Sparsity != symmetry.

It's simple to say "Ah well, it's sparse" that doesn't mean anything and doesn't explain anything.

Symmetries are equivalent to a conserved quantity. They exist because something else is invariant with respect to some transformation and vice versa. We didn't discover arbitrary constraints we found a conserved quantity & the implied symmetry.

"There are integers", "the numbers should be small" all of these are nothing like what works normally. They aren't symmetries. At most they're from some anthropic argument about collections of universes being more or less likely, which is its own rabbit hole that most people stay away from.