Foundation Freudenthal Triangulation Cert Freudenthal Count

The six tetrahedra

The unit cube, the familiar three-dimensional box with edges of length one, has a well-known subdivision into six congruent tetrahedra. This is the Freudenthal triangulation, named after the Dutch mathematician Hans Freudenthal. The theorem freudenthal_count in the framework's machine-checked library of formal theorems establishes that this count is exactly six. It is a statement about a simple combinatorial fact, not about physics or geometry beyond the counting itself.

The cube has 8 vertices, 12 edges, and 6 faces. The Freudenthal decomposition introduces 13 new hinges: 12 face diagonals and 1 body diagonal. The six tetrahedra all share the body diagonal. Each tetrahedron meets the body diagonal at a dihedral angle of π/3, and six such angles sum to 2π, leaving no deficit. The face diagonals are each shared by four tetrahedra, and their angles also sum to 2π. The declaration freudenthal_count certifies the number of tetrahedra, while companion declarations in the same file certify the vertex, edge, and face counts, the number of new hinges, and the zero deficit angle condition.

The formal statement is a definitional equality: freudenthalTetCount is defined to be 6, and the theorem proves that this definition equals 6 by reflexivity. The proof is immediate, requiring no axioms beyond the standard logical ones. The certificate bundles these facts into a single structure, FreudenthalCert, which records the cube data, the tetrahedron count, the hinge count, the angle sum, and the zero deficit condition. The declaration freudenthalCert constructs this certificate from the individual proofs.

What this declaration does not claim is important. It does not prove that the Freudenthal triangulation is unique, nor that it is the only way to decompose a cube into six tetrahedra. It does not establish any physical property of space, such as the number of spatial dimensions. The zero deficit condition is stated as a symbolic check on rational numbers, not as a geometric theorem about actual angles in Euclidean space. The declaration is a combinatorial certificate, a precise record of a counting fact, not a derivation of any deeper structure.

theorem freudenthal_count : freudenthalTetCount = 6 := rfl

theorem cubeVertices_eq : cubeVertices = 8 := rfl

theorem cubeEdges_eq : cubeEdges = 12 := rfl

theorem cubeFaces_eq : cubeFaces = 6 := rfl

theorem newHinges_decomp : newHinges = 12 + 1 := rfl

/-- Body diagonal shared by all 6 tetrahedra. -/
def bodyDiagonalTetrahedra : ℕ := 6

/-- Sum of 6 × (1/6) = 1 (symbolic angle check). -/
theorem body_diagonal_full_angle : 6 * (1 : ℚ) / 6 = 1 := by norm_num

What this page does not claim

The declaration does not prove the Freudenthal triangulation is unique. The declaration does not establish any physical property of space, such as the number of dimensions. The zero deficit condition is a symbolic check on rational numbers, not a geometric theorem about actual angles.

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/Foundation/FreudenthalTriangulationCert.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.

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