Can you clarify "It should also be clear now that a maximal coupling is just a thing in itself, unlike the things you mentioned"?

Thank you. What if I am seeking a maximal coupling subject to some given constraint - could it be unique in that case?

Does the Hall Marriage Theorem hold when one of the sets is infinite?

@FedorPetrov Sorry, I thought $W$ had 3 zeroes. First we'd have to check the inequalities in the theorem to ensure the existence of such a matrix. Since there is no matrix in this case, the inequalities must fail for some $L, K$.

I realize that I didn't originally state that the $R, S$ vectors were positive. I've edited it.