Cosmology Dark Energy Amplitude Derivation Attenuated Amplitude Nonneg

The amplitude envelope

Dark energy is the name cosmologists give to the unknown agent accelerating the expansion of the universe. In the Recognition Science (RS) framework, its dynamic amplitude is not a free number but an attenuation fraction: a fraction θ of a fixed ceiling J(φ), where θ is allowed to range from 0 to 1. The declaration attenuatedAmplitude_nonneg proves, in the framework's machine-checked library of formal theorems, that any admissible attenuation fraction yields a nonnegative amplitude. The proof is short: it multiplies the nonnegativity of θ by the positivity of the ceiling J(φ).

The statement is a theorem, not a hypothesis. It is one of three results that together form an amplitude envelope certificate. The second, attenuatedAmplitude_le_ceiling, proves that any admissible amplitude stays at or below the ceiling J(φ). The third, attenuatedAmplitude_saturation, shows that the ceiling is reached exactly when θ = 1, the saturation case. The nonnegativity theorem is the simplest of the three: it rules out negative amplitudes, which would be unphysical, and it does so from the definition of an admissible attenuation fraction alone.

The framework then applies this envelope to data. A separate definition, impliedOccupancy, takes a sharp bound from cosmic aging data and divides it by the ceiling J(φ), yielding the attenuation fraction implied by that data. Two further theorems, impliedOccupancy_pos and impliedOccupancy_lt_one_sixth, prove that this implied fraction is positive and strictly less than 1/6. The meaning is concrete: the data-implied amplitude sits below J(φ)/6, so the full ceiling is at least six times too large to be the actual dynamic amplitude. Saturation is excluded.

What the declaration does not do is just as important. It does not derive the value of θ. The attenuation fraction remains an open scalar, to be derived from lower RS primitives. The theorem only fixes the envelope: amplitudes are nonnegative, bounded above by J(φ), and, for the current data-implied case, strictly below one sixth of that ceiling. The framework's own docstring states this plainly: this proves the correct envelope and turns the remaining problem into one scalar.

/-- 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

/-- 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

This theorem does not derive the value of the attenuation fraction θ. This theorem does not claim that the dynamic amplitude equals J(φ); saturation is only the special case θ = 1. This theorem does not establish that the data-implied bound is the true amplitude, only that it lies below one sixth of the ceiling.

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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