Gravity Regge Component Theorem3 Dproof Genuine Component Dirichlet Reduction Fr

The Dirichlet reduction

Regge calculus is a discrete approach to general relativity in which spacetime is approximated by a triangulated manifold, and the gravitational field is encoded in the lengths of its edges. The action, a quantity whose minimization gives the equations of motion, is built from these edge lengths. In the weak-field limit, the second-order part of this action is a quadratic form in small variations of the edge lengths, represented by a coefficient matrix.

The theorem genuine_component_dirichlet_reduction_from_final establishes a precise identity for this quadratic form. It states that for any triangulation K with a consistent incidence geometry, and for any variation ε of the edge lengths, the second-order Regge action equals (1/2) times the Dirichlet form of ε with respect to a weight function derived from the edge lengths. In plain terms: the second-order action is a Dirichlet form, a standard type of quadratic form associated with a graph Laplacian, where the weights are the edge lengths themselves.

The proof constructs the weights independently from the incidence geometry, not by negating the Regge Hessian matrix. These weights are symmetric and nonnegative. The theorem then shows that the off-diagonal components of the Hessian match these weights, and that the row sums satisfy the Schläfli condition. This construction is the "genuine component" that the theorem's name refers to.

In Recognition Science, this theorem is a step toward showing that the weak-field Regge action has the structure of a Dirichlet form, which is a natural object in the framework's discrete geometry. The theorem is proved in the framework's machine-checked library of formal theorems, meaning the derivation is verified by a computer.

The theorem does not claim that the full Regge action, beyond second order, is a Dirichlet form. It does not claim that the Dirichlet form is the unique or canonical one. It does not claim that this reduction holds for arbitrary weight functions; it holds for the specific weights constructed from incidence geometry.

/-- 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

/-- Independent dual/hinge weight for a vertex pair, defined directly from
the incidence-level edge-length chart.  It is not defined by negating a
Regge Hessian coefficient. -/
def vertexPairHingeWeight
    (K : Triangulation3D) (hK : IncidenceGeometry K)
    (i j : Fin K.nV) : ℝ :=
  ∑ e : Fin K.nE, edgePairIncidenceWeight K hK i j e

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

/-- 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 claim the full Regge action, beyond second order, is a Dirichlet form. The theorem does not claim the Dirichlet form is the unique or canonical one. The theorem does not claim the reduction holds for arbitrary weight functions, only for the specific weights constructed from incidence geometry.

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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