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

2026-06-02

AI Summary

This work proposes the Kolmogorov Hardness (KH) hypothesis, which unifies several central open problems in computational complexity by reducing them to information-theoretic constraints on incompressible random strings. By integrating techniques from proof complexity, Kolmogorov complexity, model theory, and metamathematics—and introducing variants such as time-bounded and sparsely supported formulations—the KH framework systematically reframes longstanding complexity-theoretic challenges as arithmetical truths about genuine randomness that formal theories cannot verify. Under this hypothesis, key consequences follow: the polynomial hierarchy does not collapse, SAT is not in P/poly, dense families of hard tautologies of small size exist, there is no mutual randomness among axioms, and canonical disjoint NP pairs are inequivalent. These results illuminate profound connections between complexity barriers, reflection principles, and logical independence in formal systems.

Abstract

Monroe (2026) shows that the nonexistence of an optimal proof system can be read as an information constraint regarding canonical hard instances: no sound arithmetic theory simulates the extensions adjoining sufficiently large, unprovable Busy Beaver values. Furthermore, if the best-known route to simulation is also necessary -- that is, if simulation requires a relative-consistency explanation over a weak base theory -- then the same constraint holds for inaccessible Kolmogorov-randomness facts. Call this Kolmogorov Hardness (KH). We argue that open questions in computational complexity can likewise be reformulated as information constraints involving Kolmogorov-random strings. Variants of KH yield, as conditional consequences, dense families of small hard tautologies, no-mutual-help phenomena for independent random axioms, PH noncollapse with explicit dense separators at each level, $SAT\notin P/poly$, and canonical disjoint NP pairs arising from random-axiom constructions. Time-bounded and sparse-support variants extend the same template to one-way functions via Liu--Pass, derandomization, natural-proofs-style limitations, and Feige-style random refutation. This framework gives a unified working model of complexity theorists' beliefs, organized around canonical hard instances. It seems self-evident that efficient proofs in a theory should not leverage true randomness facts the theory cannot verify. Yet the structural features of KH and its variants suggest they may be formally independent of standard metatheories. They behave like reflection principles; their internal readings fail in nonstandard models even when the corresponding external readings are true; and the same information constraints may apply to the metatheories themselves. We propose a research program: extend the model, resolve questions of formal independence, and identify which principles are potential new axioms.

Problem

• Kolmogorov randomness
• information constraint
• computational complexity
• proof systems
• formal independence

Innovation

• Kolmogorov Hardness
• information constraint
• optimal proof system
• random axioms
• meta-complexity