Geometry Regge Action First Variation Conformal Schlaefli Incidence Bookkeeping

The incidence bookkeeping theorem

The declaration conformalSchlaefliIncidenceBookkeeping_of_edgeSlotBookkeeping is a theorem in the framework's machine-checked library of formal theorems. It states that if you have a triangulated three-dimensional space, and you have a structure that consistently assigns each of the six edges of each tetrahedron to a global edge of the triangulation, then a certain local identity about dihedral angles holds. That identity is the conformal Schläfli incidence bookkeeping condition, which is a bookkeeping-level version of the classical Schläfli differential identity relating changes in dihedral angles to changes in edge lengths.

The theorem is proved in the module ReggeActionFirstVariation, which is part of a larger effort to show that the first variation of the nonlinear Regge action vanishes at a flat conformal potential. The bookkeeping structure, called IncidenceEdgeSlotBookkeeping, is a record of how local edge slots in each tetrahedron correspond to global edges in the triangulation. The theorem shows that this bookkeeping is sufficient to derive the conformal Schläfli incidence condition, which is a key step in the geometric proof that uses Schläfli cancellation plus zero deficit to establish the criticality of the flat potential.

What the theorem does not claim is that the full first variation of the Regge action vanishes. That is a separate statement, called ReggeActionCriticalAtZero, which is defined as the derivative of the action at the zero potential being zero. The theorem here is a local, bookkeeping-level result. It does not by itself establish the global criticality of the flat potential, nor does it prove that the Regge action is minimized or that the flat potential is a global minimum. It also does not claim that the bookkeeping structure exists for every triangulation; it takes such a structure as an input and shows what follows from it.

The broader context is a proof strategy for the first variation of the nonlinear Regge action. The geometric proof is Schläfli cancellation plus zero deficit. The theorem here is the bookkeeping part, which ensures that the local contributions from each tetrahedron can be consistently summed over the whole triangulation. This is a necessary step for the global argument, but it is not the whole argument. The full derivative calculation is still being expanded from the closed-form local Schläfli identities, and the theorem here is one named input needed for that expansion.

theorem conformalSchlaefliIncidenceBookkeeping_of_edgeSlotBookkeeping
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (A : LocalDihedralDirectionalDerivativePackage K)
    (hBook : IncidenceEdgeSlotBookkeeping K hK) :
    ConformalSchlaefliIncidenceBookkeeping K hK A := by
  intro η
  unfold deficitDirectionalDerivFromLocalAngles
  calc
    (∑ e : Fin K.nE,
      hingeMeasureUnderConformal K hK (zeroPotential K) e *
        (-∑ τ : Fin K.nT,
          match K.edgeInTet e τ with
          | none => 0
          | some f => A.angleDeriv η τ f))
        =
      - (∑ e : Fin K.nE,
        globalEdgeLength K hK e *
          (∑ τ : Fin K.nT,
            match K.edgeInTet e τ with
            | none => 0
            | some f => A.angleDeriv η τ f)) := by
          unfold hingeMeasureUnderConformal globalEdgeLength zeroPotential
          rw [← Finset.sum_neg_distrib]
          refine Finset.sum_congr rfl ?_
          intro e _
          simp
    _ = - (∑ τ : Fin K.nT,
          ∑ f : Fin 6,
            Real.sqrt ((K.tet τ).sqEdge f) * A.angleDeriv η τ f) := by
          rw [hBook.sum_match (fun τ f => A.angleDeriv η τ f)]

/-- Incidence partition certificate: summing a local edge-slot weight over
global edges and tetrahedra is the same as summing it directly over local
tetrahedral edge slots, with matching flat edge lengths.

`IncidenceConsistent.localEdge_complete` gives existence of a global edge for
each local slot.  Exact reindexing also needs uniqueness/no-duplication, so it
is recorded here as the actual bookkeeping theorem needed downstream. -/
structure IncidenceEdgeSlotBookkeeping
    (K : Triangulation3D) (hK : IncidenceConsistent K) where
  sum_match :
    ∀ w : Fin K.nT → Fin 6 → ℝ,
      (∑ e : Fin K.nE,
        globalEdgeLength K hK e *
          (∑ τ : Fin K.nT,
            match K.edgeInTet e τ with
            | none => 0
            | some f => w τ f)) =
        ∑ τ : Fin K.nT,
          ∑ f : Fin 6,
            Real.sqrt ((K.tet τ).sqEdge f) * w τ f

/-- The nonlinear Regge action is critical at the flat potential. -/
def ReggeActionCriticalAtZero
    (K : Triangulation3D) (hK : IncidenceConsistent K) : Prop :=
  fderiv ℝ (reggeAction K hK) (zeroPotential K) = 0

What this page does not claim

The theorem does not prove that the first variation of the Regge action vanishes at the flat potential. The theorem does not establish that the flat potential is a global minimum of the Regge action. The theorem does not assert that the bookkeeping structure exists for every triangulation.

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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