The cumulative path selects one carry branch
The cumulative primary path, 9170+3430, shares the regular path’s reversed total of 10620. Its original total of 12600, however, allows two aggregate possibilities within the same two-component decimal register. This identifies a specific source condition behind the convergence.
Let H, T, and U again be the sums of the two core digit columns. The total 12600 forces U=10 and 10H+T=125. The permitted digit bounds leave exactly two solutions: (12,5,10), giving reversed total 10620, and (11,15,10), giving 11610. The literal cores 917 and 343 select the first. Their units generate one carry into the tens column, but the resulting tens sum 6 generates no further carry. The other aggregate branch has a tens-to-hundreds carry and changes the reversed measurement by 990.
Thus total and register determine the regular result uniquely, while cumulative convergence additionally requires the source-selected carry branch. Neither branch is a probability estimate or an invitation to replace the source breakpoint.
Appending the already declared 1400 tail, whose one-pass value is 4100, gives completed durations 14720 or 15710. Only the first has an integral 25/23 completion. Within this precise two-branch domain, integral Priestly completion and agreement with the regular primary path select the same condition. Their agreement supplies a useful connection between modules, without becoming two independent observations.
Sources: File52c §§3.3–3.4; C940–C942; C947; Interface packet IQ04 and calculations.json:branch_equivalence.