Mathematics Fourier Analysis From Rs Dft8 Fundamental Pos

The fundamental frequency

Fourier analysis is the standard mathematical tool for breaking a signal into its frequency components. In its discrete form, the discrete Fourier transform (DFT), a signal is sampled at a fixed number of points, and those samples are re-expressed as a sum of sine and cosine waves at equally spaced frequencies. The lowest nonzero frequency in such a decomposition is called the fundamental. For an eight-point transform, the fundamental is one-eighth of the sampling rate; the other frequencies are integer multiples of it.

In the Recognition Science framework, the eight-point transform is tied to an eight-step recognition cycle, and the framework's library of machine-checked theorems records that the fundamental frequency is defined as 5φ/8, where φ is the golden ratio, approximately 1.618. This evaluates to about 1.006 hertz. The library proves a single fact about this number: it is greater than zero. The proof is a short argument that the numerator 5φ is positive and the denominator 8 is positive, so the fraction is positive.

That positivity result is the entire content of the declaration dft8Fundamental_pos. It does not claim the frequency is physically realized, that it matches any measured signal, or that it has any operational meaning beyond being a defined positive number. The framework's library also records that there are five canonical Fourier-related operations (DFT, FFT, convolution, correlation, power spectrum) and that the eight-point transform has eight modes, but those are separate declarations, not consequences of the positivity proof.

What the positivity result does establish is that the defined fundamental frequency is a valid, nonzero frequency within the framework's arithmetic. It is a consistency check: the number sits in the positive real numbers, so it can serve as a reference point in further constructions. The declaration is a small but necessary piece of the framework's internal bookkeeping, not a claim about the physical world.

/-- DFT-8 fundamental = 5φ/8 Hz ≈ 1.006 Hz. -/
noncomputable def dft8Fundamental : ℝ := 5 * phi / 8

theorem dft8Fundamental_pos : 0 < dft8Fundamental := by
  unfold dft8Fundamental
  apply div_pos
  · apply mul_pos (by norm_num) phi_pos
  · norm_num

theorem fourierOperationCount : Fintype.card FourierOperation = 5 := by decide

theorem dft8_eq_8 : dft8ModeCount = 8 := by decide

What this page does not claim

The positivity proof does not claim the frequency corresponds to any observed physical signal. The declaration does not claim that the fundamental frequency is derived from the recognition cycle; it is defined. The theorem does not establish that the eight-point transform is the only valid transform in the framework.

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/FourierAnalysisFromRS.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.

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