Balanced Saias transition budget at the canonical row scale

Write $L=\log n$, $Y_n=\lfloor n^{2/9}\rfloor$, $h=\lceil cn/\log n\rceil$, and $K=K_0+1$. Fix natural W and $K_0$, real $\beta$, and $c>0$. Let $n\ge2$, $0<r\le n$, $y\ge2$, $L\ge1$, $\log y\ge L/5$, $h\le2cn/L$, and $Kh\le n$. Put $x=\lfloor n/r\rfloor$, use the source head density and balanced alpha, and take the four physical endpoints $\lfloor(2n+h)/r\rfloor$, $\lfloor2n/r\rfloor$, $\lfloor(2n-Kh)/r\rfloor$, and x. The source physical Saias transition budget $\mathcal S$, using the inverse-log-square endpoint rate at the sharp Saias defect constant, satisfies

Here $C_{\rm trans}$ is the sharp transition-row scale constant.

The three weighted sharp Saias pair budgets have the same uniform scale as the Dickman contribution.