phi5_eq
plain-language theorem explainer
φ⁵ equals 5φ + 3 for the golden ratio φ. Cosmologists working on RS-derived dark energy and Hubble tension models cite the identity to evaluate Λ_RS = 8φ⁵/45 and obtain the interval (1.88, 2.03). The proof reduces the claim to the base quadratic via three intermediate power relations and closes with linear arithmetic.
Claim. $φ^5 = 5φ + 3$ where $φ = (1 + √5)/2$ satisfies $φ^2 = φ + 1$.
background
The module Cosmological Constant from RS — S3 Depth states that Λ_RS = 8φ⁵/45 lies in (1.88, 2.03) and supplies the dimensionless ratio Ω_Λ = Λ_RS / (3H₀²) ≈ 0.685. This builds on the prior OmegaLambdaBITKernelBand result that already fixes the expression for Λ_RS; the present identity renders the constant numerically explicit in RS-native units where c = 1 and ħ = φ^{-5}.
proof idea
The tactic proof first obtains the quadratic identity φ² = φ + 1 from phi_sq_eq. It derives φ³ = 2φ + 1 and φ⁴ = 3φ + 2 by nlinarith, then closes the target equality with a final nlinarith step.
why it matters
The identity is required by darkEnergyEoSDepthCert to certify deltaBound bounds and by hubbleTensionCert to bound the Hubble tension pipeline. It supplies the algebraic step for computing Λ_RS = 8φ⁵/45 in the S3 depth module and links to the phi fixed point in the forcing chain. The result is fully proved with zero remaining scaffolding.
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