Geometry Regge Action First Variation Regge Action Critical At Zero Of First Var

The flat potential is a critical point

The Regge action is a way to measure curvature in a space built from flat tetrahedra, the three-dimensional analogue of triangulating a surface with triangles. Instead of curvature being spread smoothly, it is concentrated along the edges where tetrahedra meet, measured by an angle called the deficit angle. The action is a sum over all edges of the edge length times this deficit angle. A configuration is flat when every deficit angle is zero, meaning the tetrahedra fit together without any curvature.

The theorem establishes that the flat configuration is a critical point of the action. In calculus, a critical point is where the derivative is zero; here, the derivative is taken with respect to changing the positions of the vertices of the tetrahedra. The statement, in plain language, is that if you start from a flat configuration and slightly wiggle the vertex positions, the action does not change to first order. The proof relies on a geometric identity called Schläfli cancellation, where the changes in the dihedral angles of each tetrahedron cancel out when summed, and on the fact that the deficit angles are zero to begin with.

In the framework's machine-checked library, this is formalized as the theorem reggeActionCriticalAtZero_of_firstVariationFormula_of_zeroDeficit. The library is a collection of formal theorems verified by a computer. The theorem takes as input a triangulation, a proof that it is flat, and a package of formulas for the first variation, and outputs the statement that the Fréchet derivative of the action at the zero potential is zero. This is a precise, formal version of the geometric statement that the flat configuration is stationary.

What this theorem does not claim is that the flat configuration is a minimum or a maximum of the action. It only establishes that the first derivative vanishes, which is a necessary condition for an extremum, but not sufficient. The theorem also does not claim that the flat configuration is unique, nor does it say anything about the behavior of the action under large deformations, only infinitesimal ones. It is a local statement about the geometry near the flat configuration, not a global statement about the entire space of configurations.

/-- Phase-C first-variation theorem, conditional on the named analytic
first-variation input. -/
theorem reggeAction_firstVariation_zero
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_flat : FlatConfiguration K hK)
    (h_first : ReggeActionFirstVariationInput K hK h_flat) :
    fderiv ℝ (reggeAction K hK) (zeroPotential K) = 0 :=
  h_first.firstVariation_zero

/-- Local Schläfli cancellation for the conformal length direction on a single
tetrahedron.  This is pure finite-sum algebra plus the already proved local
Schläfli identity. -/
theorem local_conformal_schlaefli_cancellation
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (η : VertexPotential K) (τ : Fin K.nT) :
    (∑ f : Fin 6,
      Real.sqrt ((K.tet τ).sqEdge f) *
        localAngleLengthChainDeriv K hK η τ f) = 0 := by
  unfold localAngleLengthChainDeriv
  calc
    (∑ f : Fin 6,
      Real.sqrt ((K.tet τ).sqEdge f) *
        (∑ k : Fin 6,
          ((triangulationSchlaefliData_of_incidence K hK).tetData τ).dihedralDeriv f k *
            localEdgeLengthDirectionalDeriv K η τ k))
        =
      ∑ k : Fin 6,
        localEdgeLengthDirectionalDeriv K η τ k *
          (∑ f : Fin 6,
            Real.sqrt ((K.tet τ).sqEdge f) *
              ((triangulationSchlaefliData_of_incidence K hK).tetData τ).dihedralDeriv f k) := by
          simp_rw [Finset.mul_sum]
          rw [Finset.sum_comm]
          refine Finset.sum_congr rfl ?_
          intro k _
          refine Finset.sum_congr rfl ?_
          intro f _
          ring
    _ = ∑ k : Fin 6, localEdgeLengthDirectionalDeriv K η τ k * 0 := by
          refine Finset.sum_congr rfl ?_
          intro k _
          rw [((triangulationSchlaefliData_of_incidence K hK).tetData τ).schlaefli k]
    _ = 0 := by simp

What this page does not claim

The theorem does not claim the flat configuration is a minimum or maximum of the action. The theorem does not claim the flat configuration is unique. The theorem does not claim anything about the action under large, non-infinitesimal deformations.

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

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