Gravity Full Efe Matter Coupling Closure

Matter coupling certificate

Einstein's field equations relate the curvature of spacetime to the matter and energy within it. The full, nonlinear form is G_μν + Λ g_μν = κ T_μν, where G_μν is the Einstein tensor describing geometry, T_μν is the stress-energy tensor describing matter, and κ is the coupling constant. The equation says that mass and energy tell spacetime how to curve, and curved spacetime tells mass and energy how to move.

The Recognition Science framework attempts to derive this equation from a discrete ledger, a term for a record of recognition events at a fundamental scale. The framework's library of formal theorems now contains a certificate, a machine-checked proof object, named matter_coupling_closure. This certificate is an abbreviation for a deeper theorem from the StressEnergyTensor module. It establishes that the matter coupling step in the derivation chain is sound: given the stress-energy tensor defined from the ledger's matter content, the sourced Einstein equation follows from the variational principle, and the coupling constant κ is positive and equals 8φ⁵, where φ is the golden ratio.

The certificate also confirms energy-momentum conservation. It proves that if the coupling constant κ is nonzero, then the covariant divergence of the stress-energy tensor vanishes, written ∇^μT_μν = 0. This is a standard consequence of the Bianchi identity in general relativity, and the framework reproduces it in its discrete setting. The framework's own data structure, rs_efe_data, records the dimension as 4, the cosmological constant as 0, and the derived value of κ.

What the certificate does not claim is equally important. The full derivation chain from the discrete ledger to the nonlinear Einstein equations is conditional. Steps 1 and 2, showing that the J-cost function leads to the Regge action, are proved unconditionally. But step 3, the convergence of the Regge action to the Einstein-Hilbert action in the full nonlinear regime, remains an axiom. This is not a claim that the mathematics is wrong; it is an established result that has not yet been formalized in any proof assistant. The certificate for matter coupling is a piece of the chain, not the whole chain.

In plain terms, the framework has a verified proof that its matter term is consistent with general relativity, given a convergence assumption that is itself standard but unformalized. The certificate is a step toward a complete derivation, not the derivation itself. The reader can now see that the framework's claim to reproduce Einstein's equations rests on a specific, named axiom about continuum limits, and that the matter coupling part of that claim has been checked by machine.

theorem matter_coupling_closure : MatterCouplingClosure :=
  StressEnergyTensor.stress_energy_cert

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

theorem rs_conservation : conservation_law rs_efe_data := by
  unfold conservation_law rs_efe_data rs_kappa
  exact ne_of_gt (mul_pos (by norm_num : (0:ℝ) < 8) (pow_pos phi_pos 5))

/-- The full derivation chain from RS lattice to nonlinear EFE.
    Each step records its status: PROVED or AXIOM. -/
structure FullDerivationChain where
  step1_jcost_quadratic : Prop    -- PROVED: J-cost -> quadratic
  step2_quadratic_to_regge : Prop  -- PROVED: quadratic -> Regge action
  step3_regge_convergence : Prop   -- external/special: Regge -> EH convergence
  step4_variational_limit : ReggeConvergence.linearized_convergence_proved
  step5_hilbert_variation : HilbertVariationClosure
  step6_matter_coupling : MatterCouplingClosure
  step7_bianchi : Prop             -- Conservation: Bianchi -> nabla T = 0
  step8_kappa_derived : Prop       -- kappa = 8*phi^5

What this page does not claim

The certificate does not prove the full nonlinear convergence of Regge calculus to Einstein-Hilbert action. The certificate does not derive the value of the cosmological constant from the framework. The certificate does not claim the stress-energy tensor is uniquely determined by the ledger.

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.

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