Geometry Triangulation3 Dconsistency Global Schlaefli From Geometry

The global Schläfli theorem

In 3D geometry, a triangulation is a way to divide a shape into tetrahedra, the simplest 3D solids with four triangular faces. The Schläfli identity is a relation between how the dihedral angles of these tetrahedra change and how their edge lengths change. In a triangulated space, one can ask whether the sum of local changes on each tetrahedron cancels to a global identity. The theorem global_schlaefli_from_geometry establishes that this cancellation holds whenever the triangulation satisfies a condition called incidence consistency, which means that the local edge slots of each tetrahedron agree with the global edge endpoints.

The key achievement is that the global identity follows from pure geometric incidence data alone. Earlier versions of the framework required each tetrahedron to carry a stored proof of its local Schläfli relation. The new theorem removes that requirement: once the local closed-form Schläfli theorem is available, the global identity is constructed automatically from the incidence geometry. This is a structural simplification, not a new physical law. The theorem is proved in the machine-checked library of formal theorems, meaning the reasoning is verified step by step by a computer.

In Recognition Science, this result supports the framework's treatment of 3D space as a discrete structure. The framework models space as a ledger, a discrete record of events, and this theorem shows that a triangulated 3D space has a consistent global geometric description derived from local data. The theorem does not claim that physical space is actually a triangulation, nor does it derive the three-dimensionality of space. It is a mathematical result about a specific class of 3D triangulations, not a statement about the physical universe.

The practical consequence is that building a global Schläfli identity no longer requires redundant bookkeeping on each tetrahedron. A programmer or mathematician working with the framework's triangulation scaffold can construct the global identity from incidence data alone, provided the local theorem holds. This makes the framework's 3D geometry module cleaner and more directly usable, and it removes a barrier to further formalization of geometric results.

/-- 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'

/-- Once the local closed-form Schläfli theorem is proved globally, pure
incidence geometry constructs local Schläfli data on every tetrahedron with
no stored local Schläfli field. -/
def triangulationSchlaefliData_of_geometry
    (K : Triangulation3D) (_hK : IncidenceGeometry K)
    (hLocal : SchlaefliTetrahedronClosedFormTarget) :
    TriangulationSchlaefliData K where
  tetData := fun τ =>
    tetraSchlaefliDerivativeData_closedForm (K.tet τ) (hLocal (K.tet τ))

What this page does not claim

The theorem does not claim that physical space is a triangulation. The theorem does not derive the three-dimensionality of space from the framework. The theorem does not state that incidence consistency is physically necessary for a 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/Triangulation3DConsistency.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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