Geometry Triangulation3 Dconsistency Global Schlaefli From Incidence

Global Schläfli from incidence data

The Schläfli identity is a classical result in geometry: it says that the derivative of the volume of a polyhedron with respect to its dihedral angles satisfies a particular sum. In a triangulated three-dimensional space, the identity can be stated locally on each tetrahedron and then summed globally. The theorem global_schlaefli_from_incidence in the framework's machine-checked library of formal theorems proves that this global sum works under a clean condition: the local edge data on each tetrahedron must agree with a global edge chart, up to orientation. That agreement is called incidence consistency, a discrete record of which edge belongs to which tetrahedron and in which slot.

The theorem states that if a triangulation satisfies this incidence consistency, then the global left-hand side of the Schläfli identity equals the global right-hand side. It constructs the required local data on every tetrahedron from a closed-form proof, without the caller having to supply a separate package of local Schläfli data. In plain terms: once you know the edges match across tetrahedra, the global identity follows automatically. A companion theorem shows that the same conclusion holds using only pure incidence geometry plus the local closed-form Schläfli theorem, once that theorem is available globally.

This result does not prove the local Schläfli theorem itself. It assumes a closed-form local proof exists and shows how to lift it to a global statement. It also does not claim that any particular triangulation exists, only that if one satisfies the incidence condition, the identity holds. The theorem is a structural bridge: it converts a local geometric fact into a global one, provided the incidence data is consistent.

/-- 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'

/-- 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)

/-- Pure incidence geometry plus the local theorem gives global Schläfli. -/
theorem global_schlaefli_from_geometry
    (K : Triangulation3D) (hK : IncidenceGeometry K)
    (hLocal : SchlaefliTetrahedronClosedFormTarget) (e' : Fin 6) :
    globalSchlaefliLHS K (triangulationSchlaefliData_of_geometry K hK hLocal) e' =
      globalSchlaefliRHS K (triangulationSchlaefliData_of_geometry K hK hLocal) e' :=
  global_schlaefli_of_local K (triangulationSchlaefliData_of_geometry K hK hLocal) e'

What this page does not claim

The theorem does not prove the local Schläfli identity for a tetrahedron. The theorem does not assert that any specific triangulation of space exists. The theorem does not claim the framework derives the fine-structure constant or the Riemann Hypothesis.

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)

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