phi_tenth_eq
plain-language theorem explainer
The golden ratio satisfies φ^{10} = 55φ + 34. Researchers building the Recognition Science constant ladder or mass formulas on the phi-ladder cite this step in the Fibonacci recurrence chain. The proof is a short calc that multiplies the ninth-power identity by φ, distributes, substitutes the quadratic relation, and collects terms.
Claim. Let φ be the golden ratio. Then $φ^{10} = 55φ + 34$.
background
The Constants module normalizes the fundamental RS time quantum to one tick. The golden ratio φ satisfies the quadratic x² - x - 1 = 0. Key identity: φ² = φ + 1 (from the defining equation x² - x - 1 = 0). The predecessor states Key identity: φ⁹ = 34φ + 21 (Fibonacci recurrence). These facts sit inside the Recognition framework where φ appears as the self-similar fixed point.
proof idea
The tactic proof uses a calc block. It rewrites φ^{10} as φ · φ^9, substitutes the ninth-power identity, expands by ring, replaces the quadratic term via the square identity, and finishes with ring simplification to obtain the linear and constant coefficients.
why it matters
This lemma supplies the tenth term in the power identities and is invoked by the eleventh-power result. The chain realizes the Fibonacci recurrence forced by the quadratic minimal polynomial of φ, which underpins the phi-ladder used in mass formulas and the eight-tick octave structure of the forcing chain at T6.
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