Geometry Periodic Freudenthal Torus Freudenthal Tet Sq Edge Eq Periodic Disp Sq

Edge lengths in a periodic tiling

A periodic tiling of space with tetrahedra, the Freudenthal tiling, is a standard way to break up a volume into a repeating mesh of simplexes. The declaration freudenthalTet_sqEdge_eq_periodicDispSqEdge_localEdgeOf is a machine-checked theorem in the framework's library of formal theorems. It proves that the squared length of any edge in this periodic tiling is determined by a small table of seven numbers, indexed by the edge's direction.

The seven possible squared lengths are 1, 1, 1, 2, 2, 2, and 3. The theorem states that for any tetrahedron in the tiling, the squared length of a local edge equals the value in this table that corresponds to the edge's displacement. This is a structural fact about the geometry of the Freudenthal tiling, not a physical law. It holds for any periodic tiling that follows the Freudenthal pattern, regardless of the overall size of the repeating cell.

In the Recognition Science framework, this theorem is a building block. The framework models the universe as a discrete ledger of recognition events, and it uses such geometric facts to construct a consistent picture of space. The theorem ensures that the edge lengths in the tiling are consistent with the framework's model of space as a discrete structure. It is a formal guarantee that the geometry of the tiling behaves as expected.

The theorem does not claim that the Freudenthal tiling is the only possible discrete geometry of space. It does not establish that the seven edge lengths are physically meaningful or that they correspond to any measured property of the universe. It also does not prove that the tiling can be embedded in a finite, concrete mesh; that remains an open target in the framework's development.

/-- Squared edge length determined only by the positive displacement class. -/
def periodicDispSqEdge : Fin 7 → ℝ
  | 0 => 1
  | 1 => 1
  | 2 => 1
  | 3 => 2
  | 4 => 2
  | 5 => 2
  | 6 => 3

/-- Squared edge length determined only by the positive displacement class. -/
def periodicDispSqEdge : Fin 7 → ℝ
  | 0 => 1
  | 1 => 1
  | 2 => 1
  | 3 => 2
  | 4 => 2
  | 5 => 2
  | 6 => 3

theorem localEdgeOf_endpoints_match_tetVerts
    {Nx Ny Nz : ℕ} [NeZero Nx] [NeZero Ny] [NeZero Nz]
    (cell : Vertex Nx Ny Nz) (tet f : Fin 6) :
      let edge := localEdgeOf cell tet f
      let ev := edge.endpoints
      let tv := ReggeRigorousFoundation.edgeVertices f
      (addVertexBits cell (FreudenthalCubeTriangulation.tetVerts tet tv.1) = ev.1 ∧
        addVertexBits cell (FreudenthalCubeTriangulation.tetVerts tet tv.2) = ev.2) ∨
        (addVertexBits cell (FreudenthalCubeTriangulation.tetVerts tet tv.1) = ev.2 ∧
          addVertexBits cell (FreudenthalCubeTriangulation.tetVerts tet tv.2) = ev.1) := by
  fin_cases tet <;> fin_cases f <;>
    simp [localEdgeOf, PeriodicEdge.endpoints,
      FreudenthalCubeTriangulation.localEdgeOf,
      FreudenthalCubeTriangulation.tetVerts,
      ReggeRigorousFoundation.edgeVertices,
      cubeEdgeBase, cubeEdgeDisp, dispBits, vertexBits, addVertexBits, addBits]

What this page does not claim

The Freudenthal tiling is the only possible discrete geometry of space. The seven edge lengths correspond to any measured physical property. A finite, concrete mesh encoding the typed periodic torus has been constructed.

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

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