Geometry Freudenthal Cube Triangulation Local Sq Edge Eq Global

The edge-length consistency theorem

The Freudenthal triangulation divides a unit cube into six tetrahedra, each with six edges. The edges come in three kinds: three unit-length cube edges, two face diagonals of length √2, and one body diagonal of length √3. The theorem local_sqEdge_eq_global states a consistency property: when a tetrahedron and a global edge of the cube are matched, the squared length recorded in the tetrahedron equals the squared length recorded in the global list. The proof checks all 19 global edges, all 6 tetrahedra, and all 6 local edge slots, a finite case analysis of 684 combinations.

The declaration lives in a machine-checked library of formal theorems, part of the Recognition Science framework's geometry foundation. The framework models physical structure through a discrete record of events, and this theorem belongs to the bookkeeping layer that ensures a triangulation is internally coherent. The theorem does not derive the cube's geometry from first principles; it takes the cube's vertex coordinates as given and verifies that the six-tetrahedron decomposition assigns consistent squared lengths.

What the theorem does not claim is broader. It does not assert that the Freudenthal triangulation is the only way to divide a cube into six tetrahedra, nor does it say anything about the recognition cost function or the forcing chain that produces constants like the golden ratio. The theorem is a local consistency check, not a statement about why space has three dimensions. It also does not claim that the squared edge lengths are physically measured; they are definitional choices for the unit cube.

What the theorem establishes is a precise, verified fact: in this triangulation, the local squared-edge tuple for every tetrahedron is (1, 2, 3, 1, 2, 1), and the Cayley-Menger determinant for each tetrahedron equals 8, meaning the tetrahedron is nondegenerate. The consistency theorem guarantees that when the framework's incidence bookkeeping matches a global edge to a local slot, the squared lengths agree. This is the kind of foundational check that lets the framework build larger geometric claims on solid ground.

theorem local_sqEdge_eq_global
    (e : Fin 19) (τ f : Fin 6) (h : edgeInTet e τ = some f) :
    freudenthalTet.sqEdge f = globalSqEdge e := by
  fin_cases e <;> fin_cases τ <;> fin_cases f <;>
    simp [edgeInTet, freudenthalTet, freudenthalTetSqEdges, globalSqEdge] at h ⊢

/-- The local squared-edge tuple for every Freudenthal tetrahedron in the unit
cube: three unit step edges, two face diagonals, and one body diagonal. -/
def freudenthalTetSqEdges : CayleyMengerPolynomial.SqEdges
  | 0 => 1
  | 1 => 2
  | 2 => 3
  | 3 => 1
  | 4 => 2
  | 5 => 1

theorem cm3_freudenthalTetSqEdges :
    CayleyMengerPolynomial.cm3 freudenthalTetSqEdges = 8 := by
  unfold freudenthalTetSqEdges CayleyMengerPolynomial.cm3
  norm_num

What this page does not claim

The theorem does not claim the Freudenthal triangulation is the only decomposition of a cube into six tetrahedra. The theorem does not claim anything about the recognition cost function or the forcing chain that produces constants like the golden ratio. The theorem does not claim the squared edge lengths are physically measured; they are definitional for the unit cube.

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/FreudenthalCubeTriangulation.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: