Geometry Freudenthal Two Cube Strip Local Edge Complete

The complete edge slots

A tetrahedron, the simplest three-dimensional solid, has six edges. When you split a cube into six tetrahedra, you get a small combinatorial bookkeeping problem: each tetrahedron's edges must line up with its neighbors' edges. The declaration localEdge_complete settles this bookkeeping for a specific shape, the two-cube Freudenthal strip, which is two unit cubes sharing one square face, each cube split into six tetrahedra with matching face triangulations.

The strip has 12 vertices, 33 distinct global edges, and 12 tetrahedra. Each tetrahedron has six local edge slots, so there are 72 slots in total. The theorem proves that for every tetrahedron and every one of its six slots, there exists a global edge that fills that slot. In other words, no slot is empty, and no slot is a phantom: every local position in every tetrahedron corresponds to a real edge of the whole strip. The proof is constructive: it names the specific global edge for each slot, and a companion theorem confirms that the mapping is consistent, so the same global edge is recognized from both sides of any shared face.

This is the first multi-cube incidence instance beyond a single-cube sanity check. The declaration is a theorem in the framework's machine-checked library of formal theorems, and it is part of a larger structure that defines what it means for a triangulation to have consistent edge bookkeeping. The practical consequence is that the two-cube strip passes a fundamental consistency test: its local geometry and its global geometry agree on every edge, with no gaps and no double-counting after the five shared face edges are deduplicated.

The theorem does not claim that the two-cube strip is the only shape with complete edge slots, nor does it say anything about the physical meaning of the strip in a broader theory. It is a purely combinatorial and topological statement about a specific finite object. It also does not assert that the strip is a valid Regge triangulation or that it satisfies any geometric regularity conditions beyond the incidence consistency that the definitions encode.

theorem localEdge_complete (τ : T) (f : Fin 6) :
    ∃ e : E, edgeInTet e τ = some f := by
  exact ⟨localEdgeOf τ f, (edgeInTet_iff_localEdgeOf (localEdgeOf τ f) τ f).2 rfl⟩

def twoCubeStrip : Triangulation3D where
  nV := 12
  nE := 33
  nT := 12
  edgeVerts := edgeVerts
  tetVerts := tetVerts
  edgeInTet := edgeInTet
  tet := fun _ => FreudenthalCubeTriangulation.freudenthalTet

theorem edgeInTet_iff_localEdgeOf (e : E) (τ : T) (f : Fin 6) :
    edgeInTet e τ = some f ↔ e = localEdgeOf τ f := by
  native_decide +revert

theorem edgeInTet_vertices
    (e : E) (τ : T) (f : Fin 6) (h : edgeInTet e τ = some f) :
      let ev := edgeVerts e
      let tv := ReggeRigorousFoundation.edgeVertices f
      (tetVerts τ tv.1 = ev.1 ∧ tetVerts τ tv.2 = ev.2) ∨
        (tetVerts τ tv.1 = ev.2 ∧ tetVerts τ tv.2 = ev.1) := by
  have he : e = localEdgeOf τ f := (edgeInTet_iff_localEdgeOf e τ f).1 h
  subst e
  fin_cases τ <;> fin_cases f <;>
    simp [localEdgeOf, edgeVerts, tetVerts,
      ReggeRigorousFoundation.edgeVertices] at h ⊢

What this page does not claim

The theorem does not claim the two-cube strip is the only triangulation with complete edge slots. It does not assert any physical or dynamical meaning for the strip. It does not establish that the strip satisfies geometric regularity conditions beyond incidence consistency.

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

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