Geometry Regge Triangulation3 D Triangulation3 D

The finite scaffold

In computational geometry, a Regge triangulation approximates curved spacetime by flat tetrahedra joined along their faces. The Recognition Science framework's Triangulation3D declaration is a machine-checked definition of the finite combinatorial data such a triangulation needs. It records the number of vertices, edges, and tetrahedra; which vertices bound each edge; which vertices bound each tetrahedron; and how edges sit inside tetrahedra. Each tetrahedron carries a tuple of squared edge lengths that must be nondegenerate, meaning the tetrahedron has positive volume.

The definition deliberately separates the combinatorial scaffold from any choice of coordinates. A global edge variation, the derivative of edge lengths, is represented locally on each tetrahedron rather than through a global chart. This separation avoids prematurely committing to a particular coordinate system, which matters when the goal is to lift local tetrahedral identities to the whole triangulation.

In Recognition Science, this structure is the finite incidence scaffold used to lift local tetrahedral identities to arbitrary 3D Regge triangulations. The framework's library proves no theorem about physical space from this definition alone. The declaration establishes only the combinatorial object, not that any particular physical geometry exists or that three dimensions are forced. The framework's separate claim that three spatial dimensions are forced by linking is a different result, with its own proof obligations.

What the declaration does not claim is as important as what it defines. It does not assert that any tetrahedral complex satisfying these fields corresponds to a physical spacetime. It does not define a metric or a curvature. It does not prove that Regge calculus converges to general relativity. The definition is a tool: a precise combinatorial container that later theorems can use, not a physical statement on its own.

/-- A finite 3D Regge triangulation with abstract incidence data and a
nondegenerate squared-edge tuple on every tetrahedron. -/
structure Triangulation3D where
  nV : ℕ
  nE : ℕ
  nT : ℕ
  edgeVerts : Fin nE → Fin nV × Fin nV
  tetVerts : Fin nT → Fin 4 → Fin nV
  edgeInTet : Fin nE → Fin nT → Option (Fin 6)
  tet : Fin nT → NonDegenerateTet

/-- A finite 3D Regge triangulation with abstract incidence data and a
nondegenerate squared-edge tuple on every tetrahedron. -/
structure Triangulation3D where
  nV : ℕ
  nE : ℕ
  nT : ℕ
  edgeVerts : Fin nE → Fin nV × Fin nV
  tetVerts : Fin nT → Fin 4 → Fin nV
  edgeInTet : Fin nE → Fin nT → Option (Fin 6)
  tet : Fin nT → NonDegenerateTet

/-- A finite 3D Regge triangulation with abstract incidence data and a
nondegenerate squared-edge tuple on every tetrahedron. -/
structure Triangulation3D where
  nV : ℕ
  nE : ℕ
  nT : ℕ
  edgeVerts : Fin nE → Fin nV × Fin nV
  tetVerts : Fin nT → Fin 4 → Fin nV
  edgeInTet : Fin nE → Fin nT → Option (Fin 6)
  tet : Fin nT → NonDegenerateTet

What this page does not claim

The declaration does not assert that any tetrahedral complex corresponds to a physical spacetime. The declaration does not define a metric, curvature, or any geometric quantity beyond squared edge lengths. The declaration does not prove that Regge calculus converges to general relativity.

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/ReggeTriangulation3D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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