Gravity Full Efewith Dark Energy Lambda Efe Kappa

The preserved coupling

Einstein's field equations tie the curvature of spacetime to its energy and matter content. In the standard form, a term proportional to the metric, the cosmological constant Λ, represents the energy of empty space itself. Adding this term changes the gravitational response on the largest scales, but a separate quantity, the coupling constant κ, sets the overall strength of gravity. The declaration lambda_efe_kappa concerns what happens to κ when Λ is introduced.

The framework's machine-checked library of formal theorems proves that in its extended Einstein data, the coupling constant remains exactly κ = 8φ⁵, where φ is the golden ratio. This is not an approximation or a numerical coincidence; it is a theorem with no unproved assumptions. The value is identical to the baseline case with no cosmological constant. The extension changes only the cosmological-constant slot, leaving the gravitational coupling untouched.

In Recognition Science, this result is part of a larger derivation. The framework models the universe as storing a discrete record of recognition events, and from that starting point it forces a chain of physical constants. The coupling κ = 8φ⁵ emerges from that chain, and the dark-energy extension preserves it. The declaration also proves the extended data remains four-dimensional, and that the vacuum energy density is positive when the Hubble parameter is positive.

The declaration does not claim that dark energy itself is derived. It takes the dimensionless dark-energy fraction ΩΛ as an input, with a value of 11/16 − α/π, where α is the fine-structure constant. The proof shows that with this input, the vacuum term is positive and the equation of state is exactly w = −1, meaning the vacuum pressure is the negative of its density. It also proves the vacuum term is covariantly conserved, consistent with the Bianchi identity, because the metric is covariantly constant.

What the theorem establishes is structural consistency: a positive, conserved vacuum term can enter the Einstein equations without disturbing the derived gravitational coupling. The absolute scale of the cosmological constant still depends on the measured Hubble parameter; the theorem does not predict that number. It shows that the framework's gravity sector can accommodate dark energy while preserving its central derived constant.

theorem lambda_efe_kappa (H0sq : ℝ) :
    (rs_efe_data_with_lambda H0sq).kappa = 8 * phi ^ 5 := FullEFE.rs_efe_kappa

/-- The full EFE data carrying the forced cosmological constant `Λ_RS(H₀²)`, with the same
derived coupling `κ = 8φ⁵` and dimension 4 as the baseline `rs_efe_data`. -/
def rs_efe_data_with_lambda (H0sq : ℝ) : FullEFE.FullEFEData where
  dimension := 4
  dim_eq := rfl
  kappa := FullEFE.rs_efe_data.kappa
  kappa_pos := FullEFE.rs_efe_data.kappa_pos
  cosmological_constant := Lambda_RS H0sq

theorem lambda_efe_dimension (H0sq : ℝ) :
    (rs_efe_data_with_lambda H0sq).dimension = 4 := rfl

theorem lambda_efe_lambda_pos {H0sq : ℝ} (h : 0 < H0sq) :
    0 < (rs_efe_data_with_lambda H0sq).cosmological_constant := Lambda_RS_pos h

/-- **VACUUM EQUATION OF STATE IS `w = −1`.** The static vacuum fluid has `p/ρ = −1`. -/
theorem vacuum_eos (rho : ℝ) (h : rho ≠ 0) : vacuum_pressure rho / rho = -1 := by
  unfold vacuum_pressure
  rw [neg_div, div_self h]

What this page does not claim

The declaration does not derive the value of the dark-energy fraction ΩΛ; it takes it as an input. The declaration does not predict the numerical value of the cosmological constant from first principles. The declaration does not establish that the framework's dark-energy model matches astronomical observations.

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/Gravity/FullEFEWithDarkEnergy.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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