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
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.