Minimal Nodal Complexity for Detecting the Primitive Middle Hodge Class on a Smooth Quadric Fourfold

This preprint studies a concrete generalized-Thomas-section benchmark on a smooth complex quadric fourfold. Let X = Q^4 ⊂ P^5, let h = c1(O_X(1)), and let ζ = 2[Π] − h^2 be the primitive middle Hodge class associated with a maximal plane Π ⊂ X. For reduced divisors D ∈ |O_X(2)| with only ordinary double points (ODP), the paper asks for the minimal number of nodes required for the restriction ζ|_D to be nonzero. The central working proposition is that this minimal nodal detection complexity equals three. The upper bound is obtained from a general quadric section containing Π: a Chern-class calculation gives three singular points, transversality yields ordinary double points, and Π ⊂ D directly detects ζ. The lower-bound analysis separates two-nodal complete intersections of two quadrics into noninduced and induced-defect cases, using defect theory and the primitivity of ζ to rule out detection in both cases. The note is a proof-candidate preprint intended for independent mathematical checking and prior-art comparison. It does not claim a solution of, or direct advance on, the Hodge Conjecture. Novelty of the exact minimization statement has not been independently established, and the manuscript has not undergone peer review. Subtitle: A working note on generalized Thomas sections, defect, and the Q⁴ benchmark