alt Hacker NewsI'm not assuming they have to be small integers—I'm saying that if the universe is built on discrete rather than continuous foundations, then small integers and coincidences at the bottom-turtle theory-of-everything become much less surprising. You're treating the space of possible charge values as if it's the reals, or at least some enormous range, but I consider that unlikely.
Consider: in every known case where we have found a deeper layer of explanation for a "coincidence" in physics, the explanation involved some symmetry or conservation law that constrained the values to a small discrete set. The quark model took seemingly arbitrary coincidences and revealed them as consequences of a restrictive structure. auntienomen's point about anomaly cancellation is also exactly this kind of thing. The smallness of the set in question isn't forced, but it is plausible.
But I actually think we're agreeing more than you realize. You're saying "this can't be a coincidence, there must be a deeper reason." I'm saying the deeper reason might bottom out at "the consistent discrete structures are sparse and this is one of them," which is a real explanation, but it might not have the form of yet another dynamical layer underneath.
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.