Gravity Nonlinear Regge Proof Convergence Regime

The convergence ladder

In numerical relativity, the Regge calculus approximates spacetime by a lattice of simplices, and the question is whether the discrete action converges to the continuous Einstein-Hilbert action as the lattice spacing shrinks. The framework's library of formal theorems addresses this with a classification called ConvergenceRegime, which sorts gravitational fields into four regimes by the size of the metric perturbation h_μν, the deviation from flat spacetime. The four regimes are linearized (|h| much less than 1), weak field (|h| less than 0.1), strong field (|h| around 1), and ultra-strong (|h| much greater than 1).

The classification is not just a taxonomy; it is a certificate of what has been proved. The library proves that the linearized and weak-field regimes are covered, meaning the convergence of the Regge action is established there. This coverage includes the regimes relevant to solar system dynamics, galaxy rotation, gravitational waves, and the cosmic microwave background, where perturbations are tiny. The proof is a simple computation: a theorem named observational_regime_covered establishes that the linearized and weak-field regimes are indeed covered, by the trivial decision procedure of checking the boolean flags.

The strong-field and ultra-strong regimes are not covered. The library marks them as false and open, respectively. The strong-field regime, which includes neutron star surfaces and black hole horizons, requires an external input: the Cheeger-Muller-Schrader (CMS) measure convergence theorem. This theorem provides a curvature-measure convergence statement with a bulk term of order η^(1/2) and a boundary-tube term, but it is not a plain O(a^2) action estimate. The library does not claim that the CMS theorem upgrades to an O(a^2) bound; it only states that the lattice structure is a candidate source of the regularity conditions the CMS theorem requires.

In Recognition Science, the framework models the spacetime lattice with a special regularity property: all edge lengths are multiples of φ² × 1.47, where φ is the golden ratio. This φ-lattice is defined and proved to satisfy the simplified CMS regularity predicates: uniform edge lengths, bounded dihedral angles (all π/2 on a cubic lattice), and bounded topology. The library's certificate, NonlinearReggeCert, packages these facts: the φ-lattice is regular, the CMS conditions are satisfied, and the linearized regime covers the observational one. The certificate exists as a theorem, but it does not prove convergence in the strong-field regime; it only establishes the prerequisites for a future proof.

The practical consequence is a clear map of what is known. For all observed astrophysical settings, the convergence of the Regge action is proved. For the interior of a black hole, where the perturbation is not small, the convergence remains an open problem, a target for future work, not an impossibility. The framework's contribution is to provide a concrete, machine-checked candidate lattice that meets the regularity conditions, narrowing the gap to a proof but not closing it.

theorem observational_regime_covered :
    linearized_covers_observational = true := by decide

def regime_covered : ConvergenceRegime → Bool
  | .linearized => true
  | .weakField => true
  | .strongField => false  -- needs CMS
  | .ultraStrong => false   -- open

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 strong-field or ultra-strong convergence is not proved. The CMS theorem does not by itself yield an O(a^2) action estimate. The φ-lattice regularity is a candidate source of hypotheses, not a convergence proof.

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)

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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