TITLE: Have we found a Turing Machine for which halting/non-halting is unprovable?
QUESTION [5 upvotes]: The undecidability of the Halting Problem implies that there exist Turing Machines such that you can't prove whether they halt or not in whatever logical system you're using (let's say ZFC)$^1$.
Have we found any such machines, and proven their halting's independence of ZFC?
It's easy to come up with problem specifications for which no proof could be found either way, but I'm looking for concrete programs you could write in any programming language. Put another way, I'm looking for a constructive proof of the Halting Problem.
$^1$ Proof: Consider a program H which takes a number N, enumerates all statements in ZFC searching for a proof that the Nth Turing Machine halts or does not halt. If every TM has  such a proof, then H will eventually find an answer, and thus will solve the halting problem (unless there's a contradiction in ZFC, in which case the proof of halting for any machine can be found trivially).

REPLY [6 votes]: As announced in postings on Scott Aaronson's blog (The 8000th Busy Beaver number eludes ZF set theory and 
Three announcements), here are two explicit Turing machines whose halting/nonhalting behavior is claimed to be independent of ZFC: 
7918 states (by Adam Yedidia and Scott Aaronson)
1919 states (by Stefan O’Rear)