phi5_eq
The golden ratio satisfies φ⁵ = 5φ + 3 as a Fibonacci identity. Cosmologists modeling inflaton potentials and dark energy equations cite this to fix slow-roll parameters ε = 1/(2φ⁵) and η = 1/φ⁵. The proof is a short tactic sequence that reduces the claim to the base quadratic identity φ² = φ + 1 via three applications of nlinarith.
claim$φ^5 = 5φ + 3$, where $φ = (1 + √5)/2$ is the golden ratio.
background
The module examines the RS inflaton potential V(χ) with five structural regimes corresponding to configDim D = 5: slow-roll plateau, slow-roll slope, hilltop decline, reheating, and post-reheating radiation. Slow-roll parameters are defined as ε = 1/(2φ⁵), η = 1/φ⁵, with n_s - 1 = -2/45 and r = 2/(45 φ²); the e-fold count is fixed at 44. This identity rests on the golden ratio φ defined by the quadratic x² - x - 1 = 0 whose positive root satisfies φ² = φ + 1. The upstream lemma phi_sq_eq from Constants and NumberTheory.PhiLadderLattice establishes the base case, which is lifted to higher powers in the phi-ladder lattice.
proof idea
The proof imports the base identity phi_sq_eq : φ² = φ + 1. It derives the auxiliary relations φ³ = 2φ + 1 and φ⁴ = 3φ + 2 by nlinarith. The final nlinarith step combines these to obtain φ⁵ = 5φ + 3.
why it matters in Recognition Science
This supplies the φ⁵ Fibonacci identity required by downstream certificates darkEnergyEoSDepthCert and hubbleTensionCert. It anchors the A2 depth analysis of the inflaton potential where φ⁵ appears in the slow-roll denominators. Within Recognition Science it supports the phi-ladder used for mass formulas and connects to the self-similar fixed point φ forced at T6 of the unified forcing chain.
scope and limits
- Does not derive the inflaton potential V(χ) from first principles.
- Does not prove bounds on the number of e-folds beyond the structural count of 44.
- Does not address the uniqueness of the golden ratio solution.
- Does not extend the identity to non-real numbers.
Lean usage
have h5 : phi ^ 5 = 5 * phi + 3 := phi5_eqformal statement (Lean)
44theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
proof body
Tactic-mode proof.
45 have h2 := phi_sq_eq
46 have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
47 have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
48 nlinarith
49