UPDATE March 6th, 2026: I diverged the systems of the x&E into two systems: one that maps to the limit of extended Buchholz's function, and this one that take account more effectively to the Bashicu matrix system (BMS) and Y sequence. Instead of going straight to &_2, we have to define &^&^&^...^&^&^& = &^^# = &(1), &(1)^^# = &(2), &(2)^^# = &(3), etc. based off the unofficial extension of Madore's OCF that the limit of &(&(&(...&(&(&(1)))...))) is &(&_2). Similarly, after &(&_2^&_2^&_2^...), we have &(&_2(1)), &(&_2(&_3)), &(&_2(&_3(1))), &(&_2(&_3(&_4(1)))), and so on, until we reach the limit at #{/}# = #{/(1)}# = #{&(&_2(&_3(&_4(...&_{n-2}(&_{n-1}(&_n(1)))...))))}#, which has the Buchholz's ordinal level (ψ_0(Ω_ω) using Buchholz's OCF), and is also equal to (0,0,0)(1,1,1) in Bashicu matrix system (BMS) and (1,2,4,8) in Y sequence.

The Sneak Peek

First of all, we observe the pattern:

#{&^^#}# = #{&^&^&^&^&^&^&^&^...}#

#{&^^#}#[n] = #{&^&^&^&^...^&^&^&^&}# with n &'s

So, #{&^^#}# = #{&_2}# in x&E1

But, in the second system of extended Collapsing-E notation (x&E2), we have:

#{&^^#}# = #{&(1)}#, where &(0) = &

So:

#{&_2}# in x&E1 = #{&(1)}# in x&E2