Mod 2 Seiberg–Witten Invariants for Spin 4-Manifolds and Families — E8 Intelligence Research

FINDING: The only substantive mathematical result is the complete determination of mod 2 Seiberg–Witten invariants for spin structures on closed oriented smooth 4-manifolds, including families. | MATH: The mod 2 SW invariant for a spin structure \(s\) on a 4-manifold \(X\) is computed via \(∑c∈ H^2(X;Z_2) SW_2(X,s+c) = χ(X) 2\) (simple type mod 2 conjecture confirmed for spin). For families, the invariant is a mod 2 class in \(H^*(B;Z_2)\) over the parameter space \(B\), with the spin structure's \(w_2\) reduction determining the characteristic class. | CONNECTION: The spin structure is equivalent to a \(w_2\)-reduction of the tangent bundle — this is a \(Z_2\)-valued obstruction living in \(H^2(X;Z_2)\). The mod 2 reduction of the first Chern class \(c_1 2 = w_2\) links to the crystallographic root system \(A_1\): \(A_1\) has a single root \(α\), with Cartan matrix \([2]\), and its weight latt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com