BusyBeaver(6) Is Quite Large

I wonder if the visible universe is large enough to write down the exact value of BB(6).

It's not.

It definitely isn't. The amount of information you can store in the universe is something like 10^120 bits. Even if I'm off by a trillion orders of magnitude it doesn't matter.

If you treat the observable universe as a closed system, you could try to apply the Bekenstein bound using - R ≈ 46.5 billion light-years (radius of the observable universe) - E ≈ total mass-energy content of the observable universe

The mass-energy includes ordinary matter, dark matter, and dark energy. Current estimates suggest the observable universe contains roughly 10^53 kg of mass-energy equivalent.

Plugging these into S ≤ 2πER/ℏc gives someting on the order of 10^120 bits of maximum information content.

S ≤ 2πER/ℏc

S ≤ (2 × 3.141593 × 3.036e+71 × 4.399e+26)/(1.055e-34 × 299792458)

S ≤ 2.654135e+124

S ≤ 10^120

So, no.

I want some easier to comprehend number for BB(6), in decimal notation. But it's such a massive number I would need to invent a new notation to express that. I love this new (to me) concept of tetration number representation. 10-million sub 10, what is the number?

Look at 3 sub 10 = which is (10^(10^10)). So that is 10 to the power of 10 billion. In regular decimal notation, that is a "1" with 10 billion "0"s following it. It takes 10 gigabytes of ram to represent the number in decimal notation, naively.

The number of atoms in the universe is only 10^80, or 1,000...000 (80 zeroes). 10-million sub 10 is so huge, how much ram to represent it.

This example is from https://www.statisticshowto.com/tetration-function-simple-de...

Just the starting number in the article is ¹⁵10. That means it's 10^(¹⁴10). That means it has ¹⁴10 digits. So no, you can't.

You’re probably referring to a state where all parts of the complete representation exist at the same time. Because if they don’t have to exist at the same time, then it might be possible to “write it down” if the universe has unbounded duration (“might” because I don’t know how the heat death plays into that). However, “at the same” time isn’t well-defined in relativistic spacetime. The sibling comments are definitely right with respect to the reference frame implied by the CMB. But I’m wondering if it wouldn’t be possible to slice spacetime in a way that actually makes a representation possible “at the same time” in some reference frame?