When does Busy Beaver surpass TREE(3)?

Question

I saw a question which asked, "When does Busy Beaver surpass TREE(n)?" I am asking a somewhat different question, about a specific value of TREE. I know that TREE(3) is an unimaginably vast number, far more than even Graham's number. I would be satisfied with even a good upper bound when Busy Beaver surpasses TREE(3).

Answer 1

As far as I know, there is no known upper bound for $TREE(3)$. To have at least a slight idea how huge this gap is , you should study the "fast growing hierarchy".

Graham's number has level $\omega+1$.

Conway chains far surpass Graham's number, already $4\to4\to4\to4$ is vastly larger. The level of Conway chains of length $n$ is about $\omega^2$

Bowers arrays far surpass Conway chains. $[3,3,3,3,3]$ is already inexpressible with Conway chains.

Even Bowers planes only reach $\omega^{\omega^2}$-level and with $3$ dimensions $\omega^{\omega^3}$.

This is still far away from the $\epsilon_0$ level and this is FAR surpassed by the $\Gamma_0$ level.

And even this level is FAR too low for $TREE(3)$.

I think you begin to understand that $TREE(3)$ is not just another league , the difference in magnitude is barely comprehensible even in googology-standards.

The busy beaver function however should surpass the TREE - function relatively early, but I have no idea where. I think, it does it below $n=100$.

Answer 2

With numbers this big, it's actually surprisingly hard to get a rigorous lower and upper bounds on the value.

It's usually significantly easier to compare growth rates of functions, instead of comparing specific numbers obtained by the functions. This is kind of like big O notation in computer science, it's easy to show that $f_1$ grows less quickly than $f_2$ because $f_1 \in \Theta(n)$ and $f_2 \in \Theta(2^n)$, but not as easy to show that $f_1(7)$ uses fewer computer instructions than $f_2(7)$.

Though for really fast growing functions, it usually is the case that if $f_1$ grows less quickly than $f_2$, then for $k$ reasonably big, like $k \geq 3$, we have $f_1(k) < f_2(k)$. Let's define the term "almost definitely bigger", where if the above conditions are true then we say $f_2(k)$ is almost definitely bigger than $f_1(k)$. We can now discuss numbers that are almost definitely bigger than TREE(3).

There are many claims about TREE(3) of varying reputability, but if we use the results from StackExchange users, https://www.googology.fandom.com, and similar, we can piece together an argument about a busy beaver number almost definitely bigger than TREE(3).

Firstly, there is the claim that TREE(n) has growth rate similar to $\theta(\Omega^\omega \omega, 0)$, in other words, the growth rate of the function is "slightly" above the Small Veblen Ordinal on the fast-growing hierarchy. However, according to this page we should be careful about such comparisons, since the fundamental sequences of ordinals about $\Gamma_0$ don't have a standard definition, and that the proofs connecting the infinite ordinals to the growth rate of the functions aren't rigorous. Nevertheless, let's accept the claim.

According to this page, a turing machine can be made to play Buchholz's Hydra game, with 2 labels, using 96 states and 7 colors (symbol alphabet size). The number of steps in Buchholz's Hydra game with 2 labels is believed to grow faster than the Bachmann-Howard ordinal on the fast growing hierarchy, which is above $\theta(\Omega^\omega \omega, 0)$. The turing machine can be modified to play the game on a very large hydra, starting from an empty tape, by adding 19 states. This would mean $\Sigma(115, 7)$ is almost definitely bigger than TREE(3).

With the result in this page, a 7 color turing machine can be converted into a 2 color turing machine. Using this routine, we can show BB(2645) = $\Sigma(2645, 2)$ is almost definitely bigger than TREE(3).

Of course, the busy beaver numbers probably beat TREE(3) much sooner than that. According to this page, BB(16) is greater than Graham's number. TREE(3) is significantly harder to reach than Graham's number, as seen by how hard it is to define its fast growing hierarchy ordinal, but still, I suspect a busy beaver of only a few hundred states is enough exceed it.