rather than FFLO phase. A similar characterisation exercise to that described above with comparison to exact diagonalization was conducted on both systems and the communal state indices determined. The system with a 3:1 ratio is most closely described by a $({N_\uparrow},{N_\downarrow})=(3,1)$ state and that with a 4:1 ratio by a $({N_\uparrow},{N_\downarrow})=(4,1)$ state.

![Summary plot of the ratio of communal state indices ${N_\uparrow}/{N_\downarrow}$ to the ratio of densities of states at the respective Fermi surfaces, ${\nu_{\uparrow}}/{\nu_{\downarrow}}$. The red triangles indicate data taken on triangular momentum space lattices and the blue squares indicate data taken on square momentum space lattices. The bracketed number pairs indicate particle numbers for selected systems, with the colour corresponding to the respective lattice types. The line ${N_\uparrow}/{N_\downarrow}={\nu_{\uparrow}}/{\nu_{\downarrow}}$ is indicated green.[]{data-label="nvsnu"}](nvsnu4){width="0.9\linewidth"}

### Relationship between ${N_\uparrow}/{N_\downarrow}$ and ${\nu_{\uparrow}}/{\nu_{\downarrow}}$

We have shown the results from four exemplar systems in detail to demonstrate the emergence of spatially modulated pairing and provided evidence that the form of the spatial modulation observed is characteristic of communal pairing. The analysis was also repeated for 14 systems with other Fermi surface ratios, for both the triangular and square lattices and at different system sizes, and the results analysed to ascertain the communal state indices, ${N_\uparrow}$ 