Foundation Face Winding Reversed Swaps Endpoints

The reversed cycle

A Hamiltonian cycle on the three-dimensional cube Q₃ is a closed loop that visits each of the cube's eight vertices exactly once before returning to the start. The framework's recognition cycle, a discrete record of eight events, is such a loop. The declaration reversed_swaps_endpoints proves a simple property of this loop: if you take the cycle and traverse it in the opposite direction, the starting vertex of each directed edge becomes the ending vertex of the corresponding reversed edge, and vice versa. In plain terms, reversing the direction of travel swaps the two ends of every edge.

The proof is computational. The cycle is defined by an explicit list of eight directed edges, and the reversed cycle is defined by another explicit list. The theorem checks, for each of the eight steps, that the source of the forward edge equals the destination of the reversed edge and that the destination of the forward edge equals the source of the reversed edge. The machine-checked proof simply evaluates all eight cases. This is a theorem in the framework's library, a machine-checked collection of formal theorems, and it is proved with no additional axioms beyond the standard logical foundations.

In Recognition Science, this endpoint-swapping property is part of the geometric foundation for describing asymmetry. The cycle's winding around each face of the cube, the signed count of how many times the loop circles a face in one direction versus the other, is what the framework uses to model a distinction between forward and backward. The reversed cycle provides the formal counterpart to that forward traversal. The theorem does not by itself establish that any physical process is asymmetric; it only establishes the graph-theoretic fact about the reversed loop.

The declaration does not claim that the reversed cycle is the same as the forward cycle, nor that reversing the cycle leaves any winding number unchanged. It does not claim that the framework's model of time reversal is physically correct. It establishes only the local, edge-by-edge relationship between the forward and reversed directed edges. The broader physical significance, the connection to CP violation, is a separate claim that depends on additional structure in the framework.

/-- Reversing the cycle reverses all edge directions. -/
theorem reversed_swaps_endpoints (step : Fin 8) :
    let fwd := cycleEdges step
    let bwd := reversedCycleEdges (⟨(7 - step.val), by omega⟩)
    fwd.src = bwd.dst ∧ fwd.dst = bwd.src := by
  fin_cases step <;> simp [cycleEdges, reversedCycleEdges, DirectedEdge.mk.injEq]

What this page does not claim

The reversed cycle is not claimed to be identical to the forward cycle. The theorem does not claim that any physical process is asymmetric. The theorem does not establish the framework's model of CP violation.

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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