Rogers--Ramanujan identities from the geometry of $X^a=Y^b$

We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers–Ramanujan and Andrews–Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1<a<b\mathrm{.}$ For a prime power $q$, let ${\mathcal{N}\mathcal{C}}_n^{a,b}({\mathbb{F}}_q)$ denote the set of pairs of commuting nilpotent $n\times n$ matrices $(A,B)$ over ${\mathbb{F}}_q$ satisfying $A^a=B^b$. We establish the threefold equality between their normalized counts, the HJO $q$-series $Z_{a,b}$, and the explicit infinite product $P_{a,b}$: $$\underset{\text{point count}}{\underbrace{\phantom{\mid }\prod _{m\ge 1}(1-q^{-m})\left(\sum _{n=0}^{\mathrm{\infty }}\frac{\mid {\mathcal{N}\mathcal{C}}_n^{a,b}({\mathbb{F}}_q)\mid }{\mid {\mathrm{GL}}_n({\mathbb{F}}_q)\mid }\right)}}=\underset{q\text{-series}}{\underbrace{\phantom{\mid }{Z}_{a,b}(q^{-1})}}=\underset{\text{infinite product}}{\underbrace{\phantom{\mid }{P}_{a,b}(q^{-1})}}\mathrm{.}$$ Our main result is a stronger finite identity: the rank $N$ HJO sum equals $(q;q{)}_N$ times the generating function for balanced cylindric partitions with entries bounded by $N$. Taking $N\to \mathrm{\infty }$ yields the HJO conjecture. The proof combines the compositional rational shuffle theorem of Bergeron–Garsia–Leven–Xin and Mellit with a multiplicativity theorem for slope operators and a determinantal model for bounded cylindric partitions, linked by a common $q$-difference equation. The finite identity and the HJO conjecture have been formalized in Lean by AxiomProver, conditional on two stated literature inputs.