Foundation Freudenthal Triangulation Cert Cube Vertices Eq

The cube's vertex count

A cube, the familiar three-dimensional box with six square faces, has eight corners. This is one of the first facts a child learns about solid geometry. The declaration cubeVertices_eq in the Recognition Science framework's machine-checked library of formal theorems states this fact as a formal theorem: the defined constant for the cube's vertex count equals 8. The proof is immediate, a definitional equality, meaning the statement holds by the very way the constant is defined.

The declaration belongs to a larger certificate about the Freudenthal triangulation, a standard way to divide the unit cube into six congruent tetrahedra. This decomposition is a classical construction in geometry and topology, often used in numerical methods and piecewise-linear approximations. The certificate records the cube's basic combinatorial data: 8 vertices, 12 edges, and 6 faces. It also records that the Freudenthal decomposition produces 6 tetrahedra, that these share a common body diagonal, and that the angles around the new internal edges sum to a full 2π, giving a zero deficit angle.

In Recognition Science, this certificate acts as a combinatorial checkpoint within a larger forcing chain that derives physical structure from a ledger of recognition events. The framework models reality as a discrete record of events, and the cost of recognition is forced by a proved uniqueness theorem. The cube's vertex count is a small piece of the geometric scaffolding that the framework uses to connect its abstract cost function to concrete spatial structure. The certificate's role is to verify that this scaffolding is internally consistent, a formal check that the numbers used in the framework's geometric arguments are correct.

The theorem does not claim that physical space is made of cubes, nor that the cube's vertex count has any direct physical consequence. It is a statement about a definition in a formal library, not an empirical discovery. The certificate does not prove that the Freudenthal triangulation is the only way to decompose a cube, nor does it connect the zero deficit angle to any physical principle. The declaration establishes a combinatorial fact and nothing more. Its value lies in being a verified, exact step within a larger formal structure, not in any standalone geometric insight.

theorem cubeVertices_eq : cubeVertices = 8 := rfl

theorem cubeVertices_eq : cubeVertices = 8 := rfl

theorem cubeEdges_eq : cubeEdges = 12 := rfl

theorem cubeFaces_eq : cubeFaces = 6 := rfl

theorem freudenthal_count : freudenthalTetCount = 6 := rfl

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

def zeroDeficitCert : ZeroDeficitCert where
  body_diagonal_deficit_zero := trivial
  face_diagonal_deficit_zero := trivial
  new_hinge_count := rfl

What this page does not claim

The theorem does not claim that physical space is composed of cubes or that the vertex count has direct physical meaning. The certificate does not prove that the Freudenthal triangulation is the only valid decomposition of the cube. The zero deficit angle is recorded as a combinatorial fact, not derived from any physical principle.

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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