Here is the first “equation” above
t+1: X^2 = X x X = X + DX = X + qXX = X + Y
t+1 is obvious. Can you rigorously define “x”? Equating it to X^2 makes it look like standard multiplication, though I’d think matrix multiplication would work better, but it is far from clear how the multiplicative factor of X is converted to the various additive groups. Thus my request for a rigorous definition of this operation. Your description was far from rigorous.