Cosmology Dark Energy Theta Status Theta Status Cert

The status certificate

In cosmology, the amplitude of dark energy is often summarized by a single number, the equation of state parameter w. Recognition Science writes that parameter as a product: δw0 = θ · J(φ), where J(φ) is a fixed, derived cost value and θ is a fraction between 0 and 1 that attenuates it. The framework's machine-checked library of formal theorems proves the envelope for this amplitude: the implied θ lies strictly between 0 and 1/6, and the resulting amplitude is positive and never exceeds the J(φ) ceiling. These are the sharp bounds currently established.

What the library does not yet have is a first-principles derivation of the exact value of θ. The declaration ThetaStatusCert is the honest record of that gap. It defines the theorem-shaped input a future derivation must supply: a real number θ with 0 < θ ≤ 1. It then proves that once such a θ is provided, all amplitude obligations reduce to already proved envelope theorems. This prevents the common mistake of replacing the missing derivation with a fitted value while still making the remaining algebra fully theorem-checked.

In Recognition Science, the status certificate is a ledger, a discrete record of what is proved and what is not. The certificate is inhabited, meaning the record exists and is internally consistent. It states three things: the exact theorem target is ThetaFromFirstPrinciples; the current data-envelope θ is positive and below 1/6; and any future first-principles θ immediately supplies a positive amplitude below J(φ). The certificate does not claim that the exact θ has been derived, nor that the envelope value is a derivation. It marks the exact θ as an open target.

This matters because it separates two very different claims. The envelope bound, that θ is between 0 and 1/6, is a proved theorem about the data. The exact value of θ, whatever it turns out to be, is not yet derived from first principles. A reader who sees the certificate knows precisely which parts of the dark-energy amplitude story rest on machine-checked proof and which parts remain a target for future work.

/-- 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⟩

/-- 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

What this page does not claim

The exact value of θ has been derived from first principles. The envelope value of θ is a derivation rather than a data-envelope value. The status certificate itself provides the missing first-principles derivation.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: