Foundation Freudenthal Triangulation Cert Cube Faces Eq

The cube's six faces

A cube is a familiar solid: eight vertices, twelve edges, and six flat square faces. The declaration cubeFaces_eq in the Recognition Science framework's machine-checked library of formal theorems states that the framework's count of faces on the unit cube equals 6. This is not a discovery about geometry; it is a formal bookkeeping statement, a definitional identity that the computer verifies by unfolding the definition of cubeFaces, a count the framework assigns to the unit cube.

The unit cube [0,1]³ decomposes into 6 congruent tetrahedra, a classical fact named after Freudenthal. Each of those tetrahedra shares the cube's body diagonal, and the 12 face diagonals plus that 1 body diagonal form 13 new hinges. The framework's certificate records these counts: 8 vertices, 12 edges, 6 faces, 6 tetrahedra, 13 new hinges. The theorem cubeFaces_eq is the smallest piece of that certificate: it pins down the face count alone.

In plain terms, the declaration proves that the framework's own symbol for the number of faces, cubeFaces, equals the natural number 6. The proof is by reflexivity: the definition of cubeFaces is literally the numeral 6, so the equality holds by definition. This is a tautology in the formal sense, not a theorem about the physical world. The framework uses such counts as ingredients in a larger certificate about zero deficit angles, but this particular declaration does not itself establish any geometric property.

The declaration does not claim that the cube has six faces in any physical sense, nor does it prove that a Freudenthal triangulation exists. It does not assert that the 6 tetrahedra have zero deficit angle; that claim lives in a separate structure, ZeroDeficitCert, which the framework records as a certificate with trivial proofs. cubeFaces_eq is a definitional equality, a bookkeeping entry that says: when the framework counts faces, it counts six.

theorem cubeFaces_eq : cubeFaces = 6 := rfl

/-- Unit cube face count. -/
def cubeFaces : ℕ := 6

What this page does not claim

The declaration does not prove that a physical cube has six faces. It does not establish the existence of a Freudenthal triangulation. It does not assert that the tetrahedra have zero deficit angle; that is a separate certificate.

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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