Trying to follow this, but I’m lost at a few points.

If you say you have a file where +L-R = 0, then L = R? That’s just a mono file?

L/R stereo and M/S stereo conversion works like this. M = L+R, S = L-R (or S = R-L, either one is fine).

If you expand out your expression M=Si+L+R, where Si=-(L-R)=-L+R, you have M=-L+R+L+R, which means you are saying M=+R+R. This doesn’t seem right.

If you just want an actual M, that’s just a mono conversion of L+R. For sure, if the recording is M/S, the two sides cancel out and you will extract the mid channel, which would often contain song - but there’s no reason to do all the repeated inversion and mixing to do it!