Cosmology Dark Energy Amplitude Derivation

The amplitude envelope

In cosmology, dark energy is the name for whatever is making the expansion of the universe accelerate. One way to quantify it is through an amplitude: a number that sets how strong the effect is in the equations. The Recognition Science framework approaches this amplitude by first deriving a ceiling, a maximum possible value, and then comparing that ceiling against what astronomical data imply.

The framework's starting point is a cost function, a measure of how expensive a recognition event is, which is forced by five plain conditions to take the form J(x) = (x + 1/x)/2 - 1. Evaluated at the golden ratio phi, this cost gives a number J(phi). An earlier mistake in the framework treated J(phi) directly as the dark energy amplitude. The corrected theorem status is sharper: J(phi) is the ceiling, not the amplitude itself.

The physical dynamic amplitude is an attenuation fraction, written as theta times J(phi), where theta is a real number between 0 and 1. The framework proves that any such admissible fraction gives a nonnegative amplitude below the ceiling. Saturation, the special case theta = 1, recovers exactly J(phi). This is the amplitude envelope certificate: a machine-checked guarantee that the true amplitude cannot exceed J(phi).

Against this ceiling, the framework places the sharp Omega-Lambda-gap-implied amplitude, a maximum value derived from current cosmological data. The data-implied attenuation fraction, called impliedOccupancy, is proved to be strictly positive and strictly below 1/6. This means the full J(phi) ceiling is at least six times too large to be the actual dynamic amplitude. The data-implied amplitude is exactly impliedOccupancy times J(phi), and it sits strictly below the ceiling.

What this establishes in plain language: the framework proves a rigorous upper bound on dark energy's amplitude, and it proves that the value implied by current data is far below that bound. What it does not do is derive the attenuation fraction theta itself. The module turns the remaining problem into one scalar: derive the cosmic-Z coupling fraction theta from lower primitives. That derivation remains open.

/-- Dynamic dark-energy amplitude as an attenuation fraction of the phantom-Carnot ceiling. -/
def attenuatedAmplitude (theta : ℝ) : ℝ := theta * Cost.Jcost phi

/-- An admissible amplitude attenuation fraction. -/
structure AmplitudeAttenuation where
  theta : ℝ
  theta_nonneg : 0 ≤ theta
  theta_le_one : theta ≤ 1

/-- Any admissible attenuation fraction gives a nonnegative amplitude. -/
theorem attenuatedAmplitude_nonneg (a : AmplitudeAttenuation) :
    0 ≤ attenuatedAmplitude a.theta := by
  unfold attenuatedAmplitude
  exact mul_nonneg a.theta_nonneg (le_of_lt Constants.Jcost_phi_pos)

/-- Any admissible attenuation fraction lies below the `J(φ)` ceiling. -/
theorem attenuatedAmplitude_le_ceiling (a : AmplitudeAttenuation) :
    attenuatedAmplitude a.theta ≤ Cost.Jcost phi := by
  unfold attenuatedAmplitude
  calc
    a.theta * Cost.Jcost phi ≤ 1 * Cost.Jcost phi := by
      exact mul_le_mul_of_nonneg_right a.theta_le_one (le_of_lt Constants.Jcost_phi_pos)
    _ = Cost.Jcost phi := by ring

/-- The data-implied attenuation fraction is positive. -/
theorem impliedOccupancy_pos : 0 < impliedOccupancy := by
  unfold impliedOccupancy
  exact div_pos (by linarith [CosmicAgingAmplitudeSharp.delta_w_implied_max_gt])
    Constants.Jcost_phi_pos

/-- The data-implied attenuation fraction is strictly below `1/6`. This is the precise
sub-saturation statement: the full `J(φ)` ceiling is at least six times too large to be the
gap-implied dynamic amplitude. -/
theorem impliedOccupancy_lt_one_sixth : impliedOccupancy < (1 : ℝ) / 6 := by
  unfold impliedOccupancy
  rw [div_lt_iff₀ Constants.Jcost_phi_pos]
  nlinarith [CosmicAgingAmplitudeSharp.jPhi_dominates_implied]

/-- The sharp implied amplitude is exactly `impliedOccupancy · J(φ)`. -/
theorem implied_amplitude_factorization :
    attenuatedAmplitude impliedOccupancy =
      CosmicAgingAmplitudeSharp.delta_w_implied_max := by
  unfold attenuatedAmplitude impliedOccupancy
  field_simp [ne_of_gt Constants.Jcost_phi_pos]

/-- The implied amplitude is strictly below the ceiling. -/
theorem implied_amplitude_below_ceiling :
    CosmicAgingAmplitudeSharp.delta_w_implied_max < Cost.Jcost phi := by
  have h := CosmicAgingAmplitudeSharp.jPhi_dominates_implied
  have hpos := CosmicAgingAmplitudeSharp.delta_w_implied_max_gt
  nlinarith

What this page does not claim

The module does not derive the numerical value of the attenuation fraction theta. The module does not claim that dark energy is literally a recognition cost or a ledger entry. The module does not claim the Omega-Lambda-gap-implied amplitude is the measured dark energy density.

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/DarkEnergyAmplitudeDerivation.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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