Cosmology Hubble Tension Dark Energy Match

The dark energy match

Dark energy is the name cosmologists give to the unknown agent that appears to accelerate the expansion of the universe. Its density is usually quoted as Ω_Λ, a fraction of the total energy budget. The Planck satellite's 2018 data give Ω_Λ = 0.6847 with an uncertainty of 0.0073. The Recognition Science framework, which models reality as a discrete record of recognition events, derives a predicted value of 0.6852 from a geometric base of 11/16 minus a small correction. The theorem dark_energy_match proves that the absolute difference between this prediction and the measured value is less than the measurement's stated error bar, meaning the prediction lands within 1σ of the observation.

The geometric base comes from counting vertices in a three-dimensional cube graph, Q₃, which has 8 vertices. The framework identifies 11 as the number of passive field elements and computes the base as 11/(2 × 8) = 0.6875. The correction subtracts α/π, where α is the fine-structure constant, approximately 0.0023. This yields the predicted 0.6852. The theorem itself is a formal statement in the framework's machine-checked library of formal theorems, verified by a proof assistant with no unproven assumptions beyond the standard logical axioms.

The same module contains a companion result for the Hubble tension, the discrepancy between early-universe and late-universe measurements of the expansion rate. The framework proposes a ratio of 13/12 between late and early values, corresponding to adding one time dimension to twelve spatial edges. The predicted late value is about 73.01, compared to the measured 73.04, a match within 0.05%. This is a separate theorem, hubble_ratio_match, proved in the same file.

What the theorem does not claim is important. It does not prove that dark energy exists or that the framework's model is physically correct. It only establishes a numerical agreement between a geometric formula and one experimental measurement. The fine-structure constant α is not derived within the framework; it is taken from an external CODATA anchor. The theorem does not explain why the universe accelerates, nor does it predict any new observations. It is a consistency check, not a physical derivation.

/-- Dark Energy matches observation to within 1 sigma.

    Omega_L_pred = 11/16 - α_CODATA/π ≈ 0.6875 - 0.00232 ≈ 0.6852
    Omega_L_exp = 0.6847
    Omega_L_err = 0.0073
    |0.6852 - 0.6847| ≈ 0.0005 < 0.0073 ✓

    Proof: From alpha/pi bounds, we establish the match. -/
theorem dark_energy_match :
    abs (Omega_L_pred - Omega_L_exp) < Omega_L_err := by
  have h_ap := alpha_over_pi_bounds
  simp only [Omega_L_pred, Omega_L_exp, Omega_L_err, dark_energy_base]
  -- Omega_L_pred = 11/16 - α_CODATA/π
  -- With α/π ∈ (0.0023, 0.0024):
  -- Omega_L_pred ∈ (0.6875 - 0.0024, 0.6875 - 0.0023) = (0.6851, 0.6852)
  -- |Omega_L_pred - 0.6847| ≤ max(|0.6851 - 0.6847|, |0.6852 - 0.6847|)
  --                        = max(0.0004, 0.0005) = 0.0005 < 0.0073 ✓
  have h_pred_lower :
      (0.6851 : ℝ) <
        (11 : ℝ) / 16 - Constants.ExternalAnchors.alpha_CODATA / Real.pi := by
    have h1 : Constants.ExternalAnchors.alpha_CODATA / Real.pi < (0.0024 : ℝ) := h_ap.2
    have h2 : (11 : ℝ) / 16 = (0.6875 : ℝ) := by norm_num
    linarith
  have h_pred_upper :
      (11 : ℝ) / 16 - Constants.ExternalAnchors.alpha_CODATA / Real.pi <
        (0.6852 : ℝ) := by
    have h1 : (0.0023 : ℝ) < Constants.ExternalAnchors.alpha_CODATA / Real.pi := h_ap.1
    have h2 : (11 : ℝ) / 16 = (0.6875 : ℝ) := by norm_num
    linarith
  rw [abs_lt]
  constructor <;> linarith

/-- Predicted Dark Energy Density.
    EM correction uses the measured fine-structure constant (canonical reading). -/
noncomputable def Omega_L_pred : ℝ :=
  (dark_energy_base : ℝ) - Constants.ExternalAnchors.alpha_CODATA / Real.pi

/-- The Dark Energy base 11/16 derives from passive edges (11) over 2*vertices (16). -/
theorem dark_energy_from_geometry :
    dark_energy_base = 11 / (2 * 8) := by
  simp only [dark_energy_base]
  norm_num

/-- The Hubble Ratio matches observation to within 0.05%.

    pred = 67.4 * (13/12) = 73.0166...
    obs  = 73.04
    |pred - obs| / obs = |73.0166 - 73.04| / 73.04 = 0.00032 < 0.0005 ✓

    This is now PROVEN, not axiomatized. -/
theorem hubble_ratio_match :
    abs (H_late_pred - H_late_exp) / H_late_exp < 0.0005 := by
  simp only [H_late_pred, H_late_exp, H_early_exp, hubble_ratio_topo]
  norm_num

What this page does not claim

The theorem does not prove dark energy exists or that the framework's cosmology is physically correct. The fine-structure constant α is not derived within the framework; it is an external input. The match does not predict any new observations beyond the two numerical agreements stated.

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/HubbleTension.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

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