phi_cubed_eq
plain-language theorem explainer
The algebraic identity φ³ = 2φ + 1 holds for the golden ratio φ satisfying φ² = φ + 1. Astrophysicists deriving tidal resonance bands and climate researchers modeling atmospheric layering cite this step in phi-ladder calculations. The proof is a five-line calc reduction that substitutes the quadratic identity twice and normalizes with ring.
Claim. Let φ = (1 + √5)/2. Then φ³ = 2φ + 1.
background
The golden ratio φ is the self-similar fixed point forced in the Recognition Science chain at T6. Its quadratic relation φ² = φ + 1 is proved in phi_sq_eq by solving the characteristic equation x² - x - 1 = 0 and simplifying the square root term. The present lemma extends that relation to the next Fibonacci power. The Constants module supplies RS-native constants with c = 1 and τ₀ = 1 tick as the base time quantum.
proof idea
A calc tactic chains five steps. Begin with φ³ = φ · φ². Rewrite the square via phi_sq_eq to obtain φ · (φ + 1). Ring normalizes to φ² + φ. Apply phi_sq_eq again to reach (φ + 1) + φ. Final ring step yields 2φ + 1.
why it matters
This identity is invoked by phi_cubed_band to produce the numerical interval (4.22, 4.24) and by J_phi_ceiling_band for the 0.11–0.13 deviation check inside the TidalLockingFromPhiResonanceCert structure. It supplies the cubic rung required for phi-ladder mass formulas and the eight-tick octave at T7. The lemma closes the recurrence step that feeds downstream resonance certificates in astrophysics and climate modules.
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papers checked against this theorem (showing 1 of 1)
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One golden-ratio curve organizes four periodic-table trends at once
"Two golden-ratio identities, IE_1(G_p)/IE_1(G_{p+1}) ≈ φ^{1/4} on three heavy noble-gas pairs and IE_1(halogen)/IE_1(alkali) ≈ φ^2 on four within-period pairs"