dft8Modes
The definition sets the number of modes in the eight-point discrete Fourier transform to two raised to the third power. Researchers certifying numerical methods in Recognition Science reference this constant when counting operations in the fast Fourier transform implementation. It is introduced by direct assignment of the arithmetic expression 2^3.
claimThe number of modes for the eight-point discrete Fourier transform is defined by $2^3$.
background
The module derives five canonical numerical methods from Recognition Science, with their count equal to the configuration dimension D set to five. DFT-8 supplies the canonical algorithm whose mode count equals 2^D, and D equals three from the spatial dimension in the forcing chain. This produces eight modes, with the fast Fourier transform reducing complexity to eight times three operations per tick.
proof idea
The declaration is a direct definition that assigns the natural number 2^3 to the constant. No lemmas or tactics are invoked; the value is obtained by evaluating the power expression.
why it matters in Recognition Science
This constant is required by the NumericalAnalysisCert structure to record eight modes alongside five methods and twenty-four FFT operations. It realizes the eight-tick octave (period 2^3) and the spatial dimension D=3 from the T0-T8 forcing chain. The definition supplies the base value used in the downstream equality proof dft8Modes_8.
scope and limits
- Does not prove the numerical value equals eight.
- Does not specify the algorithm steps of the fast Fourier transform.
- Does not derive operation counts from first principles.
- Does not link to mass formulas or coupling constants.
Lean usage
theorem dft8Modes_8 : dft8Modes = 8 := by decideformal statement (Lean)
29def dft8Modes : ℕ := 2 ^ 3