Universal two-parameter $\mathcal{W}_{\infty}$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots, N)$

We prove the longstanding physics conjecture that there exists a unique two-parameter $\mathcal{W}_{\infty}$-algebra which is freely generated of type $\mathcal{W}(2,3,\dots)$, and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type $\mathcal{W}(2,3,\dots, N)$ for some $N$ can be obtained as quotients of this universal algebra. As an application, we show that for $n\geq 3$, the structure constants for the principal $\mathcal{W}$-algebras $\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ are rational functions of $k$ and $n$, and we classify all coincidences among the simple quotients $\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ for $n\geq 2$. We also obtain many new coincidences between $\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ and other vertex algebras of type $\mathcal{W}(2,3,\dots, N)$ which arise as cosets of affine vertex algebras or nonprincipal $\mathcal{W}$-algebras