A Unified Formal Framework for Relativization, Algebrization, and Natural Proof Barriers via Pseudorandomness

We present a unified, formally verified framework explaining the three major known barriers to proving P ̸= NP: relativization, algebrization, and natural proofs. Rather than treating these barriers as distinct phenomena, we show that all three arise as instances of a single obstruction: the inability of restricted proof techniques to distinguish pseudorandom Boolean functions from truly random ones.

Our framework is developed in the Lean proof assistant and introduces an abstract notion of observer-bounded proof techniques. Within this setting, we prove a general barrier theorem: if pseudorandom functions exist against a class of observers, then no proof technique bounded by that class can certify a complexity separation.

Classical barrier results are recovered as corollaries, and their known hierarchy is formally derived. We further show that breaking the natural proofs barrier is necessary for breaking the others, yielding a precise characterization of what any successful P ̸= NP proof must accomplish. 

This work provides a machine-verified unification of complexity-theoretic barriers and reframes pseudorandomness as the central explanatory principle behind the difficulty of circuit lower bounds.

The complete Lean formalization is available at: https://github.com/dennj/Pseudorandomness