Foundation Face Winding Face Pairs Have Three Axes

The three axes of the cube

A cube has six faces, and those six faces come in three opposite pairs: left and right, front and back, top and bottom. Each pair shares a common axis, the line perpendicular to both faces. The theorem face_pairs_have_three_axes states this elementary fact formally: for every face in the cube's complete list of six, the axis index of that face is one of the three available axes, 0, 1, or 2. The proof is a direct computation over the explicit list of faces, checked by the machine. It is a small lemma, the kind of thing a mathematician would write down without a second thought.

The statement lives inside a larger construction. The framework models an eight-step cycle through the cube's eight corners, and it assigns to each face a winding number, a signed count of how many times the cycle traverses that face's boundary in one direction minus the other. A nonzero winding means the cycle distinguishes clockwise from counterclockwise on that face. The six faces, grouped by their three axes, give three independent places where this forward-backward asymmetry can appear. The framework identifies each axis with a generation in particle physics, so the three axes correspond to the three observed generations of matter. The theorem itself does not establish that identification; it only fixes the geometry underneath it.

What the theorem does not claim is just as important as what it proves. It does not prove that the winding numbers are nonzero, nor that the three axes are physically distinct, nor that the cycle's asymmetry actually produces CP violation. Those are separate statements in the library, and the physical bridge from winding numbers to observed particle behavior is a further step. The theorem is a foundation stone: it certifies that the cube's faces sort cleanly into three axes, nothing more and nothing less.

/-- The cycle has the face-pair structure: opposite faces (same axis, different
    side) are each traversed by the cycle, and the asymmetric flip schedule
    means different face-pairs experience different winding patterns. -/
theorem face_pairs_have_three_axes :
    ∀ f ∈ allFaces, f.axis.val < 3 := by
  simp [allFaces]

/-- The cycle has the face-pair structure: opposite faces (same axis, different
    side) are each traversed by the cycle, and the asymmetric flip schedule
    means different face-pairs experience different winding patterns. -/
theorem face_pairs_have_three_axes :
    ∀ f ∈ allFaces, f.axis.val < 3 := by
  simp [allFaces]

What this page does not claim

The theorem does not prove that any face has a nonzero winding number. The theorem does not establish the physical identification of axes with particle generations. The theorem does not derive CP violation from the winding numbers.

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/FaceWinding.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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