We prove that the distinction gap function g(r) = r(1−r) possesses an intrinsic Z₂ symmetry: g(r) = g(1−r). Part and remainder generate identical distinction cost. Central results: Z₂ symmetry of the gap function is exact This symmetry is topologically equivalent to the Möbius strip At the balance point φ⁻¹, inversion preserves closure Spinor structure emerges from Z₂: one circuit = inversion, two circuits = return Fermion/boson distinction follows from the algebra of distinction Core theorem: g(r) = g(1−r) → Z₂ invariance → Möbius topology → spinor structure → spin-statistics Final thesis: Spin is not a postulate of quantum mechanics. Spin is a consequence of the Z₂ symmetry of the distinction gap. This paper is part of Ontological Resolution Theory (ORT).