The modulo 9 Kanade--Russell identities and their Nahm-sum duals

Kanade and Russell initiated a family of conjectural Rogers--Ramanujan type identities of moduli $9$ and $12$, which ultimately comprised five modulo $9$ identities and eleven modulo $12$ identities. The eleven modulo $12$ conjectures were subsequently settled through the work of Bringmann, Jennings-Shaffer, and Mahlburg and of Rosengren. In this paper, we prove all five modulo $9$ Kanade--Russell sum-product identities, four individual generalized Nahm-sum dual identities, and a product formula for the natural dual companion of the fifth Kanade--Russell identity, which is expressed as a linear combination of two negative-mixed-term generalized Nahm sums. The first three individual dual identities settle Conjecture~3.6 of Wang and Wang, while the fourth proves the corresponding conjecture of Li and Wang. Our results also connect directly with the recent Dynkin-diagram framework of Sun and Wang for generalized Nahm sums. They identified the rank-two pairs $(T_1,G_2)$ and $(G_2,T_1)$ as unresolved cases whose modularity would follow, respectively, from the first modulo $9$ Kanade--Russell identity and its Wang-Wang dual. The present results prove precisely these two required identities and hence establish the corresponding modularity statements unconditionally.