I can't quite understand - did they use brute force?
> Structure of the proof. The proof of our main result, Theorem 1.1, is given in Section 6. The structure of the proof is as follows: machines are enumerated arborescently in Tree Normal Form (TNF) [9] – which drastically reduces the search space’s size: from 16,679,880,978,201 5-state machines to “only” 181,385,789; see Section 3. Each enumerated machine is fed through a pipeline of proof techniques, mostly consisting of deciders, which are algorithms trying to decide whether the machine halts or not. Because of the uncomputability of the halting problem, there is no universal decider and all the craft resides in creating deciders able to decide large families of machines in reasonable time. Almost all of our deciders are instances of an abstract interpretation framework that we call Closed Tape Language (CTL), which consists in approximating the set of configurations visited by a Turing machine with a more convenient superset, one that contains no halting configurations and is closed under Turing machine transitions (see Section 4.2). The S(5) pipeline is given in Table 3 – see Table 4 for S(2,4). All the deciders in this work were crafted by The bbchallenge Collaboration; see Section 4. In the case of 5-state machines, 13 Sporadic Machines were not solved by deciders and required individual proofs of nonhalting, see Section 5.
So, they figured out how to massively reduce the search space, wrote some generic deciders that were able to prove whether large amounts of the remaining search spaces would halt or not, and then had to manually solve the remaining 13 machines that the generic deciders couldn't reason about.
Second, you have to consider what's feasible in finite time. You can enumerate machines and also enumerate proofs, but any concrete strategy has limits. In the case of BB(5), the authors did not use naive brute force. They exhaustively enumerated the 5-state machines (after symmetry reductions), applied a collection of certified deciders to prove halting/non-halting behavior for almost all of them, and then provided manual proofs (also formalized) for some holdout machines.
You need proofs of nontermination for machines that don't halt. This isn't possible to bruteforce.
The most naive algorithm is to use the assistant to check if each length 1 coq program can prove halting with computation limited to 1 second, then check each length 2 coq program running for 2 seconds, etc till the proofs in the arxiv paper are run for more than their runtime.
... even though your actual method of discovering the programs in question was usually not purely exhaustive search (though it may have included some significant computer search components).
More precisely, we could say that if mathematicians are working in a formal system, they can't find any results that a computer with "sufficiently large" memory and runtime couldn't also find. Yet currently, human mathematicians are often more computationally efficient in practice than computer mathematicians, and the human mathematicians often find results that bounded computer mathematicians can't. This could very well change in the future!
Like it was somewhat clear in principle that a naive tree search algorithm in chess should be able to beat any human player, given "sufficiently large" memory and runtime (e.g. to exhaustively check 30 or 40 moves ahead or something). However, real humans were at least occasionally able to beat top computer programs at chess until about 2005. (This analogy isn't perfect because proof correctness or incorrectness within a formal system is clear, while relative strength in chess is hard to be absolutely sure of.)
They solved a lot of the machines with something like that, and some with more advanced methods, and "13 Sporadic Machines" that don't halt were solved with a hand coded proof.
But there is also definitely a place where your axiom systems become self-referential in the Busy Beaver and that is a qualitative change on its own. Aaronson and some of his students have put an upper bound on it, but the only question is exactly how loose it is, rather than whether or not it is loose. The upper bound is in the hundreds, but at [1] in the 2nd-to-last paragraph Scott Aaronson expresses his opinion that the true boundary could be as low as 7, 8, or 9, rather than hundreds.
ZFC is not some God given axiomatic system, it just happens to be one that mathematicians in a very niche domain have settled on because almost all problems under investigation can be captured by it. Most working mathematicians don't really concern themselves with it one way or another, almost no mathematical proofs actually reference ZFC, and with respect to busy beavers, it's not at all uncommon to extend ZFC with even more powerful axioms such as large cardinality axioms in order to investigate them.
Anyhow, just want to dispel a common misconception that comes up that somehow there is a limit in principle to what the largest BB(n) is that can be computed. There are practical limits for sure, but there is no limit in principle.
You can't categorically declare that something is unprovable. You can simply state that within some formal theory a proposition is independent, but you can't state that a proposition is independent of all possibly formal theories.
For BB(5) the proof of its value is an indirect computation. The verification process involved both computation (running many machines) and proofs (showing others run forever or halt earlier). The exhaustiveness of crowdsourced proofs was a tour de force.