Geometry Freudenthal Regge Component Concrete M Off Diag Eq Neg Area Weight

The concrete comparison

In Regge calculus, spacetime is approximated by a triangulation, a mesh of flat simplices glued together. The action is built from the lengths of edges and the dihedral angles between faces. The second variation of this action, the matrix M_ij, controls how small perturbations of the geometry behave. In a general triangulation, computing M_ij requires differentiating the dihedral angle with respect to edge lengths, a hard geometric task.

The declaration concreteM_offDiag_eq_neg_areaWeight proves a clean result for a specific finite model. The model uses eight vertices, matching the local chart of a Freudenthal subdivision. The area weight A_ij is the regular hinge-area formula, (sqrt 3 / 4) a^2 for a triangle of side length a. The theorem states that for i ≠ j, M_ij = -A_ij. That is, off the diagonal, the coefficient matrix is exactly the negative of the geometric area weight. The proof is a direct computation from the graph-Laplacian Regge data, with no new geometric axioms added.

Two companion facts complete the picture. Every row of M sums to zero, which is the consistency condition for a second-order action. The second-order Regge action itself reduces to the Dirichlet form with the concrete geometric weights: secondOrderReggeAction = (1/2) * dirichletForm. This means the weak-field bridge can consume this finite model directly.

In Recognition Science, this is the first fully concrete finite model that closes the component comparison. The framework's machine-checked library of formal theorems proves this for the regular flat-sector model. It is not a proof for arbitrary Cayley-Menger or dihedral derivative data. The full derivative computation for a general triangulation remains open. This concrete case is the exact interface a future general computation must target.

/-- Off diagonal, the concrete Regge coefficient matrix is the negative of the
geometric area/face-weight matrix. -/
theorem concreteM_offDiag_eq_neg_areaWeight
    (S : ConcreteReggeStar) (i j : LocalVertex) (hij : i ≠ j) :
    concreteM S i j = - areaWeight S i j := by
  unfold concreteM concreteWeakFieldReggeData
  rw [bilinearCoefficient_laplacianReggeData (areaWeight S) (areaWeight_symm S)]
  unfold laplacianCoefficient
  simp [hij]

/-- The concrete coefficient matrix has exact zero row sums. -/
theorem concreteM_rowSum_zero (S : ConcreteReggeStar) :
    ∀ i : LocalVertex, ∑ j : LocalVertex, concreteM S i j = 0 := by
  intro i
  unfold concreteM concreteWeakFieldReggeData
  simpa only [bilinearCoefficient_laplacianReggeData (areaWeight S) (areaWeight_symm S)]
    using laplacianCoefficient_row_sum (areaWeight S) i

/-- Concrete closure of the weak-field component comparison: the second-order
Regge action is exactly the geometric Dirichlet form for the concrete area
weights. -/
theorem concreteReggeSecondVariation_eq_jcostDirichlet
    (S : ConcreteReggeStar) (ε : LogPotential 8) :
    secondOrderReggeAction (concreteWeakFieldReggeData S) ε
      = (1 / 2) * dirichletForm (areaWeight S) ε := by
  simpa [concreteReggeComponentComparison] using
    componentComparison_gives_geometric_dirichlet
      (concreteWeakFieldReggeData S)
      (concreteReggeComponentComparison S)
      ε

What this page does not claim

This is not a proof for arbitrary Cayley-Menger or dihedral derivative data. The full derivative computation for a general triangulation is not established here. The physical recognition-to-linking bridge is not part of this declaration.

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

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