Cosmology Dark Energy Amplitude Derivation Implied Amplitude Below Ceiling

The amplitude ceiling

In cosmology, the dark-energy amplitude is a number describing the strength of the cosmic acceleration attributed to dark energy. The Recognition Science framework approaches this number through a theoretical maximum, which it calls the ceiling, a bound that no admissible amplitude can exceed. The framework's machine-checked library of formal theorems proves that the amplitude implied by current observational data sits strictly below this ceiling, and more sharply, below one-sixth of it. The theorem `implied_amplitude_below_ceiling` states this result directly: the data-implied maximum is less than the ceiling value J(φ).

The proof begins with a definition. The framework models the dynamic dark-energy amplitude as an attenuation fraction θ times the ceiling J(φ), where θ is a real number between 0 and 1. The ceiling itself comes from the framework's cost function J(x) = (x + 1/x)/2 - 1 evaluated at the golden ratio φ. The theorem `attenuatedAmplitude_le_ceiling` proves that any admissible attenuation fraction yields an amplitude at or below the ceiling. Saturation, the special case θ = 1, exactly equals the ceiling, as `attenuatedAmplitude_saturation` shows.

The sharp result uses a separate quantity: the gap-implied maximum, which the framework derives from the cosmic age and the observed matter density gap. The theorem `impliedOccupancy_lt_one_sixth` proves that the attenuation fraction corresponding to this gap-implied amplitude is strictly less than 1/6. Consequently, the full ceiling J(φ) is at least six times too large to be the actual dynamic amplitude. The framework packages these results into a certificate structure, `DynamicAmplitudeEnvelopeCert`, which records the ceiling's positivity, the admissibility bounds, and the strict sub-saturation bound.

What the framework does not claim is the value of θ itself. The theorems establish the envelope and the bound, but they do not derive the cosmic-Z coupling fraction θ from lower primitives. That remains an open target within the framework. The framework also does not claim that the ceiling J(φ) is the measured dark-energy amplitude; the whole point of the attenuation model is that the physical amplitude is smaller. The framework's library proves the envelope and the sub-saturation bound, but it does not provide a numerical prediction for the dark-energy amplitude that would match a specific observational value.

/-- 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

/-- 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 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]

What this page does not claim

The framework does not derive the numerical value of the attenuation fraction θ. The framework does not claim that the ceiling J(φ) equals the measured dark-energy amplitude. The framework does not provide a specific numerical prediction for the dark-energy amplitude.

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)

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