Torsion Trajectories from Local Discriminants to Global Obstructions

For a normal surface singularity, the discrepancy between the ordinary and dual middle-perversity intersection complexes over $\mathbb Z$ is measured by a finite group $E$. In previous work, $E$ was identified with link torsion, the exceptional-lattice discriminant group $\Lambda^\vee/\Lambda$, a resolution-neighborhood boundary quotient, and, in the hypersurface case, $\operatorname{coker}(T-\mathrm{id})_{\mathrm{tors}}$. This paper tracks the trajectory of this torsion from local singularity data to global obstruction theory. We follow the discriminant package $(E,q)$ through support cohomology, excision, global torsion, Brauer comparison, Bloch--Ogus residues, and rationalization. The method is example-driven: trajectory tables are computed for $A_1$, $A_k$, $D_4$, $E_8$, a non-ADE Brieskorn singularity, the threefold ordinary double point, nodal threefolds, nodal quintics, and the Benoist--Ottem benchmark. The computations reveal a sharp distinction: a surface $A_1$ singularity has local $\mathbb Z/2$-torsion, whereas a threefold ordinary double point has torsion-free link $S^2\times S^3$ and contributes free vanishing-cycle data instead. Thus finite discriminant torsion is naturally a codimension-two phenomenon, not a generic feature of nodes. The resulting pattern motivates a specialization problem: whether the Enriques $2$-torsion in Benoist--Ottem integral Hodge counterexamples is genuinely global, or can arise after degeneration from transverse $A_1$-type discriminant data along codimension-two strata.