phi_fourth_eq
The golden ratio satisfies φ⁴ = 3φ + 2. Researchers deriving higher powers or resonance identities in Recognition Science cite this Fibonacci recurrence. The proof reduces the fourth power by substituting the cubic and quadratic relations then applying ring normalization at each step.
claimLet φ = (1 + √5)/2 be the golden ratio. Then φ⁴ = 3φ + 2.
background
The golden ratio φ satisfies the quadratic equation x² - x - 1 = 0, which rearranges directly to φ² = φ + 1. Multiplication by φ then yields the cubic relation φ³ = 2φ + 1. The module treats constants built from this ratio in Recognition Science, taking the base time quantum τ₀ = 1 tick. Upstream phi_sq_eq records the defining identity φ² = φ + 1. Upstream phi_cubed_eq records the next step φ³ = 2φ + 1.
proof idea
The proof is a calc chain. It starts with φ⁴ = φ · φ³, replaces the cubic factor by 2φ + 1, normalizes by ring to 2φ² + φ, substitutes the quadratic identity, and normalizes again to 3φ + 2.
why it matters in Recognition Science
This supplies the exact fourth-power reduction required by the fifth-power identity φ⁵ = 5φ + 3 and by the numerical bounds 6.5 < φ⁴ < 6.9. It is invoked in the Fibonacci-phi universality theorem and in the resonance identity 3φ² = φ⁴ + 1. Within the framework it extends the self-similar fixed-point property of φ (T6) to explicit power reductions that feed the phi-ladder mass formula and the eight-tick octave.
scope and limits
- Does not derive the numerical approximation of φ.
- Does not prove that φ is the unique positive root of its minimal polynomial.
- Does not connect the identity to measured physical constants.
- Does not address extensions to non-integer powers or other algebraic numbers.
Lean usage
calc phi^5 = phi * phi^4 := by ring _ = phi * (3 * phi + 2) := by rw [phi_fourth_eq]formal statement (Lean)
126lemma phi_fourth_eq : phi^4 = 3 * phi + 2 := by
proof body
Tactic-mode proof.
127 calc phi^4 = phi * phi^3 := by ring
128 _ = phi * (2 * phi + 1) := by rw [phi_cubed_eq]
129 _ = 2 * phi^2 + phi := by ring
130 _ = 2 * (phi + 1) + phi := by rw [phi_sq_eq]
131 _ = 3 * phi + 2 := by ring
132
133/-- Key identity: φ⁵ = 5φ + 3 (Fibonacci recurrence).
134 φ⁵ = φ × φ⁴ = φ(3φ + 2) = 3φ² + 2φ = 3(φ + 1) + 2φ = 5φ + 3. -/