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def definition def or abbrev high

schumannRS

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RS supplies the n-th Schumann harmonic frequency via the zero-parameter expression f(n) = (4n - 1) φ + 3. Geophysicists and neuroscientists comparing ionospheric resonances to EEG bands cite this formula. The declaration is a direct definition that encodes the structural decomposition using D = 3 and the golden ratio φ forced by the Recognition chain.

claimThe frequency of the n-th Schumann resonance harmonic is given by $f(n) = (4n - 1)φ + 3$, where $φ = (1 + √5)/2$ is the golden ratio and $n$ is a positive integer.

background

The Earth-Brain Resonance module models Schumann cavity modes with Recognition Science constants. Here φ denotes the golden ratio from T6 self-similarity as the fixed point of the J-function, and D = 3 is the spatial dimension forced by T8. The formula decomposes as f(n) = D φ² + (n-1)(D+1)φ, which simplifies to the given expression since φ² = φ + 1 and D + 1 = 4.

proof idea

The declaration is a direct definition that substitutes the RS constants into the linear frequency expression. No lemmas or tactics are invoked; it functions as the base case for downstream numerical verifications.

why it matters in Recognition Science

This definition anchors the EarthBrainResonanceCert structure and feeds the all_harmonics_match theorem that verifies agreement with measurements. It implements the zero-parameter claim from the module, linking T6 (φ) and T8 (D=3) to observed Earth cavity spectra and their alignment with brainwave bands.

scope and limits

Lean usage

example : schumannRS 1 = 3 * phi + 3 := by rw [schumannRS]; ring

formal statement (Lean)

  96def schumannRS (n : ℕ) : ℝ := (4 * (n : ℝ) - 1) * phi + 3

proof body

Definition body.

  97
  98/-- Measured Schumann resonance harmonics (Hz). -/

used by (29)

From the project-wide theorem graph. These declarations reference this one in their body.