Foundation Freudenthal Triangulation Cert Body Diagonal Full Angle

The full angle around the body diagonal

A cube can be sliced into six congruent tetrahedra, a decomposition known as the Freudenthal triangulation. All six tetrahedra share the cube's body diagonal, the line from one corner to the opposite corner. Around that shared diagonal, each tetrahedron contributes a dihedral angle of one sixth of a full turn, so the six angles together make exactly one complete turn. The declaration body_diagonal_full_angle records this as a theorem in the framework's machine-checked library of formal theorems, stating that six times one sixth equals one.

The statement is deliberately narrow. It certifies the arithmetic of the angle sum, not the geometry of the cube itself. The theorem proves that 6 × (1/6) = 1 in the rational numbers, a fact that holds by computation. The surrounding certificate structure assembles the supporting counts: the cube has 8 vertices, 12 edges, and 6 faces; the Freudenthal decomposition uses 6 tetrahedra; the 13 new hinges consist of 12 face diagonals plus 1 body diagonal. Each of these counts is itself a proved theorem in the same file, and the full certificate packages them together with the angle sum.

What the declaration does not claim is just as important. It does not assert that the cube exists in physical space, nor that recognition events occupy such a triangulation. The theorem is combinatorial and arithmetical, a certificate that the angle bookkeeping closes: the six tetrahedra around the body diagonal leave no angular gap and no overlap. In the framework's language, the deficit angle is zero, meaning the local geometry around that hinge is flat. The certificate records this as a structural fact about the decomposition, not as a statement about any particular physical system.

The payoff for a reader is a clean example of what a formal certificate can guarantee. The arithmetic around the body diagonal is exact, checked by computation, and free of unproved assumptions. The larger framework uses such certificates to build claims about recognition and dimension, but this particular declaration stays at the level of counting and angle sums. It is a small, precise brick in a larger wall, and its value is that the brick is known to be sound.

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

theorem cubeVertices_eq : cubeVertices = 8 := rfl

theorem cubeEdges_eq : cubeEdges = 12 := rfl

theorem cubeFaces_eq : cubeFaces = 6 := rfl

theorem freudenthal_count : freudenthalTetCount = 6 := rfl

theorem newHinges_decomp : newHinges = 12 + 1 := rfl

What this page does not claim

The declaration does not assert that physical space is composed of such tetrahedra. The theorem does not establish that the deficit angle is zero in any geometric sense beyond the arithmetic sum. The certificate does not connect the cube's triangulation to the framework's recognition cycle or cost function.

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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