Foundation Freudenthal Triangulation Cert Cube Edges Eq

The cube's edge count

A cube has 8 vertices, 12 edges, and 6 faces. These are among the first facts a geometry student learns, and they are also the starting point for a classic result: the unit cube [0,1]³ can be cut into 6 congruent tetrahedra, a decomposition known as the Freudenthal triangulation. The declaration cubeEdges_eq is the framework's machine-checked way of recording the edge count: it states that the number of edges of the unit cube is 12. The proof is immediate, a definitional equality, but its value is not in the arithmetic. It is a certified foundation stone for the larger claim that this triangulation introduces 13 new internal hinges, 12 face diagonals plus 1 body diagonal, and that all of them have zero deficit angle.

The Freudenthal triangulation is a standard tool in mathematics and computer science, used to subdivide a cube into simplices for interpolation and finite element methods. The framework's library, a machine-checked collection of formal theorems, records the combinatorial facts that make the construction work: the cube's vertex, edge, and face counts, the 6 tetrahedra sharing the body diagonal, and the angle sums that confirm each new hinge is flat. Each tetrahedron meets the body diagonal at a dihedral angle of π/3, and six of those angles sum to 2π, so the deficit is zero. The face diagonals behave similarly, with four tetrahedra tiling the full 2π around each one.

In Recognition Science, this certificate is not a discovery about cubes. It is a bookkeeping check. The framework models physical structure through a discrete record of recognition events, and it needs a consistent geometry to build on. The cube's edge count being 12 is a premise that must be true for the later, more interesting theorems about three-dimensional space to hold. The declaration does not prove that space has three dimensions, nor does it derive the Freudenthal triangulation from first principles. It simply certifies one of the elementary counts on which the larger structure rests.

What the declaration does not claim is worth stating plainly. It does not assert that the Freudenthal triangulation is the only way to decompose a cube, nor does it prove that the zero deficit angle condition is sufficient for a valid physical hinge. The theorem is about the number 12 and nothing more. The deeper claims, about why this triangulation matters for the framework's model of space, are separate theorems with their own conditions and proofs.

theorem cubeEdges_eq : cubeEdges = 12 := rfl

theorem freudenthal_count : freudenthalTetCount = 6 := rfl

theorem newHinges_decomp : newHinges = 12 + 1 := rfl

/-- 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 that space has three dimensions. The declaration does not derive the Freudenthal triangulation from first principles. The declaration does not assert that the Freudenthal triangulation is the only way to decompose a cube.

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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