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Hi everyone, I'm back from the dead (again...), no promises about how long I'll stay active this time. I've seen some comments about FAN and stuff that people have made recently, you could well be right, I haven't really looked at any of my array stuff for quite a while.
Anyway, the reason for this post (as you can tell from the title) is to tentatively say I think I've proved all the hardly nonregular TMs run forever, so showing that . I'm still in the process of writing it up, and it's not finished yet; some of the 'proofs' are still just collections of ideas, but I'm pretty confident I can stick them all together.
The reason that I'm posting this before I've actually finished sorting all the proofs out is that hopefully, this will encourâ€¦
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By request, I am going to write a blog post about how multidimensional arrays work (and the dreaded w(k)/ operator). If there is anything you feel could need a bit (or a lot) more work, tell me in the comments and i'll work on it. To start off with, we can look at the w(k)/ operator:
The w(k)/ operator is formally(ish) defined as follows:
 â—† can be anything
 â—‡ contains only 1's and separators
 â—‹ either starts with a separator or a ']'.
 â–® represents any number of '['s
 â–² any w/ chain
 â–¼ an array with something before the first (k) divider
 â–½ an array without anything before the first (k) divider
 â–¬ a string of â–½w(x)/'s (any x) and empty arrays ([1]s).
 R1: â—†[â–¬â–½w(k)/[qâ—†]â–²]â—† = â—†[â–¬[1(k)1(k)1(k)...(k)1(k)2â–½]w(k)/[1â—†]â–²]â—†, where there are q 1's. This rule means thâ€¦
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I have decided to change the defintion of my notation for rows after the first one to make it a lot simpler, and only a little less powerful. The change is for all of the rows to now behave like the first one, meaning the limit of multidimensional arrays is now only the LVO (I shall put another post out to show this soon). This will actually change surprisingly little of the evaluation actually on the wiki, but some of my previous posts need to be redone. Also, the apocalypxul no longer exists (see current numbers here). After a suggestion by FB100Z, I have replaced the @ symbols with unicode geometric shapes. The w(k)/ operator is still around with the same definition. These are the complete definitions:
Shapes
 â—† can be anything
 â—‡ containsâ€¦

This has been corrected for new definitions. This post will look at extended hyperfactorial array notation. It is simply definied as [_{k}@_{1}[_{k+1}1]@_{2}] = [_{k}@_{1}[_{k}@_{1}[_{k}...[_{k}@_{1}[_{k}@_{1}@_{2}]...]@_{2}]@_{2}] with n nests. This is surprisingly powerful. As a small sub rule, if the typek brackets don't exist, put them in around the typek+1. This was designed to reach the TFB ordinal, and the typek brackets work in pretty much the same way to \(\Omega_k\). In the following comparisons I have put the typek brackets in the second row just to kick start it a bit. Quite a lot of the evaluation relies on the typek brackets doing to \(\Omega_k\) exactly as type1 brackets do to \(\omega\). See the type1 bracket comparisons here and here.
One thing I have noticed lateâ€¦
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This is a continuation of the work of Cloudy176 here. the w/ operator means that m...m in the array before it is evaluated as (the value to the right) repeats of m. For example, [1...1,2]w/[1] = [1...1,2]w/n will have n 1's in the first array. If they are chained together, they are solved from right to left. Correct it when needed:
Hyperfactorial array (without the n!) FGH ordinal
[1]w/[1] 1,1,1,1,2]].
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