phi5_eq
The golden ratio satisfies φ^5 = 5φ + 3. Dark energy modelers and Hubble tension analysts cite the relation to bound the equation-of-state deviation δ below 0.1 in the BIT kernel. The proof is a short tactic sequence that starts from the quadratic identity φ² = φ + 1 and applies linear arithmetic to obtain the higher powers.
claimLet φ denote the golden ratio. Then φ^5 = 5φ + 3.
background
The module treats w(z) on the φ-ladder with adjacent-redshift corrections of order φ^{-n}. It enumerates five canonical models for configDim D = 5: ΛCDM (w = -1), wCDM, w0wa CPL, quintessence, and phantom. The canonical BIT kernel sets w_BIT(z) = -1 + δ with δ ≤ 1/φ^5 ≈ 0.09. This identity rests on the upstream result that φ² = φ + 1, the defining quadratic relation of the golden ratio.
proof idea
The tactic proof first invokes the upstream quadratic identity to obtain φ² = φ + 1. It then applies nlinarith to derive φ³ = 2φ + 1 and φ⁴ = 3φ + 2. A final nlinarith step combines the relations to reach φ^5 = 5φ + 3.
why it matters in Recognition Science
This theorem supplies the phi5_fibonacci field in darkEnergyEoSDepthCert and hubbleTensionCert. It also anchors deltaBound_small and appears in inflaton potential and cosmological constant derivations. Within the Recognition framework it instantiates the fifth-power Fibonacci identity required for bounding the dark energy deviation and for closing the φ-ladder relations used in mass formulas.
scope and limits
- Does not compute a numerical value for φ.
- Does not extend the identity to non-integer exponents.
- Does not incorporate cosmological observations or data.
- Does not derive the full functional form of w(z).
- Does not address forcing chain steps or spatial dimensions.
formal statement (Lean)
33theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
proof body
Tactic-mode proof.
34 have h2 := phi_sq_eq
35 have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
36 have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
37 nlinarith
38