Mathematics Computational Complexity From Rs Computational Complexity Cert

A certificate, not a solution

Computational complexity theory sorts problems by the resources they need. The classes P, NP, coNP, PSPACE, and EXP are among the most studied. A certificate in the Recognition Science framework's machine-checked library of formal theorems is a structure that bundles together a few proved facts. The declaration named ComputationalComplexityCert is exactly that: it packages two statements that the library has already checked. The first is that the list of five classes is complete, meaning the count of distinct classes is five. The second is that a certain size, written as 2^3, equals 8.

The first fact is proved by a theorem that simply decides the count: the library checks that the type of complexity classes has exactly five elements. The second fact is a definition followed by a theorem. The library defines dft8Size as 2^3, and then proves that this equals 8. The certificate itself is a structure that holds these two proofs together. It is a way of saying: these two results are both established, and here they are in one package.

The docstring that accompanies the source file goes further. It connects the number five to a dimension D, and the number eight to 2^D. It suggests that P vs NP is a question about whether recognition cost can be verified in polynomial time, and it states a conjecture that P is not equal to NP because NP-complete problems have a cost structure with exponentially many basins. It also claims that the discrete Fourier transform of size 8 is computable in polynomial time in D. These are interpretations and conjectures written in prose. They are not part of the certificate.

In Recognition Science, the certificate is a small, exact object. It does not prove P ≠ NP. It does not prove any relationship between the complexity classes. It does not even define what polynomial time means. It only records two elementary facts: the count of five classes, and the equality 2^3 = 8. The larger claims in the docstring are directions for future work, not established results.

What the certificate does establish is a pattern. The number five and the number eight appear in the framework's other results, such as the forcing chain that produces three spatial dimensions. The certificate is a way of noting that these same numbers appear in a computational setting. It is a small piece of evidence for a connection, not a proof of one.

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

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

theorem dft8Size_8 : dft8Size = 8 := by decide

What this page does not claim

This certificate does not prove P ≠ NP or any other separation of complexity classes. The docstring's conjecture about NP-complete problems and cost basins is not part of the certificate. The certificate does not define or use the concept of polynomial time.

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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