This only a proof that a field with more connections is possible, not what it looks like.

I’m very out of my depth, but the structure of the proof seems to follow a pattern similar to a proof by contradiction. Where you’d say for example “assume for the sake of contradiction that the previously known limit is the highest possible” then prove that if that statement is true you get some impossible result.

They only proved that one exists; computing the actual construction is non-obvious (the naive way to construct it is computationally infeasible).

They have a "before" picture but not an "after"!

Yeah, unfortunately, they just proved there existed a better solution, they didn't construct it.

(Though in some ways that's actually more impressive.)