Strong algebraic proof systems such as IPS (Ideal Proof System; Grochow-Pitassi [GP18])
offer a general model for deriving polynomials in an ideal and refuting unsatisfiable propositional
formulas, subsuming most standard propositional proof systems. A major approach for lower
bounding the size of IPS refutations is the Functional Lower Bound Method (Forbes, Shpilka,
Tzameret and Wigderson [FSTW21]), which reduces the hardness of refuting a polynomial
equation f (x) = 0 with no Boolean solutions to the hardness of computing the function 1/f (x)
over the Boolean cube with an algebraic circuit. Using symmetry, we provide a general way to
obtain many new hard instances against fragments of IPS via the functional lower bound method.
This includes hardness over finite fields and hard instances different from Subset Sum variants,
both of which were unknown before, and stronger constant-depth lower bounds. Conversely,
we expose the limitation of this method by showing it cannot lead to proof complexity lower
bounds for any hard Boolean instance (e.g., CNFs) for any sufficiently strong proof systems...