three_phi_sq_eq_phi4_plus_1
plain-language theorem explainer
The identity 3φ² = φ⁴ + 1 holds for the golden ratio φ. Researchers deriving Schumann resonance frequencies in Recognition Science cite this to obtain the fundamental without adjustable parameters. The proof is a one-line algebraic reduction that substitutes the basic phi relations φ² = φ + 1 and φ⁴ = 3φ + 2 then applies ring normalization.
Claim. For the golden ratio φ = (1 + √5)/2, the identity 3φ² = φ⁴ + 1 holds.
background
The Earth-Brain Resonance module derives Schumann harmonics from Recognition Science quantities, with D = 3 from the forcing chain T8 and φ from T6 self-similarity. Upstream, phi_sq_eq states: 'Key identity: φ² = φ + 1 (from the defining equation x² - x - 1 = 0)'. The companion phi_fourth_eq records φ⁴ = 3φ + 2 obtained by repeated application of the quadratic relation.
proof idea
The proof rewrites both sides of the target equality using phi_sq_eq and phi_fourth_eq, then closes the resulting polynomial identity with the ring tactic.
why it matters
This theorem is invoked directly in fundamental_eq_phi4_plus_1 to establish schumannRS 1 = φ⁴ + 1. It fills the structural decomposition in the module doc-comment, linking the fundamental to dimension times self-similarity. The result supports the zero-parameter match of all five Schumann harmonics to EEG band boundaries.
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