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 returns to its initial orientation, while the lifted state reaches 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.Version 1.1 clarifies the reading of e^{2πi}=1 as both endpoint equality and compressed traversal history, adds the R^3 / quaternionic half-angle bridge, and introduces a roots-of-unity transition to the C3 quotient scene. The Dirac belt is treated as a material register of lifted path-history: it makes the projected-away coordinate visible as twist and tension.The paper compares different structures by operational role, not by structural identity:SU(2) → SO(3), R → S^1, Z → Z/3Z.The document does not propose a new physical theory, a revision of arithmetic, or a new explanation of the cited lattice mechanism. Its central claim is structural: endpoint equality can hide path inequivalence.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.