We study whether lower bounds against constant-depth algebraic circuits computing the Permanent over finite fields (Limaye-Srinivasan-Tavenas, J. ACM 2025; Forbes, CCC 2024) are hard to prove in certain proof systems. We focus on a DNF formula that expresses that such lower bounds are hard for constant-depth algebraic proofs. Using an adaptation of the diagonalization framework of Santhanam and Tzameret (SIAM J. Comput. 2025), we show unconditionally that this family of DNF formulas does not admit polynomial-size propositional AC0[p]-Frege proofs infinitely often. This rules out the possibility that the DNF family is easy, and establishes that its status is either that of a hard tautology for AC0[p]-Frege or else unprovable (not a tautology). While it remains open whether the DNFs in question are tautologies, we provide evidence in this direction. In particular, under the plausible assumption that certain weak properties of multilinear algebra, specifically those involving tensor rank, do not admit short constant-depth algebraic proofs, the DNFs are tautologies. We also observe that several weaker variants of the DNF formula are provably tautologies, and we show that the question of whether the DNFs are tautologies connects to conjectures of Razborov (ICALP 1996) and Krajicek (J. Symb. Log. 2004). Our result has two additional features. (i) Existential depth amplification: the DNF formula is parameterised by a constant depth d bounding the depth of the algebraic proofs. We show that there exists some fixed depth d such that if there are no small depth-d algebraic proofs of certain circuit lower bounds for the Permanent, then there are no such small algebraic proofs in any constant depth. (ii) Necessity: we show that our result is a necessary step towards establishing lower bounds against constant-depth algebraic proofs, and more generally against any sufficiently strong proof system.

Lower bounds against strong algebraic proof systems, and specifically fragments of the Ideal Proof System (IPS), have been obtained in an ongoing line of work. With the exception of the placeholder model, where the instance itself lacks small circuits, all existing bounds are proved only over large (or characteristic 0) fields, whereas finite fields form the more natural setting for propositional proof complexity. This work establishes lower bounds against fragments of IPS over constant-sized finite fields, resolving an open problem left by a series of prior works beginning with Forbes, Shpilka, Tzameret, and Wigderson (Theor. of Comput.’21), persisting with Behera, Limaye, Ramanathan, and Srinivasan (ICALP’25), and most recently posed by Forbes (CCC’24). We further highlight the importance of the constant-sized finite field regime in IPS by showing that any hard instance in this regime for a sufficiently strong proof system translates into a hard instance against AC0[p]-Frege, whose lower bounds remain a longstanding open problem.