Eventual two-sided slack for the balanced nonsmooth correction

Write $L=\log n$, $N=n/L$, $Y=\lfloor n^{2/9}\rfloor$, and $h=\mathrm{upperTailLength}(c,n)$. Fix $W,K_0$, $c,\delta_*,\beta_a>0$, and reals $\beta_p,\sigma$ with $\sigma\le\beta_p$. Eventually, for every realized bank and guarded certificate satisfying the central-anchor threshold and $Y$ below its cutoff, every active nonexceptional label has balanced guarded postcharge correction density $D_\lambda$ at total coefficient $\beta_p+\beta_a$ satisfying

This provides the literal endpoint slack for Proposition 8.7 without an assumed alpha box.