Cosmology Dark Energy Amplitude Derivation Dynamic Amplitude Envelope Cert

The amplitude envelope

Dark energy is the name for the unknown agent that appears to accelerate cosmic expansion. In the Recognition Science framework, its dynamic amplitude is not a free number. The framework's cost function J(φ), a forced measure of recognition cost, sets an upper bound, a ceiling, on how large that amplitude can be. The new certificate, DynamicAmplitudeEnvelopeCert, is a machine-checked collection of formal theorems that makes this envelope precise.

The central result is a correction. Earlier work treated J(φ) itself as the amplitude. The certificate proves that J(φ) is instead the ceiling, and that any admissible physical amplitude is an attenuation fraction of it: θ · J(φ), with θ between 0 and 1. Saturation, the case θ = 1, is exactly the ceiling. This is a theorem, not a choice: the structure AmplitudeAttenuation demands θ ≥ 0 and θ ≤ 1, and the library proves that any such θ yields a nonnegative amplitude below the ceiling.

The certificate also records a sharp bound from data. The current gap-implied maximum, drawn from the ΩΛ-gap analysis, corresponds to an attenuation fraction strictly below 1/6. In plain terms, the full J(φ) ceiling is at least six times too large to be the actual dynamic amplitude. The certificate proves that this implied amplitude is positive, strictly below the ceiling, and factorizes exactly as the attenuation fraction times J(φ).

What the certificate does not do is derive θ. It proves the envelope and the sub-saturation bound, then stops. The remaining problem is one scalar: derive the cosmic-Z coupling fraction θ from lower RS primitives. That derivation is open. The certificate is a theorem about the shape of the answer, not the answer itself.

/-- 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

/-- 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)

/-- 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 sharp implied amplitude is exactly `impliedOccupancy · J(φ)`. -/
theorem implied_amplitude_factorization :
    attenuatedAmplitude impliedOccupancy =
      CosmicAgingAmplitudeSharp.delta_w_implied_max := by
  unfold attenuatedAmplitude impliedOccupancy
  field_simp [ne_of_gt Constants.Jcost_phi_pos]

/-- 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

What this page does not claim

The certificate does not derive the attenuation fraction θ; that derivation remains open. The certificate does not identify dark energy with any specific physical mechanism or field. The certificate does not claim that the ceiling J(φ) is ever actually reached by the physical 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)

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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