I see it might be helpful to use symbolic solving. Don't know too much about such things.

uniform_a + uniform_b = constant

originally we had 1 equation and 3 unknowns.

with constant, we now have 1 equation and 2 unknowns.

if we consider these uniform distributions variables instead of distributions, we could use a math package to solve for one or the other, either symbolically or numerically. that could be part of the problem.

let's make it more interesting.

uniform_a + uniform_b + uniform_c + uniform_d = dist_e

dist_e = 3

uniform_c = 2

1 equation, originally with 5 unknowns, now with 3 unknowns.

Here we go. Since an equation is solved when there is only one unknown, we need one less sampled value than the total number of distributions, to guess a state.

It would be nice to guess a distribution rather than just a sample, but that could be a challenge for later. It seems reasonable to start by describing distributions with functions that could sample them.