Cosmology Dark Energy Phi Dilution Law Predicted Present Amplitude Lt Ceiling

The present amplitude

The dark energy amplitude is a number that describes how strongly dark energy acts today. In the Recognition Science framework, this amplitude is not measured and fitted; it is derived from a rule about how a unit of occupancy, a kind of ledger entry for physical events, weakens as it passes through each spatial dimension. The rule states that one factor of the golden ratio's inverse, about 0.618, is lost per dimension. Since the dark energy carrier is proved to be four-dimensional, the derived amplitude is the golden ratio raised to the minus fourth power, or about 0.146.

The declaration predictedPresentAmplitude_lt_ceiling proves that this derived amplitude is strictly less than a ceiling set by the framework's central cost function, J(phi), which equals phi minus 1.5, roughly 0.118. The proof is conditional: it assumes the dimension-uniform phi dilution law, a definitional choice about how attenuation works. Under that law, the theorem holds with zero gaps in its machine-checked derivation. The same library also proves a sharper bound, that the amplitude is less than one sixth of that ceiling, and a band placing it between 0.015 and 0.018.

The declaration does not claim that the amplitude matches any specific astronomical measurement. It does not assert that dark energy has been observed at this strength. It only establishes a mathematical relationship: the derived present amplitude sits strictly below the ceiling, and it also sits above a strong falsifier threshold and below a sharp DESI threshold, both named in the library. These are internal consistency checks, not empirical confirmations.

What the reader can take away is that the framework's dark energy amplitude is not a free parameter. It is forced by the dilution law and the proved four-dimensional carrier, and it lands in a narrow band. The ceiling comparison is one more link in that chain, showing the amplitude is bounded away from the cost function's value, not merely close to it.

/-- `δw(0)` is strictly below the phantom-Carnot ceiling `J(φ)`: the kernel does not
saturate. -/
theorem predictedPresentAmplitude_lt_ceiling :
    predictedPresentAmplitude < Cost.Jcost phi := by
  have hb := predictedPresentAmplitude_band.2
  have hlo : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
  rw [jcost_phi_closed]; linarith

/-- Closed form of the phantom-Carnot ceiling at `φ`: `J(φ) = φ − 3/2`. -/
theorem jcost_phi_closed : Cost.Jcost phi = phi - 3/2 := by
  unfold Cost.Jcost
  have hphi : phi ≠ 0 := phi_ne_zero
  have hphi_sq : phi ^ 2 = phi + 1 := phi_sq_eq
  field_simp
  nlinarith [sq_pos_of_pos phi_pos, hphi_sq]

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

/-- 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]

/-- `δw(0)` is more than six times below the ceiling, confirming `J(φ)` is a ceiling and not
the value (occupancy `θ = φ⁻⁴ < 1/6`). -/
theorem predictedPresentAmplitude_lt_ceiling_div_six :
    predictedPresentAmplitude < Cost.Jcost phi / 6 := by
  have hb := predictedPresentAmplitude_band.2
  have hlo : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
  rw [jcost_phi_closed]
  have hnum : (0.11 : ℝ) < phi - 3/2 := by linarith
  linarith

/-- 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 declaration does not claim any match to a specific measured dark energy amplitude. It does not claim the dilution law itself is derived; the law is a definitional premise. It does not claim the amplitude is the observed value, only that it is bounded above and below.

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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