Rabbit Hole: Minimum L-seams

The premise is interesting but a little unintuitive. As I understand it, an L-seam is defined by having to attach a square by two edges at a time (i.e. two of its sides already have adjacent pieces in the finished part). Consequently, it's order dependent: any sized quilt can have L-seams if you construct a L-shaped piece that needs a corner filled in, even if there is an optimal way. It follows that not every "seam that ends in the middle of the piece" is an L-seam: both ends could have been edges at the time of construction.

As an engineer with no sewing experience, I don't understand why you wouldn't sew one straight edge from the outside in, then sew the other and meet at the corner, avoiding L-turning issues. Please enlighten me.

I think you buried the lede a bit -- the "L-seams" in your diagram look like ordinary T-junctions, and I didn't figure out what you meant by the term until your later mention of maximising guillotine cuts (a term you don't explain but which I'm familiar with).

At the linked squaring.net site, the definition of a "Mrs. Perkins quilt" is a bit unclear. It says (a little offhandedly) that "An additional constraint is that the side lengths cannot have a common factor", however the example solution they provide violates this. Even if we interpret this constraint as narrowly as possible by pretending that 1 and the full side length of a square are not "factors" of its side length, there is still in their solution a 6x6 square and a 4x4 square, which share common factor 2. I guess they are looking for a way to prevent trivial solutions (e.g., any square with even side length can be partitioned into 4 equal-size squares), but either I'm misunderstanding something about their current definition or they are.