dft8ModeCount
dft8ModeCount defines the DFT-8 mode count as 2^3. Researchers deriving Fourier analysis from Recognition Science cite it to connect the eight-tick octave to three spatial dimensions. The definition is a direct power computation.
claimThe number of modes in the discrete Fourier transform on eight points is defined as $2^3$.
background
Recognition Science derives an eight-tick recognition period (T7) that, with three spatial dimensions (T8), yields exactly eight Fourier modes. The FourierAnalysisFromRS module introduces five canonical operations (DFT, FFT, convolution, correlation, power spectrum) whose count equals the configuration dimension D=5, while the DFT itself carries 8=2^D modes. This definition mirrors the upstream declaration in QuantumErrorCorrectionFromJCost whose doc-comment states 'DFT-8 harmonic scheduling: 8 = 2^D = 2^3 modes.'
proof idea
Direct definition as the natural-number value of 2 raised to the power 3.
why it matters in Recognition Science
The definition supplies the dft8_modes field required by the FourierCert structure and the dft8_count field of QECCert. It realizes the T7 eight-tick octave step, enabling the downstream theorem dft8_eq_8 that equates the count to 8 and the fundamental frequency 5φ/8 Hz. No open scaffolding remains attached to this declaration.
scope and limits
- Does not prove equality to 8; that step occurs in a separate theorem.
- Does not derive the fundamental frequency 5φ/8 Hz.
- Does not reference J-cost, error thresholds, or physical constants.
- Does not extend to other DFT sizes or non-power-of-two periods.
Lean usage
theorem dft8_eq_8 : dft8ModeCount = 8 := by decideformal statement (Lean)
32def dft8ModeCount : ℕ := 2 ^ 3