fundamental_eq_D_phi_sq
plain-language theorem explainer
The theorem equates the fundamental Schumann resonance to D times phi squared. Researchers modeling ionospheric cavities or brain-wave entrainment would cite it for its zero-parameter prediction. The proof is a direct algebraic reduction via rewriting with the first harmonic equation and the phi-squared identity, closed by ring normalization.
Claim. The fundamental Schumann resonance frequency satisfies $f(1) = D phi^2$, where $D = 3$ is the spatial dimension and $phi$ the golden ratio satisfying $phi^2 = phi + 1$.
background
In Recognition Science the spatial dimension D is fixed at 3 by the eight-tick octave construction in the forcing chain. The golden ratio phi arises as the self-similar fixed point at T6. The upstream result phi_sq_eq asserts that phi squared equals phi plus one. The module sets the Schumann fundamental as the first term in f(n) = (4n - 1) phi + 3, which reduces to D phi squared by the dimension choice. Local context ties this frequency to the theta-alpha boundary in EEG spectra, with the full set of harmonics matching observations to within 0.4 percent.
proof idea
The proof rewrites the first Schumann term using the harmonic definition, substitutes the identity phi squared equals phi plus one, casts to reals, and closes with ring. It is a one-line term proof.
why it matters
This identity populates the fundamental_is_3phi2 slot in earthBrainResonance_forced, the master theorem asserting that the entire Earth-Brain resonance spectrum follows from the Recognition Composition Law with no free parameters. It instantiates the T8 dimension and T6 self-similarity steps for the 7.85 Hz mode. The framework predicts the observed Schumann fundamental at 7.854 Hz.
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