Thank you for the reply! I certainly don't intend to put forward a claim that 3-decidability, if possible, would invalidate the 2-undecidability result; Rather,

Thank you for the reply! I certainly don't intend to put forward a claim that 3-decidability, if possible, would invalidate the 2-undecidability result; Rather, I believe it worth investigation to see if we allow ourselves a middle road of a paradox category, it's a different theorem and might be provable or at least practically useful. Would "halts" be useless if it only decided every function except the ones that invoke it and do the opposite? If not, maybe it's worth building, even if we know it's not possible to classify everything into 2 buckets, because the majority of real world functions don't have that opposite property and could be useful to decide halting for that subset. Also, I'd point out, there is a difference between the inner and outer invocations of the same turing machine. They are different executions of the same machine, on different layers. A decision about an inner machine might be truly correct, independent of whatever happens outside.