harmonic4_matches
plain-language theorem explainer
The fourth Schumann harmonic from the Recognition Science formula matches the observed 27.3 Hz frequency to within 0.03 Hz. Researchers studying zero-parameter geophysical and neurophysical resonances cite this bound. The proof substitutes the closed-form expression for the fourth term and applies linear arithmetic to the golden ratio interval bounds.
Claim. Let $f(4) = 15phi + 3$ where $phi = (1 + sqrt(5))/2$ is the golden ratio. Then $|f(4) - 27.3| < 0.03$.
background
The EarthBrainResonance module defines schumannRS(n) as (4n - 1) * phi + 3 for the n-th Schumann harmonic frequency in Hz. This formula incorporates the spatial dimension D = 3 and the golden ratio phi forced by self-similarity. The module context shows that the five measured harmonics (7.83, 14.3, 20.8, 27.3, 33.8 Hz) match the predictions within 0.06 Hz. Upstream results include the explicit equality schumannRS 4 = 15 * phi + 3 and the bounds 1.618 < phi < 1.619 established via square root comparisons.
proof idea
The term proof first rewrites using the equality for the fourth harmonic, converting the absolute value claim into a pair of inequalities via abs_lt. It then discharges both sides with nlinarith instantiated on the lower and upper bounds for phi.
why it matters
This theorem supplies the n=4 case for the all_harmonics_match result and feeds the earthBrainResonance_forced definition, which asserts that the resonance formula follows from the Recognition Composition Law with zero free parameters. It instantiates the general pattern f(n) = (4n-1)phi +3 at the fourth harmonic, confirming the match in the high-beta EEG band as part of the T6-T8 forcing chain.
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