Mathematics Computational Complexity From Rs Complexity Class Count

What the count proves

Computational complexity theory sorts problems by the resources needed to solve them. The five canonical classes are P, NP, coNP, PSPACE, and EXP. P contains problems solvable in polynomial time; NP contains problems whose answers can be checked in polynomial time; coNP is the complement of NP; PSPACE contains problems solvable with polynomial memory; EXP contains problems solvable in exponential time. These five classes form the standard backbone of the field, and the Recognition Science framework's machine-checked library of formal theorems contains a declaration, complexityClassCount, that establishes exactly this count.

The declaration proves that the set of these five classes has cardinality five. This is a trivial fact in one sense: the library defines an inductive type with five constructors, one for each class, and the theorem verifies that the type has exactly five elements. The proof is a direct computation, with no axioms and no gaps. The framework's library also defines an eight-element structure related to a discrete Fourier transform, whose size is 2 to the power of 3, and packages both facts into a single certificate object.

What the declaration does not claim is far more interesting. It does not prove that P differs from NP. The framework's own account conjectures that P is not equal to NP, on the grounds that NP-complete problems have a cost structure with exponentially many basins where the cost function equals zero. That conjecture is not a theorem in the library. The declaration also does not assign any computational power to the number five beyond the counting itself. It does not say that the five classes are distinct, that any inclusion between them is strict, or that the classification is complete in any mathematical sense.

In Recognition Science, the number five carries a structural meaning: it matches a dimension parameter D, and the eight-element Fourier structure matches 2 to the power of D. The framework models these connections as part of a larger forcing chain. But the declaration itself is a narrow, exact statement about a finite set. The honest summary is that the framework has verified a counting fact, stated a conjecture about the hardest open problem in computer science, and left that conjecture open.

theorem complexityClassCount : Fintype.card ComplexityClass = 5 := by decide

structure ComputationalComplexityCert where
  five_classes : Fintype.card ComplexityClass = 5
  dft_poly : dft8Size = 8

theorem dft8Size_8 : dft8Size = 8 := by decide

What this page does not claim

The declaration does not prove P is not equal to NP. The declaration does not show that the five classes are pairwise distinct. The declaration does not assign any computational power to the number five beyond counting.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/ComputationalComplexityFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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