Cosmology Dark Energy Equation Of State Depth Phi5 Eq

The phi-fifth identity

The golden ratio, usually written φ, is the number that solves r² = r + 1, about 1.618. A less famous but exact relation is φ⁵ = 5φ + 3. This is not an approximation; it follows from the defining quadratic by repeated substitution. The identity is a special case of the general pattern where powers of φ reduce to linear expressions in φ with Fibonacci numbers as coefficients: φ² = φ + 1, φ³ = 2φ + 1, φ⁴ = 3φ + 2, φ⁵ = 5φ + 3. The sequence 1, 1, 2, 3, 5 is the start of the Fibonacci series.

In the Recognition Science framework, this identity appears inside a machine-checked library of formal theorems, specifically in a file about the dark energy equation of state. The framework models the dark energy equation of state parameter w(z), which describes how the pressure of dark energy relates to its density, as a function of redshift. The library defines five canonical models for this parameter: ΛCDM with w = -1, wCDM with constant w, the w0wa CPL parametrization, quintessence, and phantom. A theorem in the library proves that this list of five models is complete in the framework's sense, and a separate definition sets a bound δ ≤ 1/φ⁵ ≈ 0.09 on deviation from w = -1.

The declaration phi5_eq is the formal proof that φ⁵ = 5φ + 3. Its role is purely algebraic: it rewrites the fifth power of the golden ratio into a simpler linear form. This rewrite is then used in a proof that the bound 1/φ⁵ is less than 0.1. The theorem itself says nothing about physics. It does not derive the dark energy equation of state, does not choose among the five models, and does not assert that any of them is correct. It is a computational convenience, a way to handle the number 1/φ⁵ exactly within the formal system.

What the declaration does not claim is worth stating plainly. It does not claim that dark energy actually varies according to this bound, nor that the five models are the only ones possible in nature. The completeness of the five-model list is a statement about a definitional choice in the framework, not a discovery about the cosmos. The bound δ ≤ 1/φ⁵ is a model parameter, not a measured quantity. The framework's library proves the arithmetic of the bound, but the connection between this bound and real astronomical observations remains a hypothesis, not a theorem.

theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  nlinarith

theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  nlinarith

inductive DarkEnergyModel where
  | lambdaCDM
  | wCDM
  | w0wa_CPL
  | quintessence
  | phantom
  deriving DecidableEq, Repr, BEq, Fintype

theorem darkEnergyModel_count : Fintype.card DarkEnergyModel = 5 := by decide

What this page does not claim

The declaration does not prove that dark energy varies according to the bound δ ≤ 1/φ⁵. The five-model list is a definitional choice, not a theorem about the natural world. The bound 1/φ⁵ is a model parameter, not a measured quantity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/DarkEnergyEquationOfStateDepth.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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