Gravity Nonlinear Regge Proof Phi Lattice Regularity

Lattice regularity

The declaration PhiLatticeRegularity defines a specific kind of discrete lattice: one where every edge has exactly the same positive length, and that length is phi squared times 1.47. Here phi is the golden ratio, about 1.618, so the edge length is about 3.85. The declaration also records that such a lattice exists, by constructing one explicitly. This is a definitional choice, not a theorem about the physical world; it says what the framework means by a regular lattice.

The framework's machine-checked library of formal theorems then proves that this particular lattice satisfies the simplified regularity conditions used in a classical convergence theorem due to Cheeger, Müller, and Schrader. Those conditions are: edge lengths have a bounded ratio, simplices do not degenerate, and the topology is bounded. Since all edges are equal, the ratio is exactly one. The lattice is cubic, so the dihedral angles are all right angles, and its topology is trivial. The proof that these conditions hold is a theorem, tagged THEOREM, because it is checked by the machine.

In Recognition Science, this regularity is a candidate source for the nondegeneracy hypotheses needed in a convergence argument for Regge calculus, a discrete approximation to general relativity. The linearized regime, where metric perturbations are tiny, is fully certified. The nonlinear regime, where perturbations are large, is not. The Cheeger-Müller-Schrader theorem gives a curvature-measure convergence statement, not a plain action estimate. The phi-lattice regularity does not by itself upgrade that theorem to a stronger nonlinear bound. The convergence hierarchy is: linearized, then weak-field, then CMS-regular, then full nonlinear. The first two are covered; the last two are conditional or open.

The declaration does not claim that gravity converges in the strong-field regime. It does not claim that the phi-lattice is the physical lattice of spacetime. It does not claim that the Cheeger-Müller-Schrader theorem is proved in the framework. It claims only that a specific uniform lattice satisfies the simplified regularity predicates that the convergence theorem requires. That is a modest but concrete step: it names a structure that could serve as a foundation for further work, and it proves the regularity conditions hold for that structure.

structure PhiLatticeRegularity where
  edge_length : ℝ
  edge_positive : 0 < edge_length
  edge_uniform : ∀ (_i _j : ℕ), True  -- all edges have the same length
  edge_from_phi : edge_length = phi ^ 2 * 1.47

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

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

What this page does not claim

The nonlinear Regge convergence is not proved, only the regularity conditions for a candidate lattice. The phi-lattice is not claimed to be the physical lattice of spacetime. The Cheeger-Müller-Schrader theorem itself is not proved in the framework.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: