Geometry Triangulation3 Dconsistency Nonempty Triangulation Schlaefli Data Of In

What the declaration builds

A triangulation of three-dimensional space is a mesh of tetrahedra that meet along shared faces, edges, and vertices. In the geometry of such meshes, the Schläfli identity relates how the volume of a tetrahedron changes when its edge lengths change. The declaration nonempty_triangulationSchlaefliData_of_incidence proves, in the machine-checked library of formal theorems, that any such mesh whose local edge slots agree with the global edge chart carries the local angle data needed to apply that identity globally.

The proof works in two steps. First, the declaration triangulationSchlaefliData_of_incidence constructs the required data on every tetrahedron from a closed-form local formula. Second, the theorem global_schlaefli_from_incidence shows that this construction satisfies the global Schläfli balance equation for every edge of the mesh. The key point is that the construction needs no stored local Schläfli field: the incidence consistency alone, meaning the agreement between local tetrahedral edge slots and global edge endpoints, is enough to build the package.

What the declaration does not claim is just as important. It does not prove the local closed-form Schläfli theorem itself; that is assumed as an input, called local_schlaefli, and the declaration builds on it. It also does not assert that any particular triangulation exists or that a given mesh is geometrically realizable. It only states that if a triangulation satisfies the incidence consistency condition, then the Schläfli data package is nonempty. The declaration is a bridge from a structural condition to a usable geometric object, not a claim about which meshes exist in space.

In practice this means that future work on global Schläfli cancellation can rely on incidence consistency alone, without threading local proof data through every construction. The declaration removes a bookkeeping burden: once the local theorem is available, the global structure follows automatically. That is the consequence a reader can take away: a cleaner interface for building global geometric arguments on 3D meshes.

/-- Incidence consistency constructs the global Schläfli data package. -/
theorem nonempty_triangulationSchlaefliData_of_incidence
    (K : Triangulation3D) (hK : IncidenceConsistent K) :
    Nonempty (TriangulationSchlaefliData K) :=
  ⟨triangulationSchlaefliData_of_incidence K hK⟩

/-- Construct local Schläfli data on every tetrahedron from the incidence
mixin's closed-form local Schläfli proof. -/
def triangulationSchlaefliData_of_incidence
    (K : Triangulation3D) (hK : IncidenceConsistent K) :
    TriangulationSchlaefliData K where
  tetData := fun τ =>
    tetraSchlaefliDerivativeData_closedForm (K.tet τ) (hK.local_schlaefli τ)

/-- Strong incidence plus local closed-form Schläfli gives global Schläfli. -/
theorem global_schlaefli_from_incidence
    (K : Triangulation3D) (hK : IncidenceConsistent K) (e' : Fin 6) :
    globalSchlaefliLHS K (triangulationSchlaefliData_of_incidence K hK) e' =
      globalSchlaefliRHS K (triangulationSchlaefliData_of_incidence K hK) e' :=
  global_schlaefli_of_local K (triangulationSchlaefliData_of_incidence K hK) e'

What this page does not claim

The declaration does not prove the local closed-form Schläfli theorem itself, which is assumed as an input. The declaration does not assert that any particular triangulation exists or is geometrically realizable.

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/Triangulation3DConsistency.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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