Least Squares Adjustment for Level Runs
Can a least squares adjustment be used on a level run starting on old BM 'A' and closing on old BM 'B?' In between BM 'A' and BM 'B' would be five traverse points to be used for a topo. I'm guessing an adjustment can be done only if loop 'A' and loop 'B' are run through a new BM. The new BM 'C' is then adjusted.
You can absolutely do that, mathematically speaking. Whether you want to is another story. Hold BM "A" and BM "B" and naturally any error in the relative record elevations of the 2 BM will warp the adjustment. So running a full loop is preferred.
As a trial, hold one BM and free the other in the adjustment. That will create what is called a "minimally constrained adjustment" It will tell you how for off you are on the record elevation of the freed BM. If you are satisfied with the results you can then hold the 2nd BM elevation and warp away.
Another option is to hold neither BM fixed, but to give both a small standard error value. Then your measurements will be fixed and best fit to the BM elevations.
Without redundancy, least squares is a waste of time. You have to have observations to compare. That requires a loop, ties to other control or cutoff loops to analyze and adjust.
With digital levels there is virtually no adjustment needed, and standard balancing is fine. If have a large adjustment, then something went wrong.
A 1-way run between two good benchmarks is a better check than a loop on one, because, as well as checking your work, it can indicate whether one of them has moved.
Without the redundancy of either a loop or two control points, you have no check on your work.