Cosmology Dark Energy Amplitude Derivation Attenuated Amplitude Saturation

The attenuation ceiling

In cosmology, dark energy is the name for whatever is accelerating the expansion of the universe. Its strength is often summarized by a single number, an amplitude. The Recognition Science framework approaches that number through a cost function, a measure of how expensive a recognition event is. The framework's library, a machine-checked collection of formal theorems, proves that this cost function has a specific maximum value, written J(φ).

The declaration attenuatedAmplitude_saturation establishes a precise relationship between that maximum and the physical amplitude. It proves that the amplitude is not the maximum itself, but a fraction of it. The fraction, called θ, is allowed to range from zero to one. When θ equals one, the amplitude reaches the ceiling. The theorem states this special case exactly: when θ is one, the attenuated amplitude equals J(φ). This is a formal identity, checked by the machine, and it defines what saturation would mean.

The more interesting result is that the universe does not appear to be saturated. The framework derives a maximum implied amplitude from a gap in the measured dark-energy density parameter, ΩΛ. That implied amplitude is strictly less than one sixth of the ceiling. The corresponding attenuation fraction θ is therefore positive but smaller than 1/6. The full ceiling is at least six times too large to match the data-implied value. This is a theorem, not a conjecture: the library proves the bound from the gap.

What the declaration does not claim is just as important. It does not derive the value of θ itself. The framework proves the correct envelope, the ceiling and the sub-saturation bound, but the actual coupling fraction remains an open scalar problem. The theorem also does not assert that dark energy is a recognition effect in any measurable sense. It establishes a mathematical envelope within the framework's own model, not a direct physical prediction. The classical cosmological constant problem, why the observed value is so small, is not solved by this result.

The practical consequence is a sharper target. Instead of deriving the full amplitude, the framework reduces the problem to one number, θ, which must lie between zero and one sixth. Any future derivation of θ from lower primitives would complete the picture. Until then, the ceiling and the bound stand as the proven constraints.

/-- 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 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 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

This does not claim to derive the numerical value of θ. This does not claim the framework's amplitude model is a direct physical description of dark energy. This does not claim to solve the cosmological constant problem.

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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