Geometry Regge Triangulation3 D Local Edge Variation

Local edge variation

A triangulation of three-dimensional space breaks the space into tetrahedra, the simplest solid shapes, joined face to face. In the Regge calculus approach to general relativity, physicists approximate curved spacetime by such a flat, piecewise-linear scaffold. The lengths of the edges of the tetrahedra are the dynamical variables; changing them changes the geometry. A local edge variation is the object that records such a change: for each tetrahedron in the triangulation, and for each of its six edges, it assigns a real number, the derivative of some quantity with respect to that edge's length.

The definition separates the variation from any global coordinate system. Instead of choosing one coordinate chart that covers the whole triangulation, the variation lives independently on each tetrahedron. This is a deliberate design choice. It keeps the local, tetrahedral structure of the theory explicit, and it avoids prematurely committing to a global length-coordinate chart, which may not exist or may be awkward to construct for an arbitrary triangulation. The object is defined as a function from the finite set of tetrahedra and the finite set of six edges per tetrahedron to the real numbers.

The declaration also fixes the incidence data that make the triangulation well-formed. A triangulation is not just a collection of tetrahedra; it must specify which vertices belong to which edges, which tetrahedra share which edges, and that every tetrahedron has nondegenerate squared edge lengths. The structure carries these data explicitly, so that the local edge variation can be defined on top of a precise combinatorial scaffold.

In Recognition Science, this declaration is part of the framework's library of machine-checked formal theorems. It provides the finite incidence scaffold used to lift local tetrahedral identities to arbitrary 3D Regge triangulations. The framework models the geometry of space as a discrete structure, and this declaration is one of the building blocks for that model.

What the declaration does not claim is important. It does not assert that any particular variation is physically realized, nor does it define a dynamics or an action principle. It does not claim that the triangulation approximates any specific smooth spacetime, nor that the variation is continuous or differentiable in any global sense. It is a definitional scaffold, not a physical law. The declaration establishes the language in which later statements about variations can be made, but it does not by itself make any statement about the physical content of those variations.

The practical consequence is that the declaration gives a clean, coordinate-free way to talk about how a triangulated geometry changes. Any later theorem or computation that needs to vary edge lengths, for example to derive equations of motion or to study the response of the geometry to perturbation, can build on this foundation. The separation of local variation from global coordinates is what makes the foundation flexible enough to apply to arbitrary triangulations, not just specially chosen ones.

/-- A global edge variation assigns a derivative to each local tetrahedral
edge.  This separated representation avoids prematurely choosing a global
length-coordinate chart. -/
abbrev LocalEdgeVariation (K : Triangulation3D) :=
  Fin K.nT → Fin 6 → ℝ

/-- A global edge variation assigns a derivative to each local tetrahedral
edge.  This separated representation avoids prematurely choosing a global
length-coordinate chart. -/
abbrev LocalEdgeVariation (K : Triangulation3D) :=
  Fin K.nT → Fin 6 → ℝ

/-- 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

No physical dynamics or action principle is defined by this declaration. No claim is made that the triangulation approximates any specific smooth spacetime. The declaration does not assert any global continuity or differentiability of the variation.

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)

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