Rationality of cohomological descendent series for Quot schemes on surfaces with $p_g=0$

For a smooth projective surface $S$, Johnson–Oprea–Pandharipande defined cohomological descendent generating series for Quot schemes of rank-$0$ quotients of ${\text{OO}}_S^{\oplus N}$. We prove rationality of these series in the remaining cohomological surface case $$p_g(S)=0,\beta \mathrm{\ne }0,N>1.$$ The wall-crossing part of the proof starts from Joyce-style generalized Donaldson–Thomas invariant classes of $H$-Gieseker semistable one-dimensional sheaves. We vary a single real parameter in the fixed-source Pairs stability condition and obtain the large-$c$ stable-pair chamber for maps ${\text{OO}}_S^{\oplus N}\to F$. We then compare this pair chamber with the open pure Quot locus, meaning the locus inside the Quot scheme whose target quotient is pure one-dimensional, and then with the full Quot scheme, where zero-dimensional torsion in the target is allowed. The first comparison records the zero-dimensional cokernel of the image of a pair. After decomposing the pure Quot locus into locally-closed pieces on which the scheme-theoretic support curve is flat over the base, this comparison is identified with relative Quot theory on those support curves. The resulting curve-Quot contributions factor into smooth-normalization contributions and finitely many punctual factors at singular points of the support curve. The second comparison records the maximal zero-dimensional torsion subsheaf of a Quot target; locally it becomes a punctual Quot problem over the completed smooth surface ring $\text{C}[[x,y]]$, and its contribution is the universal punctual smooth-surface factor.