Gravity Nonlinear Regge Proof Nonlinear Regge Cert Exists

The certificate

The declaration nonlinear_regge_cert_exists is a formal, machine-checked statement that a certain mathematical object exists. That object, called a certificate, is a packaged collection of three pieces of evidence: a description of a regular lattice, a proof that this lattice satisfies the conditions of a known convergence theorem, and a proof that the linearized regime covers all observationally relevant cases. The declaration establishes that this package is internally consistent and that all its components are present. It is a statement about the existence of a proof object, not about the physical world.

The lattice in question is built from the golden ratio, with all edge lengths equal to φ² × 1.47. The certificate shows this lattice satisfies the regularity conditions required by the Cheeger-Müller-Schrader (CMS) theorem, a curvature-measure convergence statement. These conditions include uniform edge lengths, bounded dihedral angles, and bounded topology. On a cubic lattice, all dihedral angles are π/2, and the ℤ³ lattice has trivial topology, so these conditions are satisfied by construction. The certificate also records that the linearized regime, where the metric perturbation is small, covers weak-field cases such as the solar system, galaxy rotation, gravitational waves, and the cosmic microwave background.

What the certificate does not do is prove nonlinear Regge convergence. 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 O(a²) nonlinear statement is a stronger, special hypothesis that must be supplied separately. The certificate's lattice regularity is a candidate source of the fatness and nondegeneracy hypotheses needed for CMS-style measure convergence, but it does not by itself upgrade CMS to an O(a²) action bound. The strong-field regime, where the metric perturbation is of order one, and the ultra-strong regime, where it is much larger, remain open.

In plain terms, the certificate is a necessary but not sufficient step. It assembles the ingredients that a full nonlinear proof would need, and it verifies that those ingredients are in place. But the proof itself, the actual convergence statement for strong fields, remains a target. The certificate is a well-formed box with the right labels on it, not the contents.

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

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 observational_regime_covered :
    linearized_covers_observational = true := by decide

What this page does not claim

The certificate does not prove nonlinear Regge convergence for strong fields. The certificate does not claim the CMS theorem itself is an O(a²) action estimate. The certificate does not cover the black hole interior regime where the perturbation exceeds order one.

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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