Copyright 2020-2021 by Louis Epstein,All Rights Reserved

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.

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
5**→**5**→**3125**→**(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 3_{s}→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":

4_{s}→' = 4_{s4}→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

4_{s}→

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

An underlined operator letter followed by a number

4_{s4}→

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

4_{ss}→

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 letter*with 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