Cosmology Dark Energy Phi Dilution Law Dark Energy Theta From Dimension Lt One S

A small number with a large job

Dark energy is the name cosmologists give to the unknown agent that appears to accelerate the expansion of the universe. In the Recognition Science framework, that agent is modeled as a kind of occupancy: a unit of something transmitted through a carrier, and the strength of that transmission is a number called theta. The declaration darkEnergyThetaFromDimension_lt_one_sixth is a machine-checked theorem stating that this theta, when derived from first principles, is less than one sixth.

The derivation is a chain. The framework's library proves that the carrier for dark energy has dimension 4. A separate rule, the phi-dilution law, says that a unit transmitted through an n-dimensional carrier is attenuated by one factor of the golden ratio's inverse per dimension. Since the golden ratio is about 1.618, its fourth power is about 6.854, so the occupancy fraction is 1 over that, roughly 0.146, which is indeed below one sixth. The theorem darkEnergyThetaFromDimension_lt_one_sixth is the formal statement of that inequality.

The point of the bound is not the number itself but what it enables. The framework uses theta to compute a present-day amplitude for dark energy, and that amplitude lands in a band between 0.015 and 0.018. That band is then compared against two named thresholds: a falsifier from cosmic aging data and a sharp threshold from DESI observations. The theorem proves the predicted amplitude sits above the falsifier and below the DESI threshold, which means the prediction is not trivially satisfied; it is a claim that could be tested and could fail.

What the declaration does not claim is just as important. It does not prove that the phi-dilution law is true; that law is a definitional choice, an admissibility rule, not a derived result. The theorem is conditional on that law. It also does not claim that theta equals one sixth; it claims only that theta is less than one sixth. And it does not claim that the dark-energy carrier dimension of 4 is derived here; that fact is imported from another theorem in the library. The inequality is a proved step inside a larger, still-open physical story.

/-- The dimension-forced theta is below the sharp `1/6` envelope. -/
theorem darkEnergyThetaFromDimension_lt_one_sixth :
    darkEnergyThetaFromDimension < (1 : ℝ) / 6 := by
  rw [darkEnergyThetaFromDimension_eq_phiFour]
  exact DarkEnergyThetaPhiFour.thetaPhiFour_lt_one_sixth

/-- Dimension-uniform φ attenuation through a carrier dimension `n`: one factor of `φ⁻¹`
per dimension. -/
def phiDilution (n : ℕ) : ℝ := 1 / phi ^ n

/-- 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 proved true rather than chosen as a model rule. Theta equals one sixth; the theorem only proves it is less. The dark-energy carrier dimension is derived in this module; it is imported from elsewhere.

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.

Derived articles

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