Refinement variance identity for regularity energy

If a set partition refines a rectangle $U\times V$ of a graph into rectangles $U_a\times V_b$, write $d=d(U,V)$ for its edge density of a bipartite graph. The weighted variance identity isIt follows by expanding the square and using ${\sum }_{a,b}|U_a||V_b|d(U_a,V_b)=|U||V|d$. In particular, equitable regularity energy cannot decrease under refinement. If a witness rectangle has relative sizes at least $\epsilon$ and edge density of a bipartite graph discrepancy greater than $\epsilon$, its contribution, after division by the squared total vertex count, exceeds ${\epsilon }^4|U||V|/N^2$.