## The Super Conway function

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

Readers of these pages are likely familiar with the Ultrex function, the entry phase of the popble.Its basic characteristic is building power towers where every exponent is the value of the entire power tower beneath. Ultrex works by exponentiating to get what we exponentiate, and in reiteration how many times we do it.

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.

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

Subsequently I created the Ultra Conway-Guy function, since it is only logical to "Conway" to get what we "Conway".

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.

And still I am accused of not being recursive enough,told that the rapidly growing terms (UCG(5) is 553125(5→5→3125) (5→5→3125→(5→5→3125)) (5→5→3125→(5→5→3125)→ (5→5→3125→(5→5→3125))) ) mean nothing compared to just a slight extension in length... when in reiteration UCG uses itself to create chains of massive length.

So,I am game for creating another function...the Super-Conway function or SC function for short (not to devalue the contribution of Richard Guy but to avoid confusion with the SubCubic Graph functions I did not invent).

As ever,notation in the Ultrex function family follows the base,operator,bound pattern,with reiterations noted by subscript, punctuation,or repetition of the operator letter. As with the various up-arrows in the HOU,SHOT,and LHOT functions and the right-arrows in the UCG function we have a designating letter:

s

Unlike the u-right-arrow,the s-right-arrow after the base n designates a chain of n phases,the first of which is CG(n) and the remainder each defined by the previous phase or phases.

The first phase of 3s→3 is therefore
3→3→3 or 3^^7625597484987
In the second of the 3 phases,this is the initial term and number of terms in the first subphase,the number of initial-layer subphases,and the number of layers of subphasing. Each subphase of the initial layer produces the initial term and number of terms in the next subphase.
The initial term and number of terms in the first subphase of the second layer is UCG(value produced by chain consisting of the entire initial layer of subphases)....this is also the number of subphases in that layer.
The initial term and number of terms in the first subphase of the third layer is UCG(UCG(value produced by chain consisting of the first two layers of subphases))....this is also the number of subphases in that layer.
The initial term and number of terms in the first subphase of the fourth layer is UCG(UCG(UCG(value produced by chain consisting of the first three layers of subphases))).... and so on to the final subphase.

The chain consisting of the entire first two phases yields the initial term and number of terms in the first subphase,the number of initial-layer subphases,and the number of layers of subphasing,in the third phase.

The final value of SC(3) is of course the chain consisting of all 3 phases.
SC(4) would have four phases with the fourth defined by the first 3 and so on.

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

4s→' = 4s4→4

(In the four iterations in this example,the first chain has four phases starting with CG(4),and each of the three subsequent chains has a number of phases equal to the value of the entire previous chain).

An underlined operator letter

4s

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

An underlined operator letter followed by a number

4s4

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

4ss

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.

As with the LHOT function calling on the SHOT function, a comparable generation rule applies for augmented arrows in the SC and UCG functions...Where the s→ calls up the u→, the s→ calls up the u→ with the result that where (chain)u→(chain) appears in the first instance of defining a layer of subphases, in the second instance this takes (chain)u→(chain) to define (with however many layers of nesting).

This continues with the letter-arrows hereafter defined.

A parenthesized operator letter

4(s)

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[s]

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{s}

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(s)

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 January 5,2021,a correction on January 6
Comparable generation rule added January 19