Counting axioms do not polynomially simulate counting gates

We give a family of tautologies whose algebraic translations have constant-degree, polynomial size polynomial calculus refutations over Z₂, but which require superpolynomial size bounded-depth Frege proofs from Count₂ axioms. This gives a superpolynomial size separation of bounded-depth Frege plus mod 2 counting axioms from bounded-depth Frege plus parity gates. Combined with another result of the authors, it gives the first size (as opposed to degree) separation between the polynomial calculus and Nullstellensatz systems.