Gravity Regge Component Theorem3 Dproof Canonical Weak Field Data Bilinear Coeff

The coefficient identity

In the Recognition Science framework, gravity is studied on a discrete 3D triangulation, a mesh of tetrahedra. Two kinds of data can be attached to such a mesh. One is the weak-field Regge data, a matrix of numbers that describes how the action, the quantity whose minimization gives the equations of motion, responds to small changes in the edge lengths. The other is a set of independent dual weights, numbers attached to pairs of vertices, built directly from the edge lengths themselves. The theorem canonicalWeakFieldData_bilinearCoefficient proves that for any triangulation satisfying the incidence consistency condition, the off-diagonal entry of the weak-field matrix equals the negative of the corresponding dual weight.

The statement is precise. For any two distinct vertices i and j, the bilinear coefficient of the canonical weak-field data at (i,j) equals the canonical Regge Hessian at (i,j). This is not an approximation or a numerical coincidence checked on examples; it is a theorem proved in the machine-checked library of formal theorems, valid for every triangulation that meets the stated condition. The proof is short: it unfolds the definition of the canonical weak-field data, applies a lemma about the Laplacian Regge data, and concludes by reflexivity.

What makes this identity significant is that it connects two constructions that were previously separate. The independent dual weights are geometric data, defined directly from the edge-length chart, not by negating the Regge coefficient matrix. The theorem shows that the canonical weak-field data, built from those weights, reproduces exactly the genuine Regge Hessian. This is the component-level match that the framework's Regge component comparison target requires. The final theorem, finalReggeComponentTarget, states that for every incidence-consistent triangulation, such a component comparison exists; the coefficient identity is the key step that discharges that target for the canonical construction.

The theorem does not claim that the weak-field data equals the Hessian for arbitrary dual weights. It holds for the canonical weights derived from the incidence geometry, not for any choice of weights. It also does not claim that the full Regge action equals the Dirichlet form; that reduction is a separate theorem, genuine_component_dirichlet_reduction_from_final, which requires the full component package, not just the coefficient identity. The identity concerns the off-diagonal entries; the diagonal entries and the row sums are handled by other theorems in the module.

theorem canonicalWeakFieldData_bilinearCoefficient
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (i j : Fin K.nV) :
    bilinearCoefficient (canonicalWeakFieldDataOfIncidence K hK) i j =
      Geometry.ReggeActionConcrete.canonicalReggeHessian K hK i j := by
  unfold canonicalWeakFieldDataOfIncidence
  rw [bilinearCoefficient_laplacianReggeData]
  rfl

theorem canonicalWeakFieldData_bilinearCoefficient
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (i j : Fin K.nV) :
    bilinearCoefficient (canonicalWeakFieldDataOfIncidence K hK) i j =
      Geometry.ReggeActionConcrete.canonicalReggeHessian K hK i j := by
  unfold canonicalWeakFieldDataOfIncidence
  rw [bilinearCoefficient_laplacianReggeData]
  rfl

/-- 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⟩

What this page does not claim

The theorem does not hold for arbitrary dual weights, only for the canonical weights derived from incidence geometry. The theorem does not by itself establish the full Dirichlet reduction, which requires the complete component package. The theorem does not address the diagonal entries or the row sums of the weak-field matrix.

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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