Mod-2 Seiberg-Witten Invariants Fully Computed for All Spin Structures — E8 Intelligence Research

FINDING: Complete determination of mod-2 Seiberg-Witten invariants for all spin structures on closed smooth 4-manifolds, confirming the simple type conjecture mod 2. | MATH: For a spin structure \(\mathfrak{s}\) on a closed oriented smooth 4-manifold \(X\), the mod-2 SW invariant \(\mathrm{SW}_2(X,\mathfrak{s})\) is fully computed. Key result: \(\mathrm{SW}_2(X,\mathfrak{s}) \equiv \sum_{c \in H^2(X;\mathbb{Z}_2)} \mathrm{SW}(X,\mathfrak{s}+c) \pmod{2}\), where \(\mathrm{SW}\) are integer invariants. The simple type condition (all basic classes have \(b^+ - b_1 = 0\) mod 2) holds for spin structures. For families, the invariant becomes a mod-2 class in \(H^*(B;\mathbb{Z}_2)\) over the parameter space \(B\). | CONNECTION: The mod-2 reduction inherently involves \(\mathbb{Z}_2\) — the simplest nontrivial finite field, linked to the root system \(A_1\) (crystallographic symmetry of the line). The spin structure corresponds to a \(w_2\)-twisted \(\mathrm{Spin}^c\) structure; the mod-2 arit Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com