Gravity Nonlinear Regge Proof Linearized Implies Weak

The weak-field bridge

In numerical relativity, the Regge action is a discrete version of the Einstein-Hilbert action, built from edge lengths of a triangulated spacetime rather than from continuous fields. The question is whether this discrete action converges to the continuous one as the triangulation is refined. The framework's machine-checked library of formal theorems addresses this question in a specific, limited way: it proves that the linearized regime, where metric perturbations are tiny, implies the weak-field regime, where perturbations are still below a tenth in magnitude.

The theorem linearized_implies_weak is a formal statement in the library's ledger, a discrete record of mathematical events. It takes as input the fact that the linearized regime is covered, meaning the convergence proof works there, and produces as output the fact that the weak-field regime is also covered. The proof is a simple case analysis: the library defines a classification of regimes, linearized, weak-field, strong-field, and ultra-strong, and the theorem checks that coverage of the first implies coverage of the second. It is a bookkeeping statement about the library's own classification, not a new physics derivation.

The classical context matters. The Cheeger-Muller-Schrader (CMS) theorem is a curvature-measure convergence statement with an η^(1/2) bulk term plus a boundary-tube term, not a plain O(a^2) action estimate. Any O(a^2) nonlinear statement is a stronger special hypothesis and must be supplied separately. The library's φ-lattice, where all edge lengths are multiples of φ² × 1.47, has a natural regularity property that satisfies simplified CMS-style conditions by construction, including uniform edge length and bounded dihedral angles. But this does not by itself upgrade CMS to an O(a^2) action bound.

What the theorem does not claim is broader. It does not prove convergence in the strong-field regime, where perturbations are of order one, nor in the ultra-strong regime. The black hole interior, where perturbations exceed order one, remains an open target. The library's own regime table shows this: solar system, galaxy rotation, gravitational waves, and CMB perturbations are all in the linearized regime and covered; neutron star surfaces and black hole horizons are conditional on CMS; the black hole interior is open. The theorem is a bridge between two adjacent rungs of the library's classification ladder, not a proof of full nonlinear convergence.

The practical consequence is modest but real. For observational regimes, the linearized and weak-field coverage together certify the convergence claim where actual measurements live. The theorem closes a small gap in the library's coverage map, and it does so with a machine-checked proof that carries no additional axioms. But the gap it closes is narrow: the strong-field regime, where gravity is genuinely nonlinear, remains exactly where it was, needing external input that the library does not yet supply.

theorem linearized_implies_weak (_h : regime_covered .linearized = true) :
    regime_covered .weakField = 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

This theorem does not prove convergence in the strong-field or ultra-strong regime. The φ-lattice regularity does not by itself upgrade CMS to an O(a^2) action bound. The theorem does not establish that the Regge action converges to the Einstein-Hilbert action in the continuum limit.

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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