Gravity Full Efe Full Gr Certificate V2

The gravity certificate

General relativity describes gravity as the curvature of spacetime. Its central equations, the Einstein field equations, link that curvature to the distribution of matter and energy. The Recognition Science framework asks whether these equations can be derived from a more basic picture, in which reality keeps a discrete record of recognition events, a ledger (a discrete record of events). The declaration full_gr_certificate_v2 is a machine-checked certificate, a formal record in the framework's library of theorems, that states exactly which parts of that derivation are proved and which parts are assumed.

The certificate proves several specific facts unconditionally. It shows that the gravitational coupling constant, the number that sets the strength of gravity in the equations, is 8 * phi ^ 5, where phi is the golden ratio. It proves this constant is positive and non-zero, which is needed for the conservation of energy and momentum to make sense. It also proves that the Einstein tensor is flat in the vacuum case, meaning empty spacetime is a valid solution, and that the Riemann curvature tensor has its required antisymmetry. These are all formal theorems, checked by the machine.

The certificate also records a linearized convergence result. This means that in the regime of weak gravitational fields, such as those around the solar system or in gravitational waves, the discrete lattice action provably converges to the continuous Einstein-Hilbert action. This is a proved step. However, the certificate explicitly marks the full nonlinear convergence, the step that says the discrete Regge action converges to the continuous one for strong fields, as an axiom. This is not a proof. It is a conditional statement, relying on the Regge convergence axioms, which are established mathematical results that have not yet been formalized in any proof assistant.

In plain language, the certificate is a precise inventory. It says: here are the parts of the bridge from the discrete ledger to general relativity that are solidly proved, and here is the one load-bearing assumption that remains. It does not claim that general relativity itself is derived. It claims that a specific formal structure, the certificate, exists and records the status of each step in the derivation chain.

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

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

theorem full_gr_certificate_v2 : FullGRCertificateV2 where
  kappa_derived := rs_kappa_value
  kappa_positive := rs_kappa_pos
  kappa_nonzero := ne_of_gt rs_kappa_pos
  hilbert_variation := hilbert_variation_closure
  matter_coupling := matter_coupling_closure
  regge_flat := regge_action_flat
  bianchi_flat := flat_bianchi
  riemann_antisymmetric := RiemannTensor.riemann_antisymmetric_last_two
  riemann_flat := RiemannTensor.riemann_flat_vanishes
  einstein_flat := RicciTensor.einstein_flat
  linearized_convergence := ReggeConvergence.linearized_convergence

What this page does not claim

This certificate does not prove the full, nonlinear Einstein field equations from the ledger. It does not claim that the Regge convergence axioms are proved within the framework. It does not derive the value of the cosmological constant, which is set to zero in the certificate's data.

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

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