Mathematics Numerical Analysis From Rs Dft8 Modes 8

The eight modes

The declaration dft8Modes_8 establishes a simple arithmetic fact: the number of modes in a certain discrete Fourier transform, called DFT-8, is exactly 8, because it is defined as 2 raised to the third power. In the Recognition Science framework, this is part of a larger claim that five canonical numerical methods (Newton's method, Euler integration, Runge-Kutta, Gaussian elimination, and the Fast Fourier Transform) correspond to a configuration dimension of 5, and that the DFT-8 algorithm is the canonical numerical algorithm with 8 modes.

The statement itself is a theorem in a machine-checked library of formal theorems, meaning it has been verified by a computer to follow from the definitions, with no unproven assumptions. The library also proves that the number of numerical methods is 5, and that the FFT operations per tick are 24 (8 times 3). These are all packaged together in a structure called NumericalAnalysisCert, which asserts these three facts together.

What the declaration does not claim is important. It does not prove that the DFT-8 algorithm is actually the canonical numerical algorithm in any physical or mathematical sense; that is a definitional choice within the framework, not a theorem. It does not claim that the number 8 has any special significance beyond being defined as 2 cubed. It does not claim that the five numerical methods are the only ones that exist, or that they are the best ones. It merely states that, given the definitions, the number of modes is 8.

The declaration is a small piece of a larger framework that derives physical constants and structures from a single cost function. But this particular declaration is about arithmetic and definitions, not about physics. It is a building block, not a conclusion.

theorem dft8Modes_8 : dft8Modes = 8 := by decide

theorem numericalMethodCount : Fintype.card NumericalMethod = 5 := by decide

theorem fftOps_24 : fftOps = 24 := by decide

What this page does not claim

The declaration does not prove that DFT-8 is the best or only numerical algorithm. The declaration does not establish any physical significance for the number 8 beyond its definition. The declaration does not claim the five numerical methods are the only possible ones.

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/NumericalAnalysisFromRS.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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