fundamental_eq_phi4_plus_1
plain-language theorem explainer
The declaration proves the Recognition Science prediction for the fundamental Schumann resonance equals φ⁴ + 1. Researchers modeling zero-parameter geophysical resonances would cite this when verifying the 7.83 Hz match to observation. The proof is a one-line term reduction that substitutes the harmonic definition, applies the auxiliary identity 3φ² = φ⁴ + 1 in reverse, invokes φ² = φ + 1, and normalizes with ring.
Claim. The first Schumann resonance frequency satisfies $f(1) = φ^4 + 1$, where $f(n) = (4n-1)φ + 3$ for $n ≥ 1$ and $φ = (1 + √5)/2$.
background
The module derives Schumann harmonics from RS constants with zero free parameters. The function schumannRS(n) is defined by $f(n) = (4n-1)φ + 3$, so the fundamental at n=1 is 3φ + 3. The golden-ratio identity φ² = φ + 1 is the defining relation from the quadratic x² - x - 1 = 0. An auxiliary result states that 3φ² = φ⁴ + 1, which directly identifies the fundamental with φ⁴ + 1. The local setting is the structural decomposition f(1) = 3φ² with spacing 4φ, both forced by T6 self-similarity and T8 spatial dimension D=3.
proof idea
The term proof rewrites schumannRS 1 via the definition of the first harmonic, applies the reverse of the identity 3φ² = φ⁴ + 1, substitutes φ² = φ + 1, and closes with ring normalization.
why it matters
This identity is invoked inside the master theorem earthBrainResonance_forced to certify that the fundamental matches φ⁴ + 1 with zero parameters. It completes the module's claim that f(1) = 3φ² = φ⁴ + 1, linking directly to the forcing chain (T6 for φ, T8 for D=3) and the Recognition Composition Law. The downstream certification also records the spacing 4φ and strict monotonicity.
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