harmonic2_eq
Recognition Science predicts the second Schumann harmonic frequency as 7φ + 3 in native units. Researchers modeling Earth cavity resonances or EEG band alignments cite this to confirm the zero-parameter formula matches the 14.3 Hz measurement. The proof is a one-line wrapper that unfolds the schumannRS definition and reduces the resulting arithmetic expression.
claimThe Recognition Science prediction for the second Schumann resonance frequency satisfies $f(2) = 7φ + 3$, where $φ = (1 + √5)/2$ is the golden ratio forced by self-similarity.
background
The EarthBrainResonance module constructs Schumann harmonics from RS-forced constants only. The definition schumannRS(n) := (4n − 1)φ + 3 encodes the structural decomposition f(n) = D φ² + (n−1)(D+1)φ with D = 3 the spatial dimension and φ the golden ratio. This rests on the upstream schumannRS definition which supplies the zero-parameter formula valid for n ≥ 1.
proof idea
The proof is a one-line wrapper. It unfolds schumannRS, applies push_cast to convert the natural number argument, and invokes the ring tactic to confirm the identity (4*2 − 1)φ + 3 = 7φ + 3.
why it matters in Recognition Science
This supplies the explicit value for the second harmonic that downstream results harmonic2_bounds and harmonic2_matches invoke to place 7φ + 3 inside (14.326, 14.333) and within 0.04 Hz of the measured 14.3 Hz. It instantiates the general formula at n = 2 inside the Earth-Brain Resonance section, connecting the eight-tick octave (T7) and D = 3 (T8) to observed EEG boundaries.
scope and limits
- Does not establish the formula for n other than 2.
- Does not derive the Schumann expression from cavity boundary conditions.
- Does not incorporate measurement uncertainties beyond the phi bounds used downstream.
- Does not address non-harmonic modes or damping effects.
Lean usage
rw [harmonic2_eq]formal statement (Lean)
113theorem harmonic2_eq : schumannRS 2 = 7 * phi + 3 := by
proof body
One-line wrapper that applies unfold.
114 unfold schumannRS; push_cast; ring
115