Trifautx (Water's Other Number)

We will be starting this page from the end of the last number, where we defined $$\hbar(1)$$ as Water's Number. As per tradition, we will go even further and develop and even more extended version of Ackermann numbers.

Though I won't be doing a repeat of the last number but with $$\hbar(n)$$, there will be some differences. This extension will start the same way that the original did, as it doesn't change until stacks.

$$\hbar^2(n)$$ = $$\hbar(\hbar(n))$$

$$\hbar^{\hbar(x)}(y)$$

$$\hbar^{\hbar^{\hbar(x)}}(y)$$

This is where the extension changes.

$$\hbar[3]x,y$$ = $$\hbar^{\hbar^{\hbar(x)}}(y)$$

(Note: Alternatively you can just use $$n$$ instead of $$x,y$$ to define both x and y.)

Now the extension mainly takes inspiration from array notations like BEAF or BAN.

$$\hbar[\hbar[\hbar[\hbar[10^{24}]]]]10$$ can be compressed into $$\hbar4, 10^{24}10$$.

$$\hbar10^{10},[[10^{10}, 10^{10}]]10$$ can be even further compressed into $$\hbar\{10^{10},[3]\}10$$

I know I didn't develop this extension as much as I did the original extension, but I believe this is already large enough of a jump.

I will define Trifautx as $$\hbar\{\hbar10^{10^{100}},\hbar(100)10^{24},[\hbar300^{600},\hbar(100)]3 \uparrow \uparrow \uparrow 10\}10^{10^{10^{100}}}$$

The symbol is $$\mu(n)$$, and once again, n is how many times you want it to repeat. No, I'm not guaranteed making ANOTHER extension, but anyway, $$\mu(1)$$ is default. This concludes Trifautx!