Geometry Periodic Freudenthal Torus Canonical Encoded Periodic Tet Equiv Eq

The encoded torus

A periodic Freudenthal torus is a repeating three-dimensional grid of tetrahedra, the kind of shape used to study how curved space can be approximated by flat pieces. The classical problem is to encode such an infinite repeating pattern into a finite list of numbers that a computer can handle. A new result in the framework's machine-checked library of formal theorems shows that this encoding is always possible in a canonical way: the finite set of tetrahedra in any periodic tiling is in one-to-one correspondence with a typed description of vertices and tetrahedron types.

The declaration canonicalEncodedPeriodic_tetEquiv_eq establishes that a specific equivalence, called canonicalPeriodicTetEquiv, is a bijection. This means that for any periodic torus with dimensions Nx, Ny, Nz, each tetrahedron in the tiling corresponds to exactly one pair: a base vertex and a number from 0 to 5 selecting one of six tetrahedron orientations. The proof is constructive: it builds the correspondence explicitly, so it is not just an abstract existence statement but a recipe that could be implemented.

The result is part of a larger effort to give the periodic Freudenthal torus a concrete finite representation. The library also proves that this encoding preserves the incidence structure needed for a nonlinear Regge first-variation theorem, a tool used in numerical relativity. The current declaration is a stepping stone: it shows the typed model is faithful, but it does not yet provide the final finite encoder into plain natural-number indices for vertices, edges, and tetrahedra.

What the declaration does not claim is equally important. It does not prove that any particular finite mesh exists; it proves that if a finite triangulation encodes the typed model, then the incidence partition holds. It does not enumerate a concrete mesh, and it does not claim that the periodic torus is the only shape that can be so encoded. The remaining work, building the finite encoder, is explicitly left open in the framework's documentation.

def canonicalPeriodicTetEquiv
    (Nx Ny Nz : ℕ) [NeZero Nx] [NeZero Ny] [NeZero Nz] :
    Fin (canonicalPeriodicTriangulation Nx Ny Nz).nT ≃
      PeriodicTet Nx Ny Nz :=
  tetFinEquiv Nx Ny Nz

def canonicalPeriodicTetEquiv
    (Nx Ny Nz : ℕ) [NeZero Nx] [NeZero Ny] [NeZero Nz] :
    Fin (canonicalPeriodicTriangulation Nx Ny Nz).nT ≃
      PeriodicTet Nx Ny Nz :=
  tetFinEquiv Nx Ny Nz

def canonicalPeriodicIncidenceConsistent
    (Nx Ny Nz : ℕ) [NeZero Nx] [NeZero Ny] [NeZero Nz] :
    IncidenceConsistent (canonicalPeriodicTriangulation Nx Ny Nz) :=
  canonicalPeriodicIncidenceConsistent_of_endpoint Nx Ny Nz
    (canonicalPeriodicEndpointIncidence Nx Ny Nz)

What this page does not claim

The declaration does not construct a concrete finite mesh for any specific torus. It does not prove that the periodic torus is the unique shape admitting such an encoding. It does not establish the physical bridge from recognition to linking.

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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