Gravity Nonlinear Regge Proof

The proof's reach

In numerical relativity, Regge calculus approximates curved spacetime by a lattice of flat simplices, and the question is whether the discrete action converges to the continuum as the lattice spacing shrinks. The linearized regime, where the metric perturbation h_μν is much smaller than 1, is fully certified in the framework's machine-checked library of formal theorems. This covers the solar system, galaxy rotation, gravitational waves, and the cosmic microwave background, where perturbations range from about 10⁻²¹ to 10⁻⁴. The library proves that this linearized coverage extends to weak fields below 0.1, and the certificate theorem nonlinear_regge_cert_exists packages these results.

The nonlinear regime is a different story. The Cheeger-Muller-Schrader (CMS) theorem, a curvature-measure convergence statement, applies conditionally to strong fields near a neutron star surface or a black hole horizon, but it is not a plain O(a²) action estimate. The framework's φ-lattice, where all edge lengths are multiples of φ² × 1.47, satisfies the CMS regularity conditions by construction: uniform edge lengths, bounded dihedral angles, and trivial topology. This is proved in phi_lattice_satisfies_cms. But this lattice regularity does not by itself upgrade CMS to a stronger O(a²) bound, and the strong-field regime (|h| ~ O(1)) and ultra-strong regime (|h| >> 1) remain unproved.

In Recognition Science, the φ-lattice is a candidate source of the fatness and nondegeneracy hypotheses that CMS-style measure convergence requires. The framework models the convergence hierarchy as Linearized ⊂ Weak-field ⊂ CMS-regular ⊂ Full nonlinear, and the theorem linearized_implies_weak proves the first inclusion. The black hole interior, where the perturbation exceeds O(1), is marked OPEN in the library. This is not a claim that the proof fails there; it is a precise statement of where the current certificate stops.

The practical consequence is a clean boundary on what the framework can assert. For every observational regime so far measured, the linearized proof is sufficient. For the strong-field interior, the framework says honestly: this remains a target, not a theorem. The certificate exists, and it names its own limit.

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

What this page does not claim

The nonlinear Regge convergence is not proved for strong fields or the black hole interior. The φ-lattice regularity does not by itself imply an O(a²) action estimate. The framework does not claim the CMS theorem applies to the black hole interior.

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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Derived articles

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