Cosmology Dark Energy Amplitude Derivation Attenuated Amplitude Le Ceiling

The amplitude ceiling

In cosmology, the amplitude of dark energy describes the strength of the force driving the accelerated expansion of the universe. The Recognition Science framework approaches this amplitude through its cost function, a mathematical object that measures the price of recognition events. The framework's library of machine-checked formal theorems establishes that this dynamic amplitude cannot exceed a specific ceiling, denoted J(φ), where φ is the golden ratio. This ceiling is not a guess or an adjustable parameter; it is a proved upper bound.

The theorem attenuatedAmplitude_le_ceiling states that any admissible amplitude, defined as the product of a fraction θ between 0 and 1 and the ceiling J(φ), is always less than or equal to that ceiling. The fraction θ represents an attenuation factor, a number that scales the ceiling down to the actual physical amplitude. The proof is direct: since θ is at most 1 and J(φ) is positive, multiplying them cannot exceed J(φ) itself. Saturation, where θ equals 1 and the amplitude equals the ceiling, is a special case that the framework also proves.

The framework goes further with data. It defines an implied occupancy fraction based on the current sharp bound on the ΩΛ-gap, a measure of the discrepancy between observed and predicted dark energy density. This implied fraction is proved to be strictly positive and strictly less than 1/6. This means the data-implied amplitude sits below J(φ)/6, so the actual attenuation is less than one-sixth of the ceiling. The full ceiling is therefore at least six times too large to be the gap-implied dynamic amplitude, a precise sub-saturation statement.

What the declaration does not claim is equally important. It does not derive the value of θ itself. The theorem proves the envelope, the correct upper bound, and the data-implied sub-saturation bound, but the remaining problem is to derive the cosmic-Z coupling fraction θ from lower RS primitives. This is left as an open target, not an impossibility. The framework explicitly states that the dynamic amplitude is not J(φ) by default; J(φ) is the ceiling, and admissible amplitudes are attenuated versions of it.

/-- 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]

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

What this page does not claim

The declaration does not derive the value of the attenuation fraction θ itself. The theorem does not state that the dynamic amplitude equals the ceiling J(φ); it only establishes an upper bound. The data-implied sub-saturation bound does not specify the exact amplitude, only that it lies below J(φ)/6.

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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