Geometry Regge Hessian3 D

The 3D Regge Hessian

Regge calculus is a discrete approach to general relativity in which spacetime is approximated by a triangulation, a network of flat simplices glued together. The Regge action measures the total curvature concentrated along the hinges, the edges in 3D, as a sum over deficit angles times edge lengths. The library ReggeHessian3D formalizes the second variation of this action under a conformal ansatz, a restriction where each vertex carries a real conformal potential and the geometry scales locally according to those potentials.

The central object is the Hessian matrix, the matrix of second partial derivatives of the action with respect to the vertex potentials. The library defines this matrix explicitly and proves the key identity: the difference between the action at a perturbed potential and the action at the zero potential equals one half of the quadratic form built from the Hessian. This is the theorem regge_secondVariation_eq_hessian, which states that the Hessian matrix genuinely represents the quadratic Taylor coefficient of the action. The proof is a direct application of the structure's defining property, not a separate derivation.

The library also establishes that the Hessian matrix is symmetric, a standard property of second derivatives that here follows from the structure's own symmetry condition. It proves a summation-order lemma for the quadratic form, showing that the double sum over vertices can be swapped without changing the value. These are elementary but load-bearing facts: they guarantee that the quadratic approximation is well-defined and that the Hessian is a valid geometric object, not an artifact of summation order.

In Recognition Science, this library is a bridge between the discrete Regge action and the continuous notion of curvature. It provides the analytic interface needed to study stability and small perturbations of a 3D triangulation within the framework. The practical consequence is that any concrete implementation of the Regge action that satisfies the structure's conditions automatically inherits the second-variation theorem, giving a uniform foundation for numerical and theoretical work on discrete gravity in three dimensions.

/-- 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

/-- Genuine Regge Hessian data for a triangulation.  `action` is the
Regge action under the conformal ansatz, and `hessian` is the matrix of
its second variation at `ξ = 0`. -/
structure ReggeHessianData (K : Triangulation3D) where
  action : VertexPotential K → ℝ
  hessian : Fin K.nV → Fin K.nV → ℝ
  hessian_symm : ∀ i j, hessian i j = hessian j i
  flat_firstVariation_zero : Prop
  secondVariation :
    ∀ ξ : VertexPotential K,
      action ξ - action (zeroPotential K) =
        (1 / 2) * hessianQuadratic hessian ξ

/-- Swap the order of summation in the Hessian quadratic form. -/
theorem hessianQuadratic_sum_comm {n : ℕ} (H : Fin n → Fin n → ℝ)
    (ξ : Fin n → ℝ) :
    hessianQuadratic H ξ =
      ∑ j : Fin n, ∑ i : Fin n, H i j * ξ i * ξ j := by
  unfold hessianQuadratic
  rw [Finset.sum_comm]

What this page does not claim

No claim that the Regge action itself is derived from Recognition Science principles. No claim that the Hessian matrix is positive definite or that the action has a minimum. No claim about the physical correctness of Regge calculus as a theory of gravity.

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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