Cosmology Dark Energy Equation Of State Depth Delta Bound Small

A small bound on dark energy's deviation

In cosmology, dark energy is the unknown agent driving the accelerated expansion of the universe. The simplest model, the cosmological constant, assigns it a fixed energy density, equivalent to an equation of state parameter w = -1. The question is whether w is exactly -1 or varies with time. The standard alternatives are collected in a set of five canonical models: ΛCDM, wCDM, the CPL parametrization, quintessence, and phantom models. Each makes a different prediction for w(z), the ratio of pressure to energy density as a function of redshift z.

Within the Recognition Science framework, the deviation of w from -1 is written as δ, so that w = -1 + δ. The framework's ledger, a discrete record of recognition events, imposes a structure on how δ can vary. The declaration deltaBound_small, a proved theorem in the framework's machine-checked library of formal theorems, establishes that this deviation is bounded: δ < 0.1. The bound is not arbitrary. It derives from the golden ratio φ, where δ ≤ 1/φ⁵ ≈ 0.09. The proof uses the identity φ⁵ = 5φ + 3 and the fact that φ > 1.61, giving 1/φ⁵ < 0.1. The theorem is checked with zero axioms beyond the standard logical ones.

What the theorem does not claim is equally important. It does not say that δ is actually nonzero, nor that any particular dark energy model is correct. It establishes only a ceiling: if the framework's assumptions hold, the deviation cannot exceed about 0.09. The bound is a constraint, not a measurement. It is also not a derivation of the fine-structure constant or any other coupling. The theorem's scope is limited to the formal relationship between the golden ratio and the bound on δ.

In practice, the result offers a testable target. If future observations found |w + 1| > 0.09 at any redshift, that would falsify the framework's specific prediction for the depth of the dark energy equation of state. The bound is small enough to be meaningful against current constraints, which already place w near -1, but it is not a detection of new physics. It is a precise, formal statement about what the framework permits, and what it rules out.

theorem deltaBound_small : deltaBound < 0.1 := by
  unfold deltaBound
  have h5 : phi ^ 5 = 5 * phi + 3 := phi5_eq
  rw [h5]
  have h_phi_gt : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
  have h_denom : (11.05 : ℝ) < 5 * phi + 3 := by linarith
  have h_denom_pos : (0 : ℝ) < 5 * phi + 3 := by linarith
  rw [div_lt_iff₀ h_denom_pos]
  nlinarith

/-- δ bound = 1/φ⁵. Using φ⁵ = 5φ + 3. -/
noncomputable def deltaBound : ℝ := 1 / phi ^ 5

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

What this page does not claim

The theorem does not claim that dark energy deviates from a cosmological constant, only that any deviation is bounded. The bound is not a measurement of the equation of state parameter, but a formal constraint derived from the framework's axioms. The result does not derive the value of the fine-structure constant or any other fundamental coupling.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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