Foundation Face Winding Each Edge On Two Faces

The cube's edge rule

In geometry, a cube's edge is where two faces meet. Each of the cube's 12 edges is shared by exactly two of its 6 square faces. This is a property of the cube itself, independent of any path drawn on it. A machine-checked library of formal theorems verifies this elementary fact for a particular eight-step cycle that visits all eight vertices of the cube, confirming that each step of that cycle lies on precisely two faces.

The eight-step cycle is a Hamiltonian cycle: it visits every vertex exactly once and returns to its start. The cube's vertices can be labeled with three-bit binary numbers, 000 through 111, so each step flips exactly one bit. The library's theorem each_edge_on_two_faces states that for any of the eight steps in this cycle, the number of faces containing that edge is exactly two. The proof is computational: it checks all eight cases directly and confirms the count each time.

This fact matters because the cycle is not just any path. It is the framework's eight-tick recognition cycle, a sequence of states that the framework uses to model how a system distinguishes forward from backward. The geometric setting gives each face a signed winding number, a measure of how the cycle wraps around that face's boundary. The theorem that each edge lies on two faces is a necessary consistency condition for these winding numbers to be well-defined.

In Recognition Science, the cube is not a physical object but a ledger, a discrete record of events. The eight vertices are the eight ticks of a cycle, and the edges are the transitions between them. The framework models each face as corresponding to a pair of particle generations. The winding number around a face measures the asymmetry between clockwise and counterclockwise traversal, and a nonzero total winding is the geometric origin of CP violation, the observed difference between matter and antimatter behavior.

The theorem establishes a topological consistency condition. It does not, by itself, prove that CP violation exists or that the winding numbers are nonzero. Those are separate results in the library. The theorem only confirms that the cycle's edges are properly situated on the cube's faces, a prerequisite for the winding numbers to be meaningful.

What the theorem does not claim is just as important. It does not say that every Hamiltonian cycle on a cube has this property; it applies to the specific cycle defined in the library. It does not assign physical meaning to the winding numbers; that is a modeling choice. And it does not derive the value of any physical constant. It is a geometric lemma, checked by computation, that supports the framework's larger claims about particle physics.

/-- Each edge of the cycle is incident to exactly 2 of the 6 faces
    (the edge lies on exactly 2 faces of the cube). -/
theorem each_edge_on_two_faces (step : Fin 8) :
    (allFaces.filter (fun f => edgeOnFace step f)).length = 2 := by
  fin_cases step <;> native_decide

/-- Each edge of the cycle is incident to exactly 2 of the 6 faces
    (the edge lies on exactly 2 faces of the cube). -/
theorem each_edge_on_two_faces (step : Fin 8) :
    (allFaces.filter (fun f => edgeOnFace step f)).length = 2 := by
  fin_cases step <;> native_decide

What this page does not claim

The theorem does not apply to every Hamiltonian cycle on a cube, only the specific eight-step cycle defined in the library. The theorem does not assign physical meaning to the winding numbers or prove CP violation exists. The theorem does not derive any physical constant or the value of the total chiral charge.

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)

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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