution of
water and, in particular, determine whether the D/H ratio is inherited from the
molecular cloud or established locally. We seek to determine the D$_2$O column
density and derive the D$_2$O/HDO ratios in the warm region toward the low-mass
Class 0 sources B335 and L483. The results are compared with astrochemical
models and previous observations to determine their implications for the
chemical evolution of water. We present ALMA observations of the D$_2$O
transition at 316.8 GHz toward B335 and L483 at $<$0.5" ($<$ 100 au)
resolution, probing the inner warm envelope gas. The column densities of
D$_2$O, HDO, and H$_{2}^{18}$O are determined by synthetic spectrum modeling
and direct Gaussian fitting, under the assumption of a single excitation
temperature and similar spatial extent for the three water isotopologs. D$_2$O
is detected toward both sources in the inner warm envelope. The derived
D$_2$O/HDO ratios is $(1.0\pm0.2)\times10^{-2}$ for L483 and
$(1.4\pm0.1)\times10^{-2}$ for B335. The high D$_2$O/HDO ratios are a strong
indication of chemical inheritance of water from the prestellar phase down to
the inner warm envelope. This implies that the local cloud conditions in the
prestellar phase, such as temperatures and timescales, determine the water
chemistry at later stages and could provide a source of chemical
differentiation in young systems. In addition, the observed D$_2$O/H$_2$O
ratios support an observed dichotomy in the deuterium fractionation of water
toward isolated and clustered protostars, namely, a higher D/H ratio toward
isolated sources
