## The Ultra Conway-Guy function

#### Part of Counting Really,Really,Really High

Among the most celebrated notations for expressing enormous numbers is the Conway Chained Arrow Notation,as expressed in the original text reproduced here and explained in detail in various places. In that text one finds the Conway-Guy Function where CG(n) is n repeated n times separated by arrows as needed for n...

1
2->2
3->3->3
4->4->4->4
etc.

I effectively incorporated this as a phase of my popble function where the input n was the result of each popble cycle's Knuth phase and the resulting value the input (number,number of polygons,and number of sides in the polygons) for the Moser phase.

In 2016 I devised the Ultrex function and made it the entry phase of the popble.Its basic characteristic is building power towers where every exponent is the value of the entire tower beneath. In April 2020 I expanded greatly on the Ultrex with the Hyper-operating Ultrex, Simple Hyper-Operating Transformer, and Ladder Hyper-Operating Transformer functions that extended the principle first to hyper-operating Ultrex exponents, then to defining the exponents by hyper-operation,and then to using the special powers created beyond basic Knuth up-arrows to evaluate the exponents going up as well as down.

Conway arrows have been described as "up-arrows on steroids"...I am not sure if the diet of energy drinks,weight gain shakes,stimulants,and fertility drugs I have fed the LHOT up-arrows outstrips the effect of the steroids...but while I await a solution to that question,something became obvious.

Ultrex works by exponentiating to get what we exponentiate, and in reiteration how many times we do it.

The 'HOT functions work by "Knuthing" to get what we "Knuth", and in reiteration how many times we do it.

So it is only logical to "Conway" to get what we "Conway".

Ultrexing creates power towers that far outstrip the value of equal-term-value tetration stacks of similar height (ultrex a base to 2000 and it's worth more than tetrating it to a centilliard). The Conway-Guy function's level values are similarly crushed if we apply the basic Ultrex principle.

In Ultrex,each rising exponent is equal to the value of the entire power tower beneath.

In Ultra Conway-Guy,each succeeding term in the chain is equal to the value of the entire chain preceding the arrow to its left.

4 u 0 = 4 = 4 = 4 u-> 0 = 4

Zero exponents,zero succeeding terms,both equal to the base.

4 u 1 = 44 = 256 = 4 u-> 1 = 4 -> 4

One exponent,equal to all below/before(nothing but the base),so both equal again.

4 u 2 = 44256 < 4 u->2 = 4 -> 4 ->256

And now the divergence from ultrex (the value works out to 4^^4)begins though the UCG value is actually well below 4 l 4 or even 4 hu 2.

But with a bound of 3 the ultrex value only reaches 4^^8...

44256(44256) << 4 -> 4 -> 256 -> (4->4->256)

which clearly crushes 4->4->4->4 into powder,just as

5→5→3125→(5→5→3125)→ (5→5→3125→(5→5→3125)) obliterates 5→5→5→5→5.

Notation in the Ultrex function family follows the base,operator,bound pattern,with reiterations noted by subscript or repetition of the operator letter...but if we're going to use the arrow symbol we won't repeat it.As with the various up-arrows in the HOU,SHOT,and LHOT functions we have a designating letter:

u

The base goes before the arrow...if there is nothing after the arrow that implies the bound is equal to the base n,that is,UCG(n). If there is an apostrophe after the arrow that follows the shorthand for the other functions,meaning "itself times to itself":

4u→' = 4u4→4

(In the four iterations in this example,the first chain has four terms after the base,and each of the three subsequent chains has a number of terms equal to the value of the entire previous chain).

This is the version now added to the 2020 definition of popble. The Knuth phase number is now iterated through the UCG function a number of times equal to itself.

An underlined operator letter

4u

means the value of the first chain is the number of iterations.

An underlined operator letter followed by a number

4u4

means after a number of iterations equal to the first chain,there are additional iterations equal to the value then reached,until this has repeated that number of times. An additional underlined operator letter

4uu

means after a number of iterations equal to the first chain,there are additional iterations equal to the value then reached,until this has repeated a number of times equal to the first chain.Any further additional underlined operators indicate a number of further repetitions equal to the chain reached after the number implied by the previous operator. A parenthesized operator letter

4(u)

means a number of underlined operators equal to the value of the first chain.Any additional parentheses indicate a number of further underlined operators equal to the chain reached after the number implied by the previous parentheses.

A bracketed operator letter

4[u]

means a number of parentheses of the operator equal to the value of the first chain.Any additional brackets indicate a number of further parentheses equal to the chain reached after the number implied by the previous brackets.

A braced operator letter

4{u}

means the iteration of the previous generation of operator a number of times equal to the value of the first chain is itself repeated a number of times equal to the value of the first chain.Any additional braces indicate a number of further generations of repeated iterations equal to the chain reached after the number implied by the previous braces.

A parenthesized operator letterwith the right parenthesis underlined

4(u)

means a number of braces equal to the first chain.

MORE MAY FOLLOW...

I had nibbled at giant rising Conway chains for years from the later Titled Number definitions...but it's high time they went live.

First published June 6,2020;extended from the parenthesized operator to the right-underlined-parenthesis operator June 13
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