Hardness as an Information Constraint: A Unifying Meta-Complexity Assumption

Abstract

Monroe (2026) shows that, if no optimal proof system exists, then every sound arithmetic theory S extending S^1_2 with polynomial-time decidable axioms fails, for all sufficiently large k, to simulate S^1_2+phi_BB(k), where phi_BB(k) asserts the exact k-state Busy Beaver value. This gives the nonexistence hypothesis an information-constraint interpretation through canonical hard instances. If the best-known route to simulation is also necessary--namely, if simulation requires a relative-consistency explanation over a weak base--then the same constraint applies to inaccessible Kolmogorov-randomness facts. We call this conjecture Kolmogorov Hardness (KH). Finite-scale and hierarchy-level forms of KH yield, conditionally, dense families of small hard tautologies, PH noncollapse with explicit dense separators, and SAT notin P/poly. Under separately stated assumptions, variants yield P-inseparability of a disjoint NP pair, no mutual help between mutually conditionally random axioms, one-way functions via Liu--Pass, derandomization, and Feige-style random-refutation hardness. Allender et al. provide a complementary calibration: full access to conventional random-string oracles supports every PSPACE computation, whereas any fixed sound computably axiomatized theory can certify only finitely many positive randomness instances. The framework organizes complexity conjectures around canonical information constraints. It seems self-evident that efficient proofs should not leverage true randomness facts unavailable to the proving theory. Yet KH may be independent of standard metatheories: it resembles reflection, can fail internally in nonstandard models even when externally true, and may constrain the metatheories themselves. We propose a research program on extensions, self-evidence, formal independence, and possible new axioms.