 Equalization with a controlled exceptional set

In a proof of the Szemerédi regularity lemma, suppose $k$ equal classes have size $L$ and witness splitting creates at most $2^k$ atoms per class. Divide every atom into blocks of size $\ell =\lfloor L/4^k\rfloor$, putting leftovers into the exceptional set. At most $k{2}^k\ell \le kL{2}^{-k}$ vertices are lost. When $L\ge 2\cdot 4^k$, at most $2k{4}^k$ blocks remain. For equitable regularity energy defined by omitting exceptional vertices, the loss from discarding a fraction $r$ is at most $2r$: only ordered pairs incident with discarded vertices are removed, and every squared edge density of a bipartite graph is at most one. This provides a quantitative way to restore equal class sizes after each energy increment.