Geometry Triangulation3 Dconsistency Local Edge Length Eq Global Edge Length

The edge length guarantee

A triangulation of three-dimensional space is a way of filling a volume with tetrahedra, the four-faced solids that are the 3D analogue of triangles. A single edge in such a mesh can border several tetrahedra. Each tetrahedron carries its own local record of the squared lengths of its six edges. The question is whether these local numbers agree with the global picture of the whole mesh.

The declaration localEdgeLength_eq_globalEdgeLength establishes that they do, under a precise condition. The condition is that the mesh is incidence-consistent: a discrete record of which local edge slots belong to which global edges, with no mismatches or duplicates. When that holds, the theorem proves that the length of an edge computed from any tetrahedron that contains it is exactly the length computed from the global edge chart. The proof is a direct consequence of the corresponding equality for squared lengths, which the framework proves first.

The statement is a theorem in the framework's machine-checked library of formal theorems. It is not a claim about how real physical space behaves. It is a statement about the internal consistency of a mathematical model. The framework uses this consistency to assemble local geometric data into a global structure, for instance when constructing the data needed for a global version of the Schläfli identity, a relation among the angles and edge lengths of a tetrahedron.

What the theorem does not claim is just as important. It does not say that any triangulation is incidence-consistent; that is a separate property that must be assumed or proved. It does not say that the global edge length is physically meaningful, only that it is well-defined within the model. It does not establish that the local and global squared-edge charts agree for edges that are not identified by the incidence map; the theorem only covers the edges that the map explicitly connects.

The practical consequence is that, within this framework, a worker can compute geometric quantities locally, inside a single tetrahedron, and trust that the result is the same as the global computation. This is the kind of guarantee that makes it possible to build larger structures, such as a global Schläfli identity, from local pieces without fear of inconsistency.

/-- The local edge length at any incident tetrahedral edge equals the global
edge length determined by the consistency chart. -/
theorem localEdgeLength_eq_globalEdgeLength
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (e : Fin K.nE) (τ : Fin K.nT) (f : Fin 6)
    (h : K.edgeInTet e τ = some f) :
    Real.sqrt ((K.tet τ).sqEdge f) = globalEdgeLength K hK e := by
  unfold globalEdgeLength
  rw [localSqEdge_eq_globalSqEdge K hK e τ f h]

/-- Local tetrahedral squared-edge slots agree with the global squared-edge
chart whenever the incidence map identifies them. -/
theorem localSqEdge_eq_globalSqEdge
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (e : Fin K.nE) (τ : Fin K.nT) (f : Fin 6)
    (h : K.edgeInTet e τ = some f) :
    (K.tet τ).sqEdge f = hK.globalSqEdge e :=
  hK.local_sqEdge_eq_global e τ f h

/-- 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 theorem does not assert that any triangulation is incidence-consistent. The theorem does not claim that the global edge length is a physically measured quantity. The theorem does not cover edges that the incidence map does not identify.

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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