Geometry Schlaefli Triangulation3 D

The global identity

In geometry, the Schläfli identity is a bookkeeping rule for shapes that change smoothly. For a single tetrahedron, the four-dimensional analogue of a triangle, it says that a certain combination of how edge lengths change and how dihedral angles change always sums to zero. Dihedral angles are the angles between two faces of the tetrahedron. The identity is named after Ludwig Schläfli, who studied it in the 1850s while working on the geometry of higher-dimensional spaces.

The classical identity applies to one tetrahedron at a time. The question is whether the rule survives when many tetrahedra are glued together to form a larger shape, a triangulation. In a finite 3D triangulation, tetrahedra meet along shared faces, and edges are shared by several tetrahedra. A naive sum of the local identities could fail because the same edge appears in many tetrahedra, and its contribution might not cancel globally.

The framework's machine-checked library of formal theorems proves that the cancellation does survive. The theorem, global_schlaefli_of_local, states that summing the local tetrahedral Schläfli identities over every tetrahedron in a finite triangulation gives zero. The left-hand side is the finite sum, over all tetrahedra and all six edges of each tetrahedron, of the edge length times the derivative of the dihedral angle. The right-hand side is zero. The proof is short: each local identity is already zero, and a finite sum of zeros is zero.

In Recognition Science, this global identity is not a new physical law. It is a structural fact about how the framework models geometry. The framework's ledger, a discrete record of events, treats a triangulation as a finite collection of tetrahedra, and the identity is the statement that the ledger balances locally and therefore globally. The theorem does not depend on any particular metric or on the shape of the triangulation; it holds for any finite 3D triangulation with the given derivative data.

The consequence is practical for anyone working with discrete geometry. When you refine a triangulation or move its vertices, the local Schläfli identity guarantees that errors do not accumulate across the whole shape. The global cancellation is exact, not approximate. This is the kind of structural guarantee that makes a discrete model trustworthy: the bookkeeping rule that holds for one tetrahedron holds for the entire assembly.

/-- 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 left-hand side, summed over tetrahedra and local
tetrahedral edges. -/
def globalSchlaefliLHS (K : Triangulation3D)
    (D : TriangulationSchlaefliData K) (e' : Fin 6) : ℝ :=
  ∑ τ : Fin K.nT,
    ∑ e : Fin 6,
      Real.sqrt ((K.tet τ).sqEdge e) * (D.tetData τ).dihedralDeriv e e'

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

What this page does not claim

The theorem does not derive any physical constant or the dimensionality of space. The identity holds for the given derivative data structure, not for arbitrary edge-length variations. The proof does not require any metric or shape assumption beyond the finite 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/SchlaefliTriangulation3D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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