phi_eighth_eq
plain-language theorem explainer
The lemma establishes that the eighth power of the golden ratio equals 21 times phi plus 13. Researchers deriving gamma-band frequencies or economic growth factors in Recognition Science cite this identity when traversing the phi-ladder. The proof reduces the expression through a direct calc chain that applies the seventh-power identity followed by the quadratic relation for phi.
Claim. Let φ be the golden ratio satisfying φ² = φ + 1. Then φ⁸ = 21φ + 13.
background
The Constants module builds successive power identities for the golden ratio φ, which obeys the quadratic relation φ² = φ + 1. These identities realize the Fibonacci recurrence φ^n = F_n φ + F_{n-1}. The present lemma continues the chain begun by the seventh-power case. Upstream, phi_seventh_eq states the key identity φ⁷ = 13φ + 8 (Fibonacci recurrence), while phi_sq_eq states φ² = φ + 1. The module sets the fundamental RS time quantum τ₀ = 1 tick.
proof idea
A calc block opens with φ^8 = φ · φ^7. It rewrites the seventh power via phi_seventh_eq, distributes coefficients with ring, substitutes the quadratic identity on the resulting φ² term, and collects like terms with a final ring step to obtain 21φ + 13.
why it matters
This identity feeds phi_ninth_eq to extend the recurrence and supports phi_eighth_in_gamma_band, which places φ^8 inside the gamma EEG band (30–100 Hz) for photobiomodulation. It also supplies the bound 46 < φ^8 < 48 used in octaveGrowth_bounds for economic cycle predictions. Within the framework it realizes the eight-tick octave scaling step.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.