Geometry Schlaefli Triangulation3 D Global Schlaefli Of Local

The global identity

In the geometry of 3D space, a triangulation is a way to divide a solid shape into tetrahedra, the simplest 3D building blocks, which meet along shared faces, edges, and vertices. The Schläfli identity is a classical result about how the shape of a single tetrahedron changes as its edges vary: it relates the change in the dihedral angles (the angles between faces) to the change in the edge lengths. The theorem global_schlaefli_of_local extends this local fact to an entire finite triangulation.

The statement is a clean equation. For any finite triangulation of a 3D region, and for any assignment of derivative data to each tetrahedron (data that tracks how dihedral angles respond to edge-length changes), the sum over all tetrahedra of a certain product equals zero. The product in question is the square root of the squared edge length times the dihedral derivative with respect to a chosen edge. The theorem, which is proved in the framework's machine-checked library of formal theorems, shows that this global sum is identically zero. The right-hand side of the equation is simply zero, because the Euclidean angle-variation term vanishes in flat space.

The proof is a direct summation. Each tetrahedron's local Schläfli identity says its own contribution is zero; the theorem sums these local cancellations over all tetrahedra. The result is a global identity that holds for any finite triangulation, without any additional conditions on the shape or the number of tetrahedra. This is a structural fact about 3D geometry: the local constraints are consistent when assembled into a whole.

In Recognition Science, this theorem is part of the framework's treatment of geometry as derived from recognition events. The framework models space as a discrete structure built from such triangulations, and this identity is a consistency condition that any such model must satisfy. The theorem itself is a pure statement about Euclidean geometry; it does not by itself assert anything about physics, such as gravity or the dimension of space. It is a lemma about the mathematics of triangulated 3D shapes, not a physical law.

The practical consequence is that any computation involving variations of a triangulated 3D shape, in any context, can rely on this cancellation. It simplifies global analysis by reducing it to local checks. The theorem does not claim that the sum is zero for any particular physical system; it claims the sum is zero for any mathematical triangulation with the given derivative data.

/-- Summing local tetrahedral Schläfli identities gives the global finite
triangulation identity. -/
theorem global_schlaefli_of_local
    (K : Triangulation3D) (D : TriangulationSchlaefliData K) (e' : Fin 6) :
    globalSchlaefliLHS K D e' = globalSchlaefliRHS K D e' := by
  unfold globalSchlaefliLHS globalSchlaefliRHS
  have hlocal : ∀ τ : Fin K.nT,
      (∑ e : Fin 6,
        Real.sqrt ((K.tet τ).sqEdge e) * (D.tetData τ).dihedralDeriv e e')
        = 0 := by
    intro τ
    exact (D.tetData τ).schlaefli e'
  simp_rw [hlocal]
  simp

/-- The global Schläfli right-hand side: the Euclidean angle-variation term
vanishes. -/
def globalSchlaefliRHS (K : Triangulation3D)
    (_D : TriangulationSchlaefliData K) (_e' : Fin 6) : ℝ :=
  0

/-- Summing local tetrahedral Schläfli identities gives the global finite
triangulation identity. -/
theorem global_schlaefli_of_local
    (K : Triangulation3D) (D : TriangulationSchlaefliData K) (e' : Fin 6) :
    globalSchlaefliLHS K D e' = globalSchlaefliRHS K D e' := by
  unfold globalSchlaefliLHS globalSchlaefliRHS
  have hlocal : ∀ τ : Fin K.nT,
      (∑ e : Fin 6,
        Real.sqrt ((K.tet τ).sqEdge e) * (D.tetData τ).dihedralDeriv e e')
        = 0 := by
    intro τ
    exact (D.tetData τ).schlaefli e'
  simp_rw [hlocal]
  simp

What this page does not claim

The theorem does not claim anything about the physical laws of gravity or the dimension of physical space. It does not claim that the sum is zero for any particular physical system, only for any mathematical triangulation with the given data. It does not claim that every triangulation of 3D space is geometrically valid; it assumes a valid triangulation is given.

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/SchlaefliTriangulation3D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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