I mean, if the extension of the relationships is indeed logical, then it can be made using some algorithm. And if that is the case, then it is isomorphic to some mathematical statement.
In other words, since the Goldbach Conjecture scans in formal English, we know it is isomorphic to some mathematical statement.
The GC is already a statement. But if the GC is true, there's no smallest counterexample (call it SCGC). The question isn't whether or not GC is mathematical (GC itself falls under the umbrella of mathematical concerns)--it's whether or not there is such a "thing" as SCGC--that is, whether SCGC conveys a real relationship, or is simply a contradictory concept.
If you interpret this in the general--there's absolutely no guarantee that GC, or some GC-ish statement, can be proven to be true if it is true. As such, there's nothing to guarantee that SCGC like relationships, given that they
don't hold, can be shown not to hold with an algorithm that runs in finite time.
Now, you can describe the SCGC--in fact, merely by talking about it, we are describing it. But the
description of the number is not the interesting thing with respect to mathematics. The interesting thing is whether or not there
is such a relationship--whether there
is a smallest counterexample. SCGC conveys something "real" if and only if (and in the sense that and only in the sense that) the GC is false.
So it doesn't help us at all to be able to
describe the SCGC. What we want to know, in terms of mathematics, is whether or not the relationship holds. We want to know if there's a finite counterexample.
The problem is, there are an infinite number of candidates, and there's nothing that gives us any guarantees that if GC is true, we should be able to in principle prove it. Now if GC is false, we're lucky (at least in principle). And if it's true, we
might could prove it. But it's not guaranteed.
Even the fact that it's a logical extension of relationships isn't sufficient to give us this guarantee.