If I understand what you are asking then you need to run a slope stability analysis of the cut.  This can be done by hand, however, one of several computer programs would be much quicker.  There are no "standard" equations for this type of analysis.

You use the term width in your post, what dimension of the cut does this apply to?

The initial cut will be a 45 degrees (1:1), descending inward from the property line from a height of 5' to 0,  5'  inside the property line. This cut is considered temporarily stable and OSHA-compliant; no SS analysis is required

Once this initial cut is achieved, it is necessary to remove and recompact this "wedge" of material along the property line in small increments, so that the adjoining parallel sidewalk at P/L remains stable. These small increments of removal and recompaction are called "slots"

Now here's the tricky part: I need to calculate the maximum width of these slot cuts. That is, as you are facing the "wedge" along the property line, how wide a swath of material can be cut, parallel to P/L, left to right, in order to maintain a 1.5 safety factor? It may be neccesary to assume a Q surcharge from across the P/L in order to make the equations "work".

Thank you for you timely response to my query.

Now I understand.  The bad news is that there is no way, that I believe, to calculate the safe length of these slots.  However, for a 5 foot cut, a length of 5 feet is generally safe by observation, if any length is going to be safe.  Beyond that, trial and error.

Since the other side has light loadings like a sidewalk, you may only use sequence A, B, A, B and so on.  You would cut all the A's 8 to 12 foot wide, depending on phi, C and gamma, then come back later and do all the B's.  The following link might come in handy.

http://www.dot.ca.gov/hq/esc/construction/Manuals/TrenchingandShoring/H-1.pdf#xml=http://dap1.dot.ca.gov/cgi-bin/texis/webinator/search/pdfhi.txt?query=shoring&db=db&pr=www&prox=page&rorder=500&rprox=500&rdfreq=500&rwfreq=500&rlead=500&sufs=0&order=r&cq=&id=43751fa017