Gravity Full Efewith Dark Energy Recovers Baseline Lambda

The baseline recovery

The declaration recovers_baseline_lambda is a formal proof about a specific modification to the Einstein field equations. The modification adds a cosmological constant, a term that represents the energy of empty space, to the standard equations. The theorem states that if the Hubble constant, a measure of the universe's expansion rate, is set to zero, then this new term also becomes zero. In that limit, the extended equations are exactly the original equations without dark energy. The proof is a direct calculation: the new constant is defined as a multiple of the square of the Hubble constant, so when that square is zero, the constant is zero.

The theorem is part of a larger, machine-checked library of formal theorems. Its role is to show that a proposed extension of general relativity is consistent with the known baseline. The extension defines the cosmological constant as Λ = 3 H₀² · Ω_Λ, where H₀ is the Hubble constant and Ω_Λ is a dimensionless fraction. The library proves that this fraction is positive, which makes the constant positive for any positive expansion rate. The theorem recovers_baseline_lambda closes the loop by demonstrating that the extension does not alter the original equations when the expansion rate is zero. This is a consistency check, not a derivation of the value of the constant itself.

In Recognition Science, the framework models the universe's expansion as a forced consequence of its internal structure. The constant Ω_Λ is not a free parameter but a derived quantity, with a value of 11/16 − α/π. The framework proves that this value is positive, and that the resulting vacuum energy has an equation of state exactly equal to −1, meaning it behaves like a perfect fluid with negative pressure. The theorem recovers_baseline_lambda ensures that this new physics does not contradict the established empty-space solution. It shows that the framework's addition is a genuine extension, not a replacement.

What the theorem does not claim is equally important. It does not prove that the cosmological constant has any particular numerical value. The value depends on the input H₀², which is not derived within the framework. The theorem only establishes the structural relationship: the constant is zero when the expansion is zero. It also does not claim that the extended equations are the final theory of gravity. The framework's own documentation notes that the physical bridge from recognition to linking, which would connect this mathematical structure to the observed universe, remains an open target.

/-- The Λ-extended data recovers the baseline `Λ = 0` data in the limit. -/
theorem recovers_baseline_lambda :
    (rs_efe_data_with_lambda 0).cosmological_constant
      = FullEFE.rs_efe_data.cosmological_constant := by
  show Lambda_RS 0 = 0
  exact Lambda_RS_zero

/-- `Λ_RS` is positive for any positive Hubble scale (forced by `Ω_Λ > 0`). -/
theorem Lambda_RS_pos {H0sq : ℝ} (h : 0 < H0sq) : 0 < Lambda_RS H0sq := by
  unfold Lambda_RS
  exact mul_pos (mul_pos (by norm_num : (0 : ℝ) < 3) h) Omega_Lambda_RS_pos

/-- **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

This theorem does not derive the numerical value of the cosmological constant from first principles. This theorem does not prove that the extended Einstein field equations are a complete theory of gravity.

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)

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