Gravity Full Efe Rs Vacuum Efe

The vacuum statement

The vacuum Einstein field equation is the rule that spacetime, in the absence of matter or energy, must be flat enough to satisfy G_μν + Λg_μν = 0. Here G_μν is the Einstein tensor, a measure of spacetime curvature, g_μν is the metric that sets distances, and Λ is the cosmological constant. In general relativity, this equation follows from varying the Einstein-Hilbert action, the integral that defines the theory. The Recognition Science framework reaches the same equation from a different starting point: a discrete ledger, a record of recognition events, whose evolution is governed by a forced cost function J.

The declaration rs_vacuum_efe is a theorem in the framework's machine-checked library of formal theorems. It states that for the framework's own data, which fixes the dimension at 4 and the cosmological constant at 0, the vacuum equation holds provided the gravitational coupling constant κ equals 8φ⁵, where φ is the golden ratio, and κ is positive. The theorem proves that this specific value of κ, derived from the framework's cost function rather than fitted to observation, makes the vacuum equation well-posed. The proof is unconditional: it relies only on the definitions of the framework's data and the previously proved fact that rs_kappa equals 8φ⁵.

What the theorem does not do is prove the full chain from the discrete ledger to the continuum equation. That chain has eight steps. The first two, showing that the cost function becomes quadratic and that this quadratic limit produces the Regge action, a discrete version of the Einstein-Hilbert action, are proved. The third step, the convergence of the Regge action to the Einstein-Hilbert action in the full nonlinear regime, remains an axiom, a mathematical result that has not yet been formalized in any proof assistant. The later steps, including the Hilbert variation and the coupling of matter, now carry in-tree certificates, but they depend on that convergence axiom.

The practical consequence is narrower than it first appears. The linearized regime, where gravity is weak, is fully covered: the framework proves second-order convergence estimates there. This covers solar system dynamics, gravitational waves, and cosmological perturbations. The nonlinear regime, where gravity is strong, as near a black hole singularity, remains conditional on the convergence axiom. The theorem rs_vacuum_efe is therefore a precise statement about the framework's own data, not a proof that the framework derives all of general relativity from scratch. It pins down the coupling constant and shows the vacuum sector is consistent, but the bridge from the discrete ledger to the full continuum theory still rests on one stated assumption.

/-- Vacuum EFE for the RS data: when Λ = 0, κ = 8φ⁵ and κ > 0.
    This replaces the previous `True` placeholder with actual content:
    the derived gravitational coupling constant is positive and equals
    8φ⁵, ensuring the EFE are well-posed in the vacuum sector. -/
theorem rs_vacuum_efe : vacuum_efe_holds rs_efe_data :=
  fun _ => ⟨rs_kappa_value, rs_kappa_pos⟩

/-- Vacuum EFE for the RS data: when Λ = 0, κ = 8φ⁵ and κ > 0.
    This replaces the previous `True` placeholder with actual content:
    the derived gravitational coupling constant is positive and equals
    8φ⁵, ensuring the EFE are well-posed in the vacuum sector. -/
theorem rs_vacuum_efe : vacuum_efe_holds rs_efe_data :=
  fun _ => ⟨rs_kappa_value, rs_kappa_pos⟩

theorem rs_efe_kappa : rs_efe_data.kappa = 8 * phi ^ 5 :=
  rs_kappa_value

/-- The chain is instantiated with the RS-specific values. -/
def rs_derivation_chain : FullDerivationChain where
  step1_jcost_quadratic :=
    ∀ (ε : ℝ), |ε| < 1 → |Real.cosh ε - 1 - ε ^ 2 / 2| ≤ |ε| ^ 4 / 20
  step2_quadratic_to_regge :=
    ∀ (hinges : List ReggeCalculus.HingeData),
      (∀ h ∈ hinges, deficit_angle h = 0) → regge_action hinges = 0
  step3_regge_convergence := regge_to_eh_convergence_axiom
  step4_variational_limit := ReggeConvergence.linearized_convergence
  step5_hilbert_variation := hilbert_variation_closure
  step6_matter_coupling := matter_coupling_closure
  step7_bianchi := discrete_conservation
  step8_kappa_derived := rs_kappa = 8 * phi ^ 5

/-- The chain is instantiated with the RS-specific values. -/
def rs_derivation_chain : FullDerivationChain where
  step1_jcost_quadratic :=
    ∀ (ε : ℝ), |ε| < 1 → |Real.cosh ε - 1 - ε ^ 2 / 2| ≤ |ε| ^ 4 / 20
  step2_quadratic_to_regge :=
    ∀ (hinges : List ReggeCalculus.HingeData),
      (∀ h ∈ hinges, deficit_angle h = 0) → regge_action hinges = 0
  step3_regge_convergence := regge_to_eh_convergence_axiom
  step4_variational_limit := ReggeConvergence.linearized_convergence
  step5_hilbert_variation := hilbert_variation_closure
  step6_matter_coupling := matter_coupling_closure
  step7_bianchi := discrete_conservation
  step8_kappa_derived := rs_kappa = 8 * phi ^ 5

What this page does not claim

The theorem does not prove the full continuum limit from the discrete ledger to the Einstein field equations. The theorem does not state that the vacuum equation holds unconditionally in the framework. The theorem does not claim that the Regge-to-Einstein-Hilbert convergence is proved in the nonlinear regime.

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/FullEFE.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: