## 4. Original-span register theorem

An exact register argument explains part of that agreement. Consider two original spans, each equal to ten times a three-digit core ending in a nonzero digit. Each span therefore has exactly one trailing zero. Let `H,T,U` be the sums of the cores’ hundreds, tens and units columns. The original and transformed totals are

\[
10(100H+10T+U),\qquad 10(H+10T+100U),
\]

with `2≤H,U≤18` and `0≤T≤18`. A total 2700 forces `U=10` and `10H+T=26`, hence uniquely `(H,T,U)=(2,6,10)` and transformed total 10620. A total 12600 instead permits `(12,5,10)` or `(11,15,10)`, giving 10620 or 11610. The literal cumulative cores 917 and 343 select the first: their units carry into the tens, but their tens produce no further carry. Regular convergence follows from total and register; cumulative convergence additionally uses the source-selected carry branch. The theorem recovers aggregate measurements, while the source supplies the actual split. Its register restriction applies to these two-component calculations. [C938–C947]


The operation is applied only to original source durations in this account. A digit reversal is distinct from a coordinate reflection or formal Mirror. The named partition is retained; no second reversal is introduced.