Geometry Periodic Freudenthal Torus Canonical Encoded Periodic K Tet Verts Eq

The encoded torus

The ledger, a discrete record of events, often takes the shape of a repeating lattice. A periodic Freudenthal torus is one such shape: a three-dimensional grid of vertices, edges, and tetrahedra that wraps around on itself in all three directions, like a video game screen that loops at its borders. The declaration canonicalEncodedPeriodic_K_tetVerts_eq concerns the finite encodings of this torus, the concrete lists of numbers that a computer would use to store such a shape.

The declaration establishes that any finite Triangulation3D encoding this periodic torus possesses the global IncidenceEdgeSlotPartition needed by the nonlinear Regge first-variation theorem. In plain terms: if you build a finite computer model of this repeating tetrahedral lattice, its edges can be organized into the precise slots that a later theorem about how the shape responds to deformation requires. The declaration proves this property holds for the canonical periodic model, not just for a hand-picked example.

The work isolates a specific remaining task. The library defines the typed periodic vertex, edge, and tetrahedron model, and proves the needed edge partition exists. What it does not do is provide the finite encoder itself: the explicit function that maps the typed periodic torus into the finite sets Fin nV, Fin nE, and Fin nT (the natural numbers indexing vertices, edges, and tetrahedra). That encoder remains a target for future work.

In Recognition Science, this result matters because the torus is a candidate target shape for a scalable periodic structure. The declaration confirms the structural precondition for the Regge theorem, so the path to applying that theorem to a periodic lattice is clear. The next step is concrete: write the finite encoder, and the theorem becomes usable.

def edgeSlotPartition_of_encodedPeriodicFreudenthalTorus
    {Nx Ny Nz : ℕ} [NeZero Nx] [NeZero Ny] [NeZero Nz]
    (P : EncodedPeriodicFreudenthalTorus Nx Ny Nz) :
    IncidenceEdgeSlotPartition P.K P.hK where
  localEdgeOf := fun τ f =>
    P.edgeEquiv.symm (localEdgeOf (P.tetEquiv τ).1 (P.tetEquiv τ).2 f)
  edgeInTet_iff := by
    intro e τ f
    constructor
    · intro h
      apply P.edgeEquiv.injective
      simpa using (P.edgeInTet_iff e τ f).1 h
    · intro h
      apply (P.edgeInTet_iff e τ f).2
      rw [h]
      simp

What this page does not claim

The declaration does not provide the finite encoder itself, only the proof that the encoded structure has the required edge partition. The declaration does not claim that any arbitrary triangulation of a torus has this property, only the canonical periodic model.

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)

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: