Geometry Regge Hessian3 D Regge Second Variation Eq Hessian

The Hessian identity

Regge calculus is a discrete approach to general relativity. Instead of a smooth spacetime, one works with a triangulation, a space built from flat tetrahedra glued along their faces. The Regge action assigns a number to each such triangulation, a discrete stand-in for the Einstein-Hilbert action. Its second variation, the Hessian, controls how the action responds to small changes in the geometry, and therefore which configurations are stable.

The theorem regge_secondVariation_eq_hessian states a precise identity. For any finite three-dimensional triangulation, and any conformal potential (a real number assigned to each vertex), the difference between the action at that potential and the action at the zero potential equals one half times the quadratic form built from the Hessian matrix. In symbols: action(ξ) − action(0) = (1/2) Σ_i,j H_ij ξ_i ξ_j. The Hessian matrix is symmetric, and the identity holds for every choice of potential.

This is a statement about the structure of the second variation, not a numerical computation. It does not say what the action is, nor what the Hessian matrix contains. Those are supplied by a concrete implementation, which must provide the action, the matrix, and a proof that the identity holds. The theorem merely packages that requirement: if you have the data, the identity follows.

In Recognition Science, the framework models physical structure as a discrete ledger of recognition events, and this Hessian identity is part of its geometric toolkit. The framework's machine-checked library of formal theorems verifies the identity for any triangulation, assuming the conformal ansatz is in force. The result is a bridge between the discrete Regge action and the continuous notion of a Hessian, a step toward connecting the framework's discrete foundations to classical gravitational physics.

The theorem is a formal statement about finite triangulations. It does not claim that the Regge action converges to general relativity in any continuum limit, nor that the conformal ansatz is physically forced. Those questions remain open. What is established is a clean algebraic fact: under the stated assumptions, the second variation is exactly the quadratic form of the Hessian.

/-- Extract the second-variation formula from a concrete Hessian package. -/
theorem regge_secondVariation_eq_hessian
    (K : Triangulation3D) (D : ReggeHessianData K) (ξ : VertexPotential K) :
    D.action ξ - D.action (zeroPotential K) =
      (1 / 2) * hessianQuadratic D.hessian ξ :=
  D.secondVariation ξ

What this page does not claim

The theorem does not specify the Regge action or the Hessian matrix; those are supplied by a concrete implementation. It does not assert that the Regge action converges to general relativity in any continuum limit. It does not claim that the conformal ansatz is physically forced or preferred.

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/ReggeHessian3D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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