Geometry Periodic Freudenthal Torus Canonical Edge In Tet Eq Some Implies

The meaning of a matched edge

A periodic Freudenthal torus is a repeating three-dimensional grid of tetrahedra, wrapped around so that opposite faces meet. The framework's machine-checked library of formal theorems builds this grid from typed vertices, edges, and tetrahedra, then proves structural facts about how they connect. The declaration canonicalEdgeSlot_eq_some_implies concerns the search for a global edge inside a single tetrahedron: it states that if the search returns a positive answer, then that global edge is exactly the local edge of that tetrahedron, with no ambiguity.

In plain terms, the theorem says that the identification of a global edge with a local one is a one-way implication. If a global edge e is found in tetrahedron cell at local face tet and local edge f, then e equals the local edge defined by those coordinates. The converse, that every local edge corresponds to a global one, is a separate property that the library proves elsewhere. The theorem is a consistency guarantee: the search function never returns a match that is not the true local edge.

This result matters because the framework uses these identifications to build a global partition of edge slots, a structure needed for the nonlinear Regge first-variation theorem. The partition requires that each edge slot be assigned to exactly one tetrahedron face. The implication theorem ensures that when the assignment is made, it is faithful to the underlying geometry. It is a small but load-bearing piece of the proof that the periodic torus has the required incidence structure.

What the theorem does not claim is also precise. It does not assert that every global edge appears in some tetrahedron; that is a surjectivity statement proved separately. It does not say that the search is efficient or that it terminates quickly; it is a mathematical statement about correctness, not complexity. And it does not establish that the periodic torus is the only shape with this property; it is a fact about this specific construction, not a uniqueness theorem.

theorem canonicalEdgeSlot_eq_some_implies
    {Nx Ny Nz : ℕ} [NeZero Nx] [NeZero Ny] [NeZero Nz]
    {e : PeriodicEdge Nx Ny Nz} {cell : Vertex Nx Ny Nz}
    {tet : Fin 6} {f : Fin 6}
    (h : canonicalEdgeSlot? e cell tet = some f) :
    e = localEdgeOf cell tet f := by
  unfold canonicalEdgeSlot? at h
  split_ifs at h with h0 h1 h2 h3 h4 h5
  · cases h
    exact h0
  · cases h
    exact h1
  · cases h
    exact h2
  · cases h
    exact h3
  · cases h
    exact h4
  · cases h
    exact h5

What this page does not claim

The theorem does not prove that every global edge appears in some tetrahedron. The theorem does not claim anything about the efficiency of the edge search. The theorem does not establish that the periodic torus is the unique shape with this incidence property.

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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Derived articles

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