An Axiomatic Proposal for Non-Orientable Spacetime: A Reformulation of Spacetime Structure via the Orientation-Reversing Map

The central claim of this paper is a proposal to change the fundamental axiom of spacetime from an orientable manifold to a non-orientable one. We axiomatize spacetime as a non-orientable manifold (w₁(M) ≠ 0) equipped with an orientation-reversing map f (an isometric involution, det(df) = −1). This axiom system is not empty: elliptic de Sitter spacetime dS₄/Z₂ provides an instance as an exact solution of the Einstein equations, and the Pin(1,3) structure required for fermions on a non-orientable manifold contains orientation-preserving and orientation-reversing transformations within a single algebra. After establishing the consistency of the axioms—the existence of models, the convention for the Pin structure, the distinction from the orientable double cover—this paper presents a research program whose goal is to derive from this axiom the origin of the negative sign of the Lorentzian metric, the description of singularities as reversal points, and the provenance of the complex structure of quantum theory, while explicitly distinguishing mathematical facts from hypotheses. In the course of this consistency check, it emerged that the Pin structure requires structurally distinct σ-sectors for scalar and fermion fields—the former real and alternating across levels, the latter quaternionic (Θ² = −1) and bisecting within each level.