Extended Data Fig. 2: Numerical simulations of the topological vortex guide by chiral-symmetric disclinations.

Propagation of the probe vortex is simulated under single-site excitation at the core (circled in a1-c1) of \({C}_{3}\), \({C}_{4}\) and C₅-symmetric disclination lattices. a1 C₃ disclination lattice. a2-a5 Output intensity distributions of an \(l=1\) (a2, a3) and \(l=2\) (a4, a5) vortex beam from the lattice at selected distances \(z=20\,{\rm{mm}}\) (a2, a4) and \(200\,{\rm{mm}}\)(a3, a5), highlighting the protected topological vortex transport. Insets are the corresponding numerical interferograms, showing the phase singularity from fringe bifurcations. b1-b5 Same layout as in a1-a5 but for the C₄-symmetric lattice. In this case, the \(l=1\) vortex is ‘doubly’ protected by topology, while the \(l=2\) vortex is not protected due to the lack of real-space topological protection depicted in Fig. 4 of the main text, breaking into a quadruple-like structure at the core with ‘tails’ populating both sublattices. c1-c5 Same layout as in a1-a5 but for the C₅-symmetric lattice, which shows again robust transport for both \(l=1\) and \(l=2\) vortices because they satisfy the requirement of vorticity-coordinated rotational symmetry.