The Ramsey community number as a renormalization-group crossing

The Ramsey community number $r_k$ is the smallest size at which a network is better described by communities than by none, under a Bayesian detection rule. On the diamond hierarchical lattice we show that $r_k$ is an exact renormalization-group crossing: the block-model sufficient statistics obey a linear map with eigenvalues $\{bs,b\}$, the degree-corrected evidence density flows to $\ln K$ at a community fixed point, and $r_k$ is the generation at which the running evidence clears the detection threshold. Degree correction advances detection by two generations. We derive $r_k(b,s;q)$ in closed form for the whole family. Finally, placing on the lattice the Reichardt--Bornholdt community Hamiltonian -- whose ground state is the partition itself -- we find an exact community-ordered phase: below the ferromagnetic critical temperature the two hubs lock into opposite communities for any resolution $\gamma>0$, a staggered order that persists as $n\to\infty$. Allowing each nested sub-community its own label, the optimal partition is a hierarchy of $q_{\rm opt}\sim\sqrt{n}$ communities, so the number of Potts states that best describes the network grows with the network. This hierarchy orders thermally level by level, through a cascade of first-order transitions whose temperatures fall as $1/\ln q$, so every stable level persists as $n\to\infty$: the emergent partition is detectable, optimal, and thermodynamically ordered.