Visible Return and Lifted Return: The Dirac Belt, Projection, and Lost Path-History

This note studies a structural loss produced by projection. Its basic distinction separates visible return from lifted return. The Dirac belt provides the minimal scene: after one full turn, the visible body has returned to its initial orientation, while the lifted state has reached the opposite endpoint. The projection SU(2) → SO(3) gives the structural form of this loss: q and −q determine the same visible rotation while remaining distinct in the lifted description. Path-inequivalence is a structural distinction preserved in the lift. Projection collapses this distinction into endpoint equality. The note uses the Dirac belt as an orienting device for later numerical scenes involving signs, phases, roots of unity, and quotient structures. The guiding question is: what returns when something returns? The answer depends on the register in which the claim of return is made. The paper stays within standard topology, group theory, and elementary geometric interpretation. Its object is the lost coordinate of path-history.

Additional notes:

This is a conceptual and diagrammatic working draft. It introduces the Dirac belt as a minimal scene for distinguishing visible return from lifted return, then compares this structure with the covering map R → S¹ and a modulo-three quotient scene in recent work on angular momentum transfer among crystal lattice modes. The document does not propose a new physical theory or a revision of arithmetic. Its central claim is structural: endpoint equality can hide path inequivalence.