phi_cubed_eq
plain-language theorem explainer
The algebraic identity φ³ = 2φ + 1 follows directly from the quadratic relation φ² = φ + 1 that defines the golden ratio in Recognition Science. Climate modelers deriving layer-boundary ratios on the φ-ladder and astrophysicists working on tidal resonances cite this step when converting rung counts into altitude or period predictions. The proof is a short tactic sequence that substitutes the quadratic identity and reduces the resulting polynomial with ring and ring_nf.
Claim. Let φ > 1 be the unique positive real solution of x² − x − 1 = 0. Then φ³ = 2φ + 1.
background
In the Atmospheric Layering module the φ-ladder supplies canonical altitudes via z(k) = z₀ · φ^k, where each integer rung corresponds to a temperature-gradient reversal. The module imports the base identity φ² = φ + 1 (phi_sq_eq) from Constants and re-uses the cubic extension already proved in TidalLockingFromPhiResonance and Constants. MODULE_DOC states that the observed stratopause/tropopause ratio ≈ 4.17 lies inside the numerical interval generated by φ³, confirming a three-rung jump on the ladder.
proof idea
One-line wrapper that applies phi_sq_eq. The tactic block introduces h : φ² = φ + 1, rewrites φ³ as φ² · φ, substitutes h, then invokes ring_nf followed by a final ring reduction to reach 2φ + 1.
why it matters
This identity is the algebraic engine behind every downstream φ³-band statement, including stratopause_tropopause_ratio_band and phi_cubed_band in the same module, and J_phi_ceiling_band plus the TidalLockingFromPhiResonanceCert structure in Astrophysics. It instantiates the self-similar fixed-point property (T6) of the forcing chain and supplies the exact multiplier needed for the three-rung atmospheric step described in MODULE_DOC. No open scaffolding remains; the result is fully closed.
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