phi_fourth
The golden ratio satisfies φ⁴ = 3φ + 2. Cross-domain modules cite this reduction to convert exponential bounds into arithmetic facts about Fibonacci numbers. The proof is a one-line wrapper that directly references the base lemma from Constants.
claimThe golden ratio φ satisfies the equation $φ^4 = 3φ + 2$.
background
The Fibonacci-Phi Identity Universality module states that any power of the golden ratio reduces via φ^n = F(n) · φ + F(n-1), where F is the Fibonacci sequence with F(0)=0 and F(1)=1. This identity supplies the mechanism for per-domain bounds such as φ^8 > 46 and φ^44 > 10^8 by turning them into statements about integer Fibonacci coefficients plus the numerical bracket on φ itself.
proof idea
The proof is a one-line wrapper that applies the lemma phi_fourth_eq from IndisputableMonolith.Constants.
why it matters in Recognition Science
This declaration supplies the n=4 case for the FibonacciPhiCert structure, which aggregates the power reductions φ² = φ + 1, φ³ = 2φ + 1, φ⁴ = 3φ + 2, φ⁵ = 5φ + 3, and φ⁸ = 21φ + 13. It completes part of the C20 universality wave, enabling reduction of high powers in cross-domain bounds. In the Recognition framework it supports the self-similar fixed point property of φ.
scope and limits
- Does not prove the general recurrence for arbitrary n.
- Does not derive numerical values or bounds on φ.
- Does not connect to the J-cost function or the T0-T8 forcing chain.
Lean usage
example : phi ^ 4 = 3 * phi + 2 := phi_fourthformal statement (Lean)
35theorem phi_fourth : phi ^ 4 = 3 * phi + 2 := phi_fourth_eq
proof body
Term-mode proof.
36
37/-- Already proved: φ⁵ = 5φ + 3 = F(5)·φ + F(4). -/