Mathematics Computational Complexity From Rs Dft8 Size 8

The eight-point DFT

The discrete Fourier transform (DFT) is a standard tool that converts a sequence of numbers into its frequency components. An eight-point DFT processes eight input values and produces eight output values. The declaration dft8Size_8 is a formal statement, checked by a machine, that this size is indeed 8, computed as 2 raised to the third power. This is a trivial arithmetic fact, but its importance lies in being part of a larger, rigorously verified framework.

In Recognition Science, the framework models computation through the lens of recognition costs. The number 8 appears here because the framework links the dimension of a recognition space to the size of a computational transform. The declaration is one small piece of a certificate that also counts five canonical complexity classes: P, NP, coNP, PSPACE, and EXP. The framework's library contains a machine-checked collection of formal theorems that together assert these five classes exist and that the eight-point DFT has size eight.

The theorem dft8Size_8 itself proves only that 2^3 equals 8. It does not prove that the DFT can be computed in polynomial time, nor does it prove anything about the P versus NP question. Those are separate conjectures within the framework, not consequences of this specific declaration. The declaration is a building block, not a result about computational complexity in the classical sense.

What the declaration does establish is a precise, verified connection between a basic mathematical object and the framework's internal structure. It shows that the framework can produce exact, machine-checked statements about standard mathematics. This is a necessary step for any larger claim the framework makes about computation, but it is only a step.

theorem dft8Size_8 : dft8Size = 8 := by decide

def computationalComplexityCert : ComputationalComplexityCert where
  five_classes := complexityClassCount
  dft_poly := dft8Size_8

What this page does not claim

The declaration does not prove that the DFT can be computed in polynomial time. It does not resolve or even address the P versus NP question. It does not establish any property of the DFT beyond its size being eight.

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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