The rule formulation of the general structure conjecture

For partitions $\mu,\nu,\lambda$, let $\mathcal H_{\mu,\nu}^{\lambda}$ be the set of channel assignments, and let $\mathcal H_{\mu,\nu}^{\lambda}(\mathbb Z)$ be its integer span. A channel factor $\bm G_\Phi$ is the corresponding product of upper and lower hook symbols, with the factors for $\mu$ and $\nu$ in the numerator and those for $\lambda$ in the denominator. Let $[\cdot]$ denote evaluation of a rule as a rational function of $\alpha$.

Rule formulation of the general structure conjecture. There exists an integer rule $\bm g_{\mu\nu}^{\lambda}\in\mathcal H_{\mu,\nu}^{\lambda}(\mathbb Z)$ of the form

$$\bm g_{\mu\nu}^{\lambda}=\sum_{\Phi\in\mathcal H_{\mu,\nu}^{\lambda}}c_\Phi\bm G_\Phi,$$

with $c_\Phi\in\mathbb Z$, such that

$$g_{\mu\nu}^{\lambda}=[\bm g_{\mu\nu}^{\lambda}].$$

This is a reformulation of the general structure conjecture in the paper's rule and channel-factor language.