It is definitely impossible to (in general) determine whether a given program is equivalent to a specific Weird Program. This is a consequence of the halting problem itself!
I think the question about “statements I care about” is, at its core, a question about aesthetics and going to be kind of subjective. For example, does the above statement about not being able to prove the equivalence of programs qualify? (Or would it be non-interesting if one of the programs being compared were sufficiently weird?)
Another statement that might or might not qualify is of the form “the 8,000th Busy Beaver number is less than N”—see The 8000th Busy Beaver number eludes ZF set theory. Though, admittedly, Yedidia and Aaronson did that one by asking whether a particular conjecture-counterexample-finding program halted, so maybe that’s also too contrived for your aesthetics?
It is definitely impossible to (in general) determine whether a given program is equivalent to a specific Weird Program. This is a consequence of the halting problem itself!
I think the question about “statements I care about” is, at its core, a question about aesthetics and going to be kind of subjective. For example, does the above statement about not being able to prove the equivalence of programs qualify? (Or would it be non-interesting if one of the programs being compared were sufficiently weird?)
Another statement that might or might not qualify is of the form “the 8,000th Busy Beaver number is less than N”—see The 8000th Busy Beaver number eludes ZF set theory. Though, admittedly, Yedidia and Aaronson did that one by asking whether a particular conjecture-counterexample-finding program halted, so maybe that’s also too contrived for your aesthetics?