= Equitable regularity energy
{title2=$q(\mathcal P)$}
For a <set partition> $\mathcal P$ of the <vertices> of a <graph> of order $n$, define
$$
q(\mathcal P)=\frac1{n^2}\sum_{A,B\in\mathcal P}|A||B|d(A,B)^2,
$$
where the sum is ordered, the adjacency indicator defines the density also on diagonal pairs, and exceptional <vertices> are treated as singleton parts. Refinement cannot decrease this energy, by the <Cauchy-Schwarz inequality>. An irregular pair of equal cells of size $m$, with witnesses of relative size at least $\varepsilon$ and density discrepancy greater than $\varepsilon$, increases the energy by more than $\varepsilon^4m^2/n^2$ for each orientation. Splitting refined atoms into equal chunks and leftover singleton parts is a further refinement, so restoring equitability causes no energy loss.
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