Cosmology Dark Energy Equation Of State Depth Dark Energy Eo Sdepth Cert

The certification

Dark energy is the name cosmologists give to the observed acceleration of the universe's expansion. The simplest explanation is a cosmological constant, a fixed energy density of empty space, usually written with equation of state w = -1. Alternatives allow w to vary with time or redshift z: wCDM keeps it constant but free, the CPL parametrization lets it drift linearly, quintessence models have w above -1, and phantom models push it below. These five are the standard working set in the literature.

The Recognition Science declaration DarkEnergyEoSDepthCert is a machine-checked certificate, a small package of four proved facts about that set. It proves there are exactly five canonical models, matching the framework's counting of configuration dimensions. It proves a Fibonacci identity for the golden ratio, phi^5 = 5*phi + 3, which is a classical algebraic fact. And it proves that a certain bound, delta = 1/phi^5, is positive and smaller than 0.1; numerically it is about 0.09. The certificate assembles these into one structure, and the whole file compiles with no unproved assumptions.

In the framework's cosmology, the bound delta is the allowed deviation of the dark energy equation of state from -1, written w_BIT(z) = -1 + delta. The certificate establishes that this deviation is small but nonzero, about 9 percent at most. That is the substantive claim: within this model, dark energy sits close to a cosmological constant but is not exactly one.

What the certificate does not do is as important as what it does. It does not derive the value of delta from deeper principles; it only bounds it. It does not compare the five models against observational data, and it does not assert that any one of them is the true description of the universe. It is a structural result about a finite list and a number, not a measurement or a prediction.

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

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 deltaBound_pos : 0 < deltaBound := by
  unfold deltaBound
  exact div_pos one_pos (pow_pos phi_pos 5)

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

What this page does not claim

The certificate does not claim that any of the five models is physically realized. It does not claim that delta equals a measured value, only that it is bounded above. It does not claim to derive the existence of dark energy itself.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: