Cosmology Dark Energy Theta Status Theta Derived Amplitude Pos

The amplitude guarantee

In cosmology, dark energy is often modeled as a correction to the vacuum energy density, written as δw0 = θ · J(φ). Here θ is an unknown attenuation fraction, a number between 0 and 1 that scales how much of the framework's base cost J(φ) actually contributes. The framework's machine-checked library of formal theorems proves a conditional guarantee about this amplitude.

The theorem thetaDerived_amplitude_pos states: if a future derivation supplies any θ that is positive and at most 1, then the attenuated amplitude θ · J(φ) is strictly greater than zero. The proof is short: it multiplies the positivity of θ by the already-proved positivity of J(φ). A companion theorem, thetaDerived_amplitude_le_ceiling, shows the same amplitude never exceeds the J(φ) ceiling. Together they mean the amplitude is always in the open interval (0, J(φ)].

This is a theorem about a conditional, not a measurement. The library does not know what θ is. It only knows that if θ comes from a first-principles derivation, the amplitude behaves correctly. The framework's current sharp bound, impliedTheta_band, is a data-envelope value: it states 0 < θ_implied < 1/6, but that value comes from fitting observations, not from derivation. The exact θ remains an open target.

The practical consequence is a clean separation. The framework proves all the algebra that follows once θ is known, and it leaves the missing derivation as a named target. This prevents the common mistake of replacing an unproved quantity with a fitted number while the remaining reasoning stays fully theorem-checked.

/-- Any first-principles theta derivation gives a positive amplitude. -/
theorem thetaDerived_amplitude_pos (T : ThetaFromFirstPrinciples) :
    0 < attenuatedAmplitude T.theta := by
  unfold attenuatedAmplitude
  exact mul_pos T.theta_pos Constants.Jcost_phi_pos

/-- Any first-principles theta derivation keeps the amplitude below the `J(φ)` ceiling. -/
theorem thetaDerived_amplitude_le_ceiling (T : ThetaFromFirstPrinciples) :
    attenuatedAmplitude T.theta ≤ Cost.Jcost phi :=
  attenuatedAmplitude_le_ceiling T.toAmplitudeAttenuation

/-- The current sharp implied theta lies in the admissible positive sub-saturation band. -/
theorem impliedTheta_band :
    0 < impliedOccupancy ∧ impliedOccupancy < (1 : ℝ) / 6 :=
  ⟨impliedOccupancy_pos, impliedOccupancy_lt_one_sixth⟩

What this page does not claim

The exact value of θ is derived from first principles; it remains open. The implied θ value of 1/6 is a derivation; it is an envelope fit to data. The theorem proves dark energy exists; it only constrains an amplitude if θ is supplied.

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/DarkEnergyThetaStatus.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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