phi_ninth_eq
The golden ratio satisfies φ^9 = 34φ + 21. Researchers deriving the phi-ladder for mass formulas or time quanta in Recognition Science cite this Fibonacci recurrence identity. The proof is a direct calc chain that multiplies the eighth-power identity by φ and reduces via the quadratic relation φ² = φ + 1.
claim$φ^9 = 34φ + 21$, where $φ$ is the golden ratio satisfying $φ^2 = φ + 1$.
background
In the Constants module the golden ratio φ is the self-similar fixed point forced in the UnifiedForcingChain at step T6. The module builds a sequence of power identities that obey the Fibonacci recurrence and feed the phi-ladder mass formula. The fundamental RS time quantum is set to one tick, τ₀ = 1.
proof idea
A calc tactic begins with φ^9 = φ · φ^8, rewrites using the eighth-power lemma φ^8 = 21φ + 13, applies ring simplification, substitutes φ² = φ + 1 from the quadratic lemma, and finishes with ring arithmetic.
why it matters in Recognition Science
This lemma is invoked directly by the tenth-power identity φ^10 = 55φ + 34. It completes one link in the Fibonacci chain that supports the eight-tick octave (T7), the phi-ladder mass formula, and RS-native constants such as ħ = φ^{-5}. The sequence connects to the Recognition Composition Law and the derivation of three spatial dimensions at T8.
scope and limits
- Does not establish irrationality of φ.
- Does not extend the identity to complex numbers.
- Does not derive physical constants such as α.
- Does not reference the J-cost function or forcing chain steps.
formal statement (Lean)
193lemma phi_ninth_eq : phi^9 = 34 * phi + 21 := by
proof body
Tactic-mode proof.
194 calc phi^9 = phi * phi^8 := by ring
195 _ = phi * (21 * phi + 13) := by rw [phi_eighth_eq]
196 _ = 21 * phi^2 + 13 * phi := by ring
197 _ = 21 * (phi + 1) + 13 * phi := by rw [phi_sq_eq]
198 _ = 34 * phi + 21 := by ring
199
200/-- Key identity: φ¹⁰ = 55φ + 34 (Fibonacci recurrence). -/