Mathematics Computational Complexity From Rs

Complexity classes as a count

Computational complexity theory studies how much time or memory an algorithm needs to solve a problem. The classic classes include P, problems solvable quickly; NP, problems whose answers can be checked quickly; coNP, the complements of NP; PSPACE, problems solvable with polynomial memory; and EXP, problems solvable in exponential time. A central open question asks whether P equals NP, that is, whether every quickly checkable problem can also be solved quickly.

In 1971, Stephen Cook and Leonid Levin independently formalized NP-completeness, identifying problems that are at least as hard as any in NP. The P versus NP question remains unresolved in classical mathematics. The five classes named above are distinct by definition, but their relationships, such as whether P is a proper subset of NP, are unknown.

In Recognition Science, the framework models these five classes as a single finite set. Its machine-checked library of formal theorems defines an inductive type with exactly five constructors, one for each class, and proves that this type has cardinality five. The framework also defines the size of an eight-point discrete Fourier transform as 2^3 and proves this equals 8. These two facts, five classes and eight points, are packaged into a certificate structure.

The framework's claim is not that P differs from NP. It conjectures that P does not equal NP because NP-complete problems would have a cost function with exponentially many basins where the cost function J equals zero. This conjecture is a prediction, not a proved theorem. The proved content is the counting: five classes and eight points, both by direct computation.

What this establishes in plain language is a structural alignment. The number five, already central to the framework's derived dimensions, matches the count of standard complexity classes. The number eight, the size of a small transform, matches 2^3. These are exact equalities in the formal library, not approximations.

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

theorem dft8Size_8 : dft8Size = 8 := by decide

What this page does not claim

This module does not prove P ≠ NP. The framework does not derive the fine-structure constant here. No claim is made that the five classes are distinct in the classical sense of set theory.

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)

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Derived articles

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