We exploit the Galois symmetries of the little disks operads to show that many differentials in the Goodwillie–Weiss spectral sequences approximating the homology and homotopy of knot spaces vanish at a prime
$p$
. Combined with recent results on the relationship between embedding calculus and finite-type theory, we deduce that the
$(n+1)$
th Goodwillie–Weiss approximation is a
$p$
-local universal Vassiliev invariant of degree
$\leq n$
for every
$n \leq p + 1$
.