MMP.12:5.2 - A circulation selected away by minimum norm

Three stores exchange material around a directed cycle. Let the nonnegative transfers be a from the first to the second, b from the second to the third, and c from the third to the first. Their stock changes are

F(a,b,c)=(c-a, a-b, b-c).

Observed zero stock changes allow every (a,b,c)=(t,t,t) with t>=0. Minimizing a^2+b^2+c^2 selects t=0. The minimizer is unique, yet a circulation of one unit gives exactly the same records. Minimum norm has supplied a nominal no-circulation choice, not evidence that no transfer occurred.

A question about net stock change is already settled. A question about gross transported amount 3*t remains unresolved unless the subject account supplies another restriction or observation. If the stores can circulate material during unchanged stocks, using the minimum-norm answer as measured throughput would erase the quantity being sought.