I'm not sure everything you're talking about, but it's relatively easy to reason about constructive logic by abandoning the idea that your logic is one of "truth and falsehood"—which are, by modern, western intuition mutually exclusive options—but instead one concerning evidence.
One might believe that the Collatz conjecture is true or false, but we certainly lack evidence for either case today. This isn't exactly three value logic, but instead something else. A logic which incorporates the notion of time and communication of proof.
> which incorporates the notion of time and communication of proof.
Certainly constructive logics (typically) incorporate a notion of evidence/proof. But constructive logics don't incorporate notions of time or communication (of proof, or in general).
One might believe that the Collatz conjecture is true or false, but we certainly lack evidence for either case today. This isn't exactly three value logic, but instead something else. A logic which incorporates the notion of time and communication of proof.