Geometry Regge Action Concrete Canonical Regge Hessian Quadratic Expanded

The expansion identity

In the framework's machine-checked library of formal theorems, the declaration canonicalReggeHessian_quadratic_expanded establishes a concrete algebraic identity. It states that, for any finite three-dimensional triangulation with consistent incidence data, the quadratic form associated with the canonical Regge Hessian equals a specific expanded expression. That expression is a sum over vertices of a diagonal term minus a sum over vertex pairs of an off-diagonal term, each weighted by a canonical dual weight. The identity is a theorem, proved in the Lean 4 proof assistant, and it is the key computational step that connects the abstract Hessian to a practical, computable form.

To understand what this means, recall the classical setting. Regge calculus is a discrete approximation to general relativity where spacetime is replaced by a triangulation, and the gravitational action is expressed in terms of edge lengths and deficit angles. The Hessian of this action, its matrix of second derivatives, governs the stability of solutions and the propagation of perturbations. The framework's declaration shows that, under a specific conformal ansatz where each vertex carries a scale factor, this Hessian's quadratic form can be written as a Dirichlet energy: a sum over edges of a weight times the squared difference of vertex potentials. This is a standard and highly tractable form, familiar from graph theory and discrete potential theory.

The declaration does not claim that this identity holds for arbitrary variations. It is specific to the conformal ansatz, where edge lengths scale by exponentials of vertex potentials. It also does not claim that the Hessian is positive definite, only that its quadratic form is nonnegative, a property proved separately. The identity itself is a stepping stone: it is used to prove that the Hessian is symmetric, that its row sums are zero, and that the quadratic form equals a nonnegative Dirichlet energy. These are the properties needed to analyze the stability of the discrete gravitational action.

In Recognition Science, this result is part of a broader program to derive physical structure from a ledger of recognition events. The Regge action is a concrete instance of a cost function, and its Hessian describes how the cost responds to small perturbations. The expansion identity is a technical but necessary link in that chain, showing that the discrete geometry behaves like a well-behaved potential theory. For the general reader, the takeaway is that a complex geometric expression can be rewritten in a simple, computable form, and that this rewrite is not an approximation but an exact theorem.

theorem canonicalReggeHessian_quadratic_eq_dirichlet
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (ξ : VertexPotential K) :
    hessianQuadratic (canonicalReggeHessian K hK) ξ =
      canonicalDirichletEnergy K hK ξ := by
  rw [canonicalReggeHessian_quadratic_expanded,
    canonicalDirichletEnergy_expanded]

theorem canonicalReggeHessian_quadratic_eq_dirichlet
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (ξ : VertexPotential K) :
    hessianQuadratic (canonicalReggeHessian K hK) ξ =
      canonicalDirichletEnergy K hK ξ := by
  rw [canonicalReggeHessian_quadratic_expanded,
    canonicalDirichletEnergy_expanded]

theorem canonicalReggeHessian_quadratic_eq_dirichlet
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (ξ : VertexPotential K) :
    hessianQuadratic (canonicalReggeHessian K hK) ξ =
      canonicalDirichletEnergy K hK ξ := by
  rw [canonicalReggeHessian_quadratic_expanded,
    canonicalDirichletEnergy_expanded]

/-- Local squared-edge data in tetrahedron `τ` under the vertex-conformal
ansatz.  The local edge `f = (u,v)` scales by `exp (ξ_u + ξ_v)`. -/
def conformalLocalSqEdge
    (K : Triangulation3D) (ξ : VertexPotential K)
    (τ : Fin K.nT) (f : Fin 6) : ℝ :=
  let uv := ReggeRigorousFoundation.edgeVertices f
  (K.tet τ).sqEdge f *
    Real.exp (ξ (K.tetVerts τ uv.1) + ξ (K.tetVerts τ uv.2))

What this page does not claim

The identity holds for arbitrary variations beyond the conformal ansatz. The Hessian is positive definite, only that its quadratic form is nonnegative. The declaration itself proves the stability of the Regge action; it provides the algebraic form needed for such an analysis.

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

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