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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 
$n \leq p + 1$.
