Closed form for the gate-family retention transition in general position

Derive the closed-form gate-family retention formula for general combinations of degeneracy and transfer parameters beyond the compatible effective-coordinate case treated in the paper.

Background

The gate-family threshold theorem assumes a compatible case in which the transfer split refines the degeneracy split, allowing the effective key length and redraw depth to be written explicitly.

The authors note that the corresponding closed form is unresolved in general position. Thus, a broader characterization of retention when degeneracy and transfer directions are not aligned remains open.