Gravity Regge Component Theorem3 Dproof Canonical Weak Field Data Off Diag Compo

A component-level identity

In the discrete geometry of triangulated surfaces, Regge calculus approximates general relativity by assigning lengths to the edges of a triangulation and building an action from the angles. The second derivative of that action, the Hessian, is a matrix that controls small perturbations. Its off-diagonal entries, the bilinear coefficients, describe how a change in one edge length affects the energy contribution of another.

The theorem canonicalWeakFieldData_offDiag_component_match proves a precise identity for a specific construction. It states that for any two distinct vertices i and j, the bilinear coefficient of the canonical weak-field data equals the negative of an independently defined dual weight. That dual weight is built directly from the edge-length chart of the incidence geometry, not by negating any Hessian coefficient. The result holds for any triangulation that satisfies the incidence consistency condition, and it is checked in a machine-checked library of formal theorems.

This identity is the off-diagonal component of a larger comparison. The framework's library assembles it into a package that also carries the Schläfli row-sum condition, which expresses a geometric consistency of the energy matrix. The package then yields a Dirichlet reduction: for a small perturbation, the second-order Regge action equals half the Dirichlet form of the edge areas. That reduction is a standard step in relating the discrete action to a simpler quadratic form.

What the theorem does not claim is broader significance. It does not prove that the dual weights are the only possible ones, nor that the identity holds for every conceivable weak-field data set. It establishes the match for the canonical incidence-based construction. It also does not assert anything about the physical correctness of Regge calculus as a theory of gravity; it is a formal statement about a discrete geometric construction.

theorem canonicalWeakFieldData_offDiag_component_match
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (i j : Fin K.nV) (hij : i ≠ j) :
    bilinearCoefficient (canonicalWeakFieldDataOfIncidence K hK) i j =
      - (independentDualWeightsOfConsistent K hK).weight i j := by
  rw [canonicalWeakFieldData_bilinearCoefficient]
  exact Geometry.ReggeActionConcrete.canonicalReggeHessian_offDiag_eq_neg_weight K hK i j hij

/-- Independent dual weights built from an `IncidenceConsistent` chart using
the canonical geometry-layer incidence weights. -/
def independentDualWeightsOfConsistent
    (K : Triangulation3D) (hK : IncidenceConsistent K) :
    IndependentDualWeights K where
  weight := Geometry.ReggeActionConcrete.canonicalDualWeight K hK
  weight_symm := Geometry.ReggeActionConcrete.canonicalDualWeight_symm K hK
  weight_nonneg := Geometry.ReggeActionConcrete.canonicalDualWeight_nonneg K hK

/-- The arbitrary-triangulation component target is discharged for the
canonical incidence/Laplacian second-order Regge data. -/
theorem finalReggeComponentTarget : FinalReggeComponentTarget := by
  intro K hK
  exact ⟨concreteComponentComparisonOfIncidence K hK⟩

/-- Once the final component package is constructed, the existing Dirichlet
reduction applies immediately. -/
theorem genuine_component_dirichlet_reduction_from_final
    {K : Triangulation3D} (C : ConcreteComponentComparison K)
    (ε : LogPotential K.nV) :
    secondOrderReggeAction C.W ε =
      (1 / 2) * dirichletForm (edgeArea C.W) ε :=
  genuine_component_dirichlet_reduction (genuineComponentPackage_of_concrete C) ε

What this page does not claim

The theorem does not prove that the dual weights are unique or that the identity holds for all possible weak-field data sets. The theorem does not assert the physical correctness of Regge calculus as a theory of gravity. The theorem does not establish the off-diagonal identity for non-incidence-consistent triangulations.

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/ReggeComponentTheorem3DProof.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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