Four-Dimensional Pin+ Bordism: Classification, Decomposition, and Spacetime Topology

An important framework paper-1 Disclaimer: We hereby release this framework paper—more related papers will be issued later—to furnish readers and specialists with a holistic overview of our Pin⁺-based framework research.I hope it will help promote broader research in pin related fields. However, in the early versions of the paper, there may be some errors, please pay attention to identifying them. We present a unified treatment of Pin⁺ bordism in four and higher dimensions, combining three complementary perspectives: classification, decomposition, and physical applications. Part I (Classification). We give a complete classification of four-dimensional bordism groups for all ten tangential structures on closed manifolds: O, SO, Spin, Spinc, Spinh, Pin+, Pin−, Pinc, Pinh−, and Pinh+. The key new results are: (1) Ω4Pinh−≅(Z2)3 via the Smith isomorphism; (2) Ω4Pinh+≅Z16⊕Z2 via AHSS with Bockstein resolution; (3) a three-layer taxonomy classifying all ten structures into infinite-layer (4), finite-cyclic-layer (3), and finite-noncyclic-layer (3). We compute 13 forgetful maps and establish the information-flow topology. Part II (Decomposition). We present a spacetime-gauge decomposition theorem for Pin⁺ bordism groups with SU(N) gauge structure: for BSU(2) and n≤7, the AHSS collapses at E2, yielding ΩnPin+(BSU(2))≅ΩnPin+(pt)⊕H4(BSU(2);Ωn−4Pin+). At n=8, we show d4=0 by the dual Cartan formula, which provides an explicit first-principles calculation showing the induced cohomology operation vanishes, extending the decomposition theorem to n=8. We compute ΩnPin+(BSU(2)) for n=0–8 and ΩnPin+(BSU(3)) for n=0–6, identifying color instanton parity as a new topological invariant. Part III (Physical Applications). We show that the α-invariant vanishes on R4 by three independent arguments, implying the complete degeneration Z16→Z1 in the α-invariant sense. This generalizes: α(Rn)=0 for all n via a heat kernel argument, and α(M)=0 for any complete non-compact homogeneous Spin manifold. The Swampland Cobordism Conjecture is automatically satisfied on such spacetimes. We analyze topological constraints on the Yang–Mills mass gap from Pin⁺ bordism and identify the precise boundary of bordism methods for the millennium problem. Part IV (Topological Physics). We develop the physical consequences of the Z16 classification for spacetime topology. We establish the Altland–Zirnbauer–Pin correspondence: AZ real classes with T2=(−1)F correspond to Pin⁺ structure, while T2=+1 corresponds to Pin⁻, mediated by Clifford algebra representation theory. We introduce the Causal Direction Index CDI(k)=sin(πk/8) and state the 4D Topological Causality Theorem, conjecturing CPT conservation as a dual foundation—the standard chain from Lorentz invariance and the topological chain from RFB=∞—which are independent and exhibit scale stratification (Conjecture 48, discussed in Section XVIII D). We show that the Klein bottle occupies sector k=10 in the Z16 classification, with domain walls acting as time-direction turning points identified with Greene–Kabat parity walls.