For what it’s worth, at least in the case you described, it is indeed a case of signed vs unsigned representations. That is, the numbers

-8577780315821624425

and

9868963757887927191

have the exact same underlying 64-bit representation, which is all that matters for S2 (you can verify this by checking their difference is exactly 2^64). The former is just interpreting the 64-bit string as a signed int, while the latter is unsigned.

Separately, the level 1 number you mention

-8358680908399640576

is fairly simple if you look at its binary representation; almost all of its trailing digits are 0.

Basically, you really shouldn’t think of S2 values as integers, but more as their underlying length 64 bit string.