Cosmology Dark Energy Amplitude Derivation Implied Occupancy Pos

The attenuation fraction

In cosmology, the dark energy amplitude is a number that measures the strength of the cosmic acceleration attributed to dark energy. The Recognition Science framework models this amplitude as an attenuation fraction: a multiplier θ between 0 and 1 applied to a ceiling value J(φ), which is the maximum possible amplitude derived from the framework's cost function. The declaration impliedOccupancy_pos is a machine-checked theorem stating that the amplitude implied by current cosmic aging data, expressed as this fraction, is strictly greater than zero.

The theorem is part of a larger result called the amplitude envelope certificate. The certificate establishes three facts: any admissible attenuation fraction yields a nonnegative amplitude, any such amplitude lies at or below the ceiling J(φ), and saturation, where θ equals 1, exactly reaches the ceiling. The specific theorem impliedOccupancy_pos proves the first of these for the data-implied fraction, using the positivity of the implied maximum from the cosmic aging data and the positivity of the ceiling J(φ).

The significance of the theorem is that it rules out a zero amplitude for dark energy within the framework. A zero amplitude would mean no dark energy effect at all, contradicting the data. The theorem guarantees that the framework's model of dark energy has a positive, nonzero strength. It does not, however, determine the exact value of θ. The framework explicitly leaves the derivation of θ as an open problem, to be derived from lower-level primitives in future work.

In Recognition Science, this theorem is part of a chain that tightens the framework's account of cosmic acceleration. The framework's library, a machine-checked collection of formal theorems, proves the envelope and the sub-saturation bound. The remaining task is to derive the cosmic-Z coupling fraction θ from more fundamental principles. The theorem impliedOccupancy_pos is a step in that direction, confirming that the data-implied amplitude is a positive fraction of the ceiling, but it does not claim to have found the value of that fraction.

/-- 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

/-- 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

/-- Saturation is the special case `θ=1`. -/
theorem attenuatedAmplitude_saturation :
    attenuatedAmplitude 1 = Cost.Jcost phi := by
  unfold attenuatedAmplitude
  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

What this page does not claim

The theorem does not determine the numerical value of the attenuation fraction θ. The theorem does not establish that the dark energy amplitude is exactly the ceiling J(φ). The theorem does not derive θ from lower RS primitives; that derivation remains open.

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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