Quarkoniumlike states above open-flavor thresholds in Born-Oppenheimer EFT

Many quarkoniumlike states have been observed above open-flavor thresholds, but their organization and internal structure remain unsettled. We study the isoscalar hidden-charm and hidden-bottom sectors in Born--Oppenheimer effective field theory (BOEFT), between the spin--isospin averaged $S+S$ and $S+P$ thresholds. At leading order, heavy-quark spin decouples, and the quarkonium static potential mixes through string breaking with the lowest tetraquark/open-flavor BO potentials of the same quantum numbers. These potentials are constrained by QCD symmetries, their short- and long-distance behavior, and lattice-QCD data. The only calibrated parameter is the lowest $1^{--}$ adjoint meson mass, fixed from the shallow multiplet associated with the $\chi_{c1}(3872)$. Using $T$-matrix, $K$-matrix, and complex-scaling methods, we determine bound states and resonance poles, their masses, pole widths from the included nonstrange $S+S$ channels, normalized pole couplings, and prescription-dependent quarkonium--open-flavor composition measures. Uncoupled hybrid BOEFT multiplets are included as reference levels. The spectrum exhibits a common heavy-quark-spin-symmetry multiplet organization. Most poles are predominantly quarkonium resonances localized at short distances, with the largest open-flavor components closest to threshold. The same equations also generate shallow, spatially extended, open-flavor-dominated states with molecular long-distance characteristics. Their binding energies, radii, and small quarkonium components are highly sensitive to the adjoint meson mass, whereas the higher spectrum is more stable. Together with the hybrid reference levels, the spectrum provides multiplet assignments for most candidates. States not naturally accommodated point to the need for hidden-strange and $S+P$ tetraquark/open-flavor BO sectors and for hybrid--tetraquark and hybrid--quarkonium mixings.