harmonic5_bounds
plain-language theorem explainer
Recognition Science predicts the fifth Schumann harmonic as 19φ + 3 Hz with φ the golden ratio. This theorem establishes the strict numerical bounds 33.742 < 19φ + 3 < 33.761, confirming agreement with the measured 33.8 Hz peak to within 0.06 Hz. Researchers modeling zero-parameter Earth-brain resonance alignments would cite the result. The term-mode proof substitutes the closed-form identity and applies nlinarith to the golden-ratio interval bounds.
Claim. $33.742 < 19φ + 3 < 33.761$, where $φ = (1 + √5)/2$ is the golden ratio and the left-hand side is the Recognition Science expression for the fifth Schumann harmonic frequency.
background
The EarthBrainResonance module defines schumannRS(n) := (4n − 1)φ + 3 for n ≥ 1, where φ is the golden ratio. This expression expands the structural form f(n) = D φ² + (n−1)(D+1)φ with D = 3 forced by the eight-tick octave. Upstream results include the identity schumannRS(5) = 19φ + 3 together with the numerical lemmas 1.618 < φ and φ < 1.619.
proof idea
The proof is a term-mode one-liner. It rewrites schumannRS 5 via the equality harmonic5_eq to reach 19φ + 3. It then splits the conjunction and applies nlinarith to each inequality, feeding in the phi_gt_1618 and phi_lt_1619 bounds on the golden ratio.
why it matters
The bound is used directly by the downstream theorem harmonic5_in_gamma to place the fifth harmonic inside the gamma EEG band [30, 100). It completes the module's verification that all five measured Schumann frequencies lie within 0.06 Hz of the RS predictions using only the forced constants D = 3 and φ. The result sits in the structural identities section and relies on T6 self-similarity together with T8 dimension forcing.
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