Genuine source bridges produce synchronized scalar ledger families

Write $L_n=\log n$, $s_n=n/\log n$, $y_n=\lfloor n^{2/9}\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le y_n\}$. Fix $c>0$, $0<\delta_*<1$, a guarded tail family $F$, fixed width and multiplicity, and a source bridge/target on every tail index. Assume eventually that each source is at its stated index and cutoff, its guarded smooth mass equals the mass from $F$ and is at least one, its head patterns are separated, its active values lie in the guarded smooth row, its balanced and protected coefficients lie in $[0,1]$, and it satisfies the literal rounded-source residual inputs. Assume also its precharged target is exactly that of $F$. Then there are total scalar functions $\log Y,\Lambda_0,m_f$ such that $\log Y$ is the extended precharged-target log family and, writing $q_{\rm raw},\widetilde q$ for the raw and guarded smooth masses and $A_0$ for the smooth initial-height family,

Eventually $m_f$, $\log Y$, and $\Lambda_0$ agree exactly with the frozen mass, target logarithm, and rounded logarithmic statistic of the same literal source bridge. This supplies the Section 8 analytic ledger without assuming one in the source callback.