Hmmm.. I'm very unsure about this. I think a < c is the very definition of underflow/needing to borrow from high word.

You are saying that it's possible that a<c, but not a<L. Let's look as follows:

Let's assume a < c.
Then a - c < 0.
This means that computing L with 32-bit arithmetic gives L = (a - c)mod2^32 = 0x100000000 + a - c
Therefore L - a = 0x100000000 - c
The value c on the right-hand side is a 32-bit integer and is 0xFFFFFFFF at max, therefore the right-hand side > 0. Therefore L - a > 0, so a < L.