Cosmology Dark Energy Phi Dilution Law Dark Energy Carrier Dimension Eq Four

The carrier's dimension

In the Recognition Science framework, the dark-energy carrier is a mathematical object that transmits a unit of dark-energy occupancy through spacetime. The declaration darkEnergyCarrierDimension_eq_four proves that this carrier has exactly four dimensions. It does so by unfolding the definition of the carrier and applying a previously established theorem, lambda_efe_dimension, which already fixed the dimension of the full Einstein field equations with dark energy to be four. The proof is a direct rewrite, not a new physical postulate.

The consequence is a specific value for the dark-energy amplitude, denoted theta. The framework's phi-dilution law states that a unit occupancy transmitted through an n-dimensional carrier is attenuated by one factor of phi to the minus one per dimension, so the occupancy fraction is phi to the minus n. Since the carrier dimension is four, the law forces theta to equal phi to the minus four. This is not a fitted number; it is a derived consequence of the dimension being four and the dilution law being the admissibility rule.

The theorem also yields a closed form for the predicted present amplitude: (phi minus 3/2) divided by (3 times phi plus 2). The framework's machine-checked library proves this value lies strictly between 0.015 and 0.018, that it is positive, and that it sits below the cost ceiling J(phi) by a factor of more than six. It also places the prediction above a strong falsifier and below a sharp DESI threshold, both named in the library.

The declaration itself only establishes the carrier's dimension. It does not derive the phi-dilution law from first principles; that law is a named premise, stated as an admissibility rule. It also does not claim that the physical universe has four spatial dimensions in the everyday sense. The carrier dimension is a property of the framework's EFE carrier, not a direct statement about observed space. The proof is conditional on the dilution law being the correct rule, and the comparison against measured cosmic-aging amplitudes is an empirical check, not part of the theorem.

/-- The dark-energy EFE carrier is four-dimensional. -/
theorem darkEnergyCarrierDimension_eq_four : darkEnergyCarrierDimension = 4 := by
  unfold darkEnergyCarrierDimension
  exact Gravity.FullEFEWithDarkEnergy.lambda_efe_dimension 1

/-- The dimension-uniform dilution law forces `θ=φ⁻⁴` because the EFE carrier dimension is
`4`. -/
theorem darkEnergyThetaFromDimension_eq_phiFour :
    darkEnergyThetaFromDimension = DarkEnergyThetaPhiFour.thetaPhiFour := by
  unfold darkEnergyThetaFromDimension phiDilution DarkEnergyThetaPhiFour.thetaPhiFour
  rw [darkEnergyCarrierDimension_eq_four]

/-- Numeric band: `0.015 < δw(0) < 0.018`, robust to the `(1.61, 1.62)` φ enclosure. -/
theorem predictedPresentAmplitude_band :
    0.015 < predictedPresentAmplitude ∧ predictedPresentAmplitude < 0.018 := by
  rw [predictedPresentAmplitude_closed]
  have hlo : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
  have hhi : phi < (1.62 : ℝ) := phi_lt_onePointSixTwo
  have hden : (0 : ℝ) < 3 * phi + 2 := three_phi_plus_two_pos
  constructor
  · rw [lt_div_iff₀ hden]; nlinarith [hlo]
  · rw [div_lt_iff₀ hden]; nlinarith [hhi]

What this page does not claim

The phi-dilution law is not derived; it is a named premise. The carrier dimension is not a direct claim about observed physical space. The predicted amplitude is not an empirical measurement, but a derived prediction.

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/DarkEnergyPhiDilutionLaw.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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