alpha_beta_ratio_near_phi
plain-language theorem explainer
The theorem shows that the absolute difference between 13/8 and the golden ratio φ is less than 0.01. Researchers modeling Schumann resonance harmonics against EEG band boundaries in Recognition Science would cite it to justify the φ approximation in the zero-parameter frequency formula. The proof is a one-line wrapper that rewrites the absolute-value inequality and discharges both sides via nlinarith using the pre-established bounds 1.618 < φ < 1.619.
Claim. $|13/8 - φ| < 0.01$ where $φ = (1 + √5)/2$ is the golden ratio forced by self-similarity.
background
The Earth-Brain Resonance module matches the five Schumann harmonics (7.83, 14.3, 20.8, 27.3, 33.8 Hz) to the formula f(n) = (4n − 1)·φ + 3 with D = 3 and φ the golden ratio. The fundamental lands at the theta/alpha boundary and the second harmonic at the beta band, so their ratio is approximated by 13/8. Upstream lemmas supply the tight interval bounds 1.618 < φ and φ < 1.619.
proof idea
The proof rewrites |13/8 − φ| < 0.01 into the pair of inequalities 13/8 − 0.01 < φ < 13/8 + 0.01, then applies the nlinarith tactic instantiated with the phi_gt_1618 and phi_lt_1619 lemmas to close both sides.
why it matters
This numerical check supports the zero-parameter Schumann-to-EEG matching claim in the Recognition Science framework, where φ is forced by T6 self-similarity and the 4φ spacing follows from the eight-tick octave and D = 3. It closes a small empirical gap without free parameters and aligns with the module's claim hygiene that all five harmonics match measured values within 0.4%.
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