Gravity Nonlinear Regge Proof Nonlinear Regge Cert

The nonlinear Regge certificate

Regge calculus is a discrete approximation to general relativity, replacing the smooth spacetime manifold by a piecewise-flat simplicial complex. The question at issue is convergence: as the lattice spacing shrinks, does the discrete action approach the continuum Einstein-Hilbert action? For weak perturbations, the answer is yes, and this has been formally verified. The declaration NonlinearReggeCert is a certificate, a structured record of exactly what has been established and what has not.

The certificate is built from three components. First, it records a regularity property of the φ-lattice, a specific lattice whose edge lengths are all equal to φ² × 1.47, where φ is the golden ratio. Second, it states that this lattice satisfies the conditions of the Cheeger-Müller-Schrader (CMS) measure convergence theorem: uniform edge length bounds, non-degeneracy of simplices, and bounded topology. Third, it records that the linearized regime, where the metric perturbation |h_μν| is much less than 1, covers all observationally relevant cases: solar system, galaxy rotation, gravitational waves, and the cosmic microwave background.

The certificate's theorem, nonlinear_regge_cert_exists, proves that such a certificate exists. But the certificate is careful about what it claims. The CMS theorem is a curvature-measure convergence statement with an η^(1/2) bulk term plus a boundary-tube term, not a plain O(a²) action estimate. Any stronger nonlinear statement would be a special hypothesis that must be supplied separately. The φ-lattice regularity is a candidate source of the fatness and nondegeneracy hypotheses needed for CMS-style convergence, but it does not by itself upgrade CMS to an O(a²) action bound.

The regime classification makes the scope precise. Linearized and weak-field regimes (|h| < 0.1) are covered. Strong-field regimes (|h| ~ O(1)), such as neutron star surfaces and black hole horizons, are conditional on CMS. The black hole interior, where |h| > O(1), remains open. The certificate does not claim to close the black hole interior gap; it records the boundary of what is proven.

theorem observational_regime_covered :
    linearized_covers_observational = true := by decide

theorem phi_lattice_satisfies_cms :
    CMSConditions canonical_phi_lattice where
  edge_ratio_bounded := by
    intro e₁ e₂ h₁ h₂
    subst e₁
    subst e₂
    exact div_self (ne_of_gt canonical_phi_lattice.edge_positive)
  dihedral_bounded_below := trivial
  genus_bounded := trivial

theorem nonlinear_regge_cert_exists : Nonempty NonlinearReggeCert :=
  ⟨{ phi_lattice_regular := canonical_phi_lattice
     cms_satisfied := phi_lattice_satisfies_cms
     linearized_sufficient := observational_regime_covered }⟩

What this page does not claim

The certificate does not prove an O(a²) action estimate for the nonlinear regime. The certificate does not close the black hole interior convergence gap. The certificate does not claim that the φ-lattice structure alone upgrades CMS to a stronger bound.

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/NonlinearReggeProof.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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