**Part ****III**.** ****C**.: ** Interlude**.** The F**.__E__.__D__.** ***Psychohistorical**-*__Dialectical__ *'**Meta**-**Equation**'** of Human**-**Social Formation*(*s*) *'***Meta**-**Evolution**'* *Series.

*Modeling the** Systematic *__Dialectic__* ***of**
‘‘‘**The Basic Operations
of Arithmetic**’’’ *using* _{C}__Q___ --

*Systematically* *Presented*** via a** **5**-**Symbol
Expression**.

Dear Readers,

Below is the third and final of the three simpler examples of dialectical models, presented as an interlude, before concluding the series on the

**F**.**E**.**D**. Psychohistorical-Dialectical 'Meta-Equation' of Human-Social Formation(s) 'Meta-Evolution', with the problematics of the

**n**ation-state social formation, and with its meta-model-predicted successor-formations.

Enjoy!

Regards,

Miguel

__Introduction__. This
model is more “__C__omplex” [pun intended] than the models of “TV-Series”,
and of ‘Modern Computerware’, presented earlier in this sub-series, because it
requires some “domain-expertise” -- or, at least, some “domain familiarity” --
with respect to the domain of the so-called “__C__omplex Numbers”, the set standardly denoted by the symbol **C**.

The ‘axioms-system’ of the arithmetic
of the **C** numbers, which we denote by __C__, is the **6**th system of arithmetic in the following standard
order of standard arithmetics, with our light-spectrum **o****r****d****i****n****a****l** color-coding added --

__N__, __W__, __Z__, __Q__, __R__,
__C__,

-- for the “**N**atural”,
“**W**hole”, “Integer”, “Rational”, “**R**eal”, and “**C**omplex” arithmetics, respectively. About the ‘Goedelian Dialectic’ of
these systems, see: http://www.dialectics.org/dialectics/Vignettes.html, Vignette **#****4**.

We will, in this blog-entry, use
the __C__omplex-Numbers-subsuming
version of the **F**.__E__.__D__. ‘first dialectical algebra’ to construct, and to
“solve”, a “heuristic”, ‘intuitional’ model of a *systematic **presentation* of the domain of “the basic
operations of arithmetic” -- encompassing both its *‘‘‘*__verse__’’’ [e.g., addition, multiplication, exponentiation] and
its *“*__in__verse” [e.g.,
subtraction, division, root-extraction]* *operations, jointly, via __q__s with **C** subscripts, which we also reference as _{C}__q__s.

The models that we usually
narrate here are constructed by interpreting the generic _{N}__Q___
version of the **F**.__E__.__D__. ‘first dialectical algebra’ [ see E.D.
Brief # 5 and its Preface
], or, at most-advanced, by interpreting
the generic _{W}__Q___ version of that algebra [ see E.D.
Brief #6 and its Preface
], with the subscripts of the _{N}__q__ or _{W}__q__
‘meta-numerals’ drawn from the number-space **N**** **__=__** {** **1**,** ****2**,** ****3**,**
****...** **}**, or from the number-space
**W**** **__=__** {** **0**,** ****1**,** ****2**,**
****3**,** ****...** **}**, respectively.

This time, the subscripts of
the _{C}__q__s will be drawn from the standard number-space

**C**** **__=__** {** **R**** ****+**** ****R****i** **}**,

wherein **R**
denotes the space of the standard so-called “**R**eal” numbers, and where the **i** unit stands for so-called "__i__maginary" unity, the
positive square root of **-****1**.

FYI: The __gene__ric
__C__omplex
number is often expressed as **a**** + ****b****i**, with **a** an element of **R**,
and also with **b** an
element of **R**, or as **z**** = ****x**** + ****y****i**, with **x** an element of **R**,
and also with **y** an element of **R**.

That is, we will be
constructing our example model using the generic _{C}__Q___
version of the **F**.__E__.__D__. ‘first dialectical algebra’.

We use the _{C}__Q___ language
this time, as it allows us to present both *‘‘‘*__verse__’’’ & *‘‘‘*__re__verse’’’ operations in a single model.

Herein we mean, by the word, *‘‘‘***systematic**’’’ in the phrase *‘‘‘***systematic** presentation’’’, a presentation of the major kinds of “entities” that
exist in this 'Human-Phenomic', 'Meme-etic' domain, the domain of the basic operations of arithmetic -- by
means of categories that classify those entities by their *“kinds”*, i.e., as *‘‘‘*[ideo-]*ontology’’’*, or as *“kinds* of [idea-]things, or "meme-things"” -- and in strict order of rising complexity, starting from
the simplest category, and moving, step-by-step, from lesser to greater, i.e.,
more inclusive, complexity, until we reach the most complex/inclusive extant
category of this domain, or for the purposes of this example.

The model that we build will
describe these categories in that strict, systematic order of rising
operational complexity / inclusivity.

This will be, once again,
like the previous two “interlude” models, a “snapshot” model, a “synchronic”
model that takes the contemporary slice of time -- or at any rate, a
recent-past slice of time -- and algorithmically generates descriptions of
categories for entities that presently exist, or that might possibly presently
exist, for the model’s domain, in their systematic order of inclusivity, as
described above.

Our model here will __not__ be a “chronology”
model, or “diachronic”
model, like the previous, major model, narrated in this series, in which the
units of *earlier* categories are
described as actually, e.g., physically, __constructing__,
through their __activity__
as *“***causal agents**”, i.e., as *“***subjects**”, the units of *later* categories, categories whose units *did not exist* until that
construction took place.

That is,
it will __not__ be a
model of a ‘self-advancing’ __historical__* *__progression__
of ontology, with each historical epoch containing both old ontology, inherited
from past historical epochs, plus new ontology, ontology that had *never appeared* *before* -- in past historical
epochs -- __until__ the *later* epoch in question, plus ‘hybrid categories’,
combining / synthesizing the old with the new.

We will apply a documented, standard
procedure
to “solve” this ‘‘‘algebratric’’’ model -- to determine what actual category
each of these generated category-descriptions refers to, and to determine
which, if any, of these category-descriptions describe “empty categories”,
i.e., represent ‘combinatorially’ *possible* entities that *actually* do
not exist "within" this domain -- at least not presently.

To get started, we must
determine the starting-point -- the point-of-departure -- for our *systematic** model*.

This starting category will
be the seed of our whole progression of generated category-descriptions,
influencing every category that follows, as the “controlling source”, and as
the “ever-present origin”, of all that follows from it.

The rule for getting started
is to ask oneself “*¿*What is
the __least__ complex kind
of thing, the __simplest__
kind of thing, the least inclusive kind of thing, which inheres in this domain*?**”* -- in our case, in the domain of ‘basic arithmetic
operations’ -- and to then find the answer to that question, based upon one’s
prior knowledge of, or familiarity with, this domain.

The answer to this starting
question that we will pursue in this example is the following: The ‘‘‘**verse**’’’
operations of “__A__ddi__t__ions”,
and its *“*__in__verse operations”, or *‘‘‘*__re__verse
operations’’’, or “__S__ubtrac__t__ions”,
are the simplest ancestors, the ultimate units, of basic arithmetical
operations, ingredient in every one of the more complex operations of that
domain.

A letter that the spelled
names of these two kinds of operations have in common is “**t**”.

Therefore, we shall
name/symbolize our starter category as _{C}__t__, or as_{ C}__q___{t}, denoting the “__C__omplex” combination of the “__A__dditions” sub-category of elementary __R__eal arithmetic basic operations, with the sub-category
of “__S__ubtractions”,
and identifying that combination of __speci__fic
sub-categories with the __gene__ric
*first* category
symbol of our __gene__ric
category-arithmetic model, namely, with the symbol --

_{}
_{C}__q___{[}_{1}_{ + }_{1}_{i}_{]},

in an “identification”, an “interpretation”, or an
“assignment” [ ‘**[---)**’ ] that we indicate by
writing:

_{}
_{C}__t__ __=__ _{C}__q___{t}** ****=**** **_{C}__q___{[}_{A}_{ }_{+}_{ }_{S}_{i}_{]}
**[---)**** ** _{C}__q___{[}_{1}_{ + }_{i}_{]}.

Our model then, will take the
form of an “interpeted”, __speci__fic
equation, assigned to the __gene__ric
equation, like this --

_{C}__)-|-(___{s}** ****=**** **_{C}__t__^{2^}^{s}** **__=__** ****(** _{C}__q___{[}_{A}_{ }_{+}_{ }_{S}_{i}_{]}_{ }**)**^{2^}^{s} **[---)**** **

_{}
_{C}__|-|-|___{h}** ****=**** **_{C}__h__^{2^}^{h}** **__=__** ****[** _{C}__q___{[}_{1}_{ }_{+}_{ 1i}_{]}_{ }**]**^{2^h}** **

-- with the variable **s** indicating the __s__tep in our *systematic
**method of** **presentation* that
the ‘accumulation of categories’, denoted by _{C}__)-|-(___{s}, represents.

We will not, here, further recount the [Marxian] method of *systematic discovery*
that was used to arrive at the starting category of this *systematic **presentation*.

For more regarding that *method of **discovery*, see Marx,
__Grundrisse__, Penguin
Books [London: **1972**], pp. **100**-**101**.

__Stage ____0__. Our
initial __s__tep -- __s__tep **s ****=**** ****0** -- contains only our starting category,

_{}

_{C}__t__ __=__ _{C}__q___{t}** ****=**** ****q**_{[}_{A}_{ + Si}_{]} --

_{C}__)-|-(___{0}** ****=**** **_{C}__t__^{2^}^{0}**
****=**** **_{C}__t__^{1}** ****=**** **_{C}__t__** ****=**** ****q**_{[}_{A}_{ }_{+}_{ Si}_{]}
**[---)**** **_{C}__q___{[}_{1}_{ }_{+}_{ }_{1}_{i}_{]}_{ }

-- because **2** “raised” to the power **0**
-- **2**^{0} -- is just **1**,

and because _{C}__t__ “raised” to the power **1** is just _{C}__t__.

__Stage ____1__. It is
when we get to the next __s__tep
after __s__tep **s ****=**** ****0**, namely, to __s__tep **s ****=**** ****1**, that our equation-model gives us back something
initially “unknown” -- and, therefore, something ‘‘‘algebraical’’’, not merely
something ‘‘‘arithmetical’’’:
something to “solve-for” --

__)-|-(___{1}** ****=**** **_{C}__t__^{2^}^{1}**
****=**** **_{C}__t__^{2}** ****=**** **_{C}__t__ **x**** **_{C}__t__** ****=**** **_{C}__q___{[}_{A}_{ + Si}_{]} **x****
**_{C}__q___{[}_{A}_{ + Si}_{]}** ****=**** **

_{C}__q___{[}_{A}_{ + Si}_{]}_{ }**+ ** _{C}__q___{[}_{AA}_{ + SSi}_{]}** ****=**** **_{C}__q___{t}_{ }**+** _{C}__q___{tt}

-- because **2** “raised” to the power **1**
-- **2**^{1} -- is just **2**, and because
our rule for multiplying a generic category, call it _{C}__q___{[}_{X + Yi}_{]}** ****=**** **_{C}__q___{Z}** **__=__** **_{C}__Z__, “by”, or “into”, itself, is, for subscripts **X** and **Y**
denoting __sub__-category
symbols, and for subscript **Z**
denoting a category-symbol, simply --

_{C}__q___{[}_{X + Yi}_{]} **x****
**_{C}__q___{[}_{X + Yi}_{]}** ****=**** **_{C}__q___{[}_{X + Yi}_{]}_{ }**+ ** _{C}__q___{[}_{XX + YYi}_{]}** ****=**** **_{C}__Z___{ }**+** _{C}__q___{ZZ}

-- and for **x** and **y** denoting
“**R**eal” numbers --

_{C}__q___{[}_{x}_{ + }_{y}_{i}_{]} **x **** **_{C}__q___{[}_{x}_{ + }_{y}_{i}_{]}** ****=**** **_{C}__q___{[}_{x}_{ + }_{y}_{i}_{]}_{ }**+ ** _{C}__q___{[}_{(}_{x}_{ + }_{x}_{) + (}_{y}_{ + }_{y}_{)i}_{]}** ****=**** **

_{}
_{C}__q___{[}_{1x}_{ + 1}_{y}_{i}_{]}_{ }**+ ** _{C}__q___{[}_{2x}_{ + 2}_{y}_{i}_{]}.

__Note again__: Herein, _{C}__q__ denotes the __gene__ric
category ‘__q__ualifier’ with
“Complex” subscripts.

The subscripts that come
after it are __speci__fic
category descriptors.

*¿*But how do we discover what the resulting, initially
“unknown”, or ''algebraical'', ‘category-description’, here _{C}__q___{tt}, __means__?

Well, the __gene__ric rule to
“solve-for” the categorial __meaning__
of such symbols is that, if we know what is meant by category _{C}__q___{Z}** ****=**** **_{C}__Z__, then the symbol _{C}__q___{ZZ} describes a category each of whose units is a ‘_{C}__Z__ __OF__ _{C}__Z__s’, that is, a
category for a different kind of units, called *‘***meta**-_{C}__Z__s’, each such unit being made up out of a __multiplicity__ of those
units of which the category of the _{C}__Z__s is made up.

To be __speci__fic with this rule, in our
example-model, _{C}__q___{ZZ} specifies a “__C__omplex” of two sub-categories.

Each of the units of the *first* sub-category,
the sub-category of the *‘‘‘***verse**’’’ operations, must be an *‘*__A__**ddition*** *__OF__
__A__**ddition**s’ that is, must be a *‘***meta**-__A__ddition’, such that each *‘***meta**-__A__ddition’ is made up out of a __multiplicity__
of “mere” __A__dditions.

Each of the units of the *second*
sub-category, the sub-category of the *‘‘‘*__in__verse’’’ operations, must be a *‘*__S__**ubtraction*** *__OF__
__S__**ubtraction**s’, that is, must be a *‘***meta**-__S__ubtraction’, such that each such *‘***meta**-__S__ubtraction’ is made up out of a __multiplicity__
of “mere” __S__ubtractions.

Our experiences of / "in" the domain of 'the basic operations of arithmetic' suggest that such operations do “presently” exist in the domain of “__R__eal” arithmetic.

“Multiplicatio__n__” is a basic arithmetical operation that is “made up
out of multiple [repeated] additions”, viz. --

**4 ****x**** 5**** = ****5 + 5 +
5 + 5**** =
****4 + 4 + 4 + 4 + 4**** = ****5 ****x**** 4**** = ****20**

-- a sum of four fives, or a
sum of five fours: either order
will do [a characteristic called “commutativity of addition”]!

In a partial reverse
likeness, “divisio__n__” is a
basic arithmetical operation that is “made up out of multiple [repeated]
subtractions”, viz., **5** "goes ["evenly", i.e., with **0** remainder] into **20**" **4** times; **4** "goes ["evenly"] into **20**" **5** times --

**20 ****÷ ****5** **=**** 4**;** ****20 ****-**** 5 ****-**** 5 ****-**** 5 ****-**** 5**** = ****0**** = ****20 ****-**** 4 ****-**** 4 ****-**** 4 ****-**** 4 ****-** 4;** ****20 ****÷ ****4** **=**** 5**** **

-- to see how many fours
there are in twenty [__not__ the same as how many twenties there are in four]; how many “times”
four “goes ["evenly"] in to” twenty, or to see how many fives there are in twenty, [__not__ the same as how many twenties there are in five]; how
many “times” five “goes ["evenly"] in to” twenty:
but, in this case, either order will __not__ do!

A letter that the spelled
names of these two kinds of operations have in common is “**n**”.

Therefore, we shall
name/symbolize our *second*
category as _{C}__n__, or as_{ C}__q___{n}, denoting the “__C__omplex” combination of the “**mu**__L__**tiplications**” sub-category of elementary __R__eal arithmetic basic operations, with the sub-category
of “**di**__V__**isions**”, and
identifying that combination of __speci__fic
sub-categories with the __gene__ric
*second*
category symbol of our __gene__ric
category-arithmetic model, namely, with the __gene__ric
category-symbol _{C}__q___{[}_{2}_{ + }_{2}_{i}_{]}.

We may “assert” our solution
as follows:

_{C}__q___{tt} **=
** _{C}__q___{n} __=__ _{C}__n__ __=__ _{C}__q___{[}_{L}_{ }_{+}_{ Vi}_{]}** ****=**** **_{C}__q___{[}_{AA}_{ }_{+}_{ SSi}_{]}** ****[---)**** **_{C}__q___{[}_{2}_{ + }_{2}_{i}_{]}.

Again, what is __dialectical__ about the
relationship between _{C}__t__ and _{C}__t__^{2}, or _{C}__t__ **x****
**_{C}__t__, or _{C}__t__ *of*** **_{C}__t__, or _{C}__t__**(**_{C}__t__**)**, the
relationship of what we call *‘***meta**-**unit**-**ization**’, or *‘***meta**-«*monad*»*-**ization**’*, between _{C}__t__ and its already presently existing, ‘**supplementary** other’, _{C}__n__, is that this relationship is a synchronic
*double**-*«*aufheben*»
relationship.

That is, each single “unit”
of the “**mu**__L__**tiplicatio**__n__**s**” sub-category of category _{C}__n__, i.e., each typical individual “multiplicatio__n__” operation, is a *negation*,
and also a *preservation*,
by way of also being an *elevation*
*to* *the* / *forming*
*the* “higher” / more inclusive “**mu**__L__**tiplicatio**__n__**s**” sub-category / level / scale, of a *whole* [*sub**-*]*group** *of *unit*__s__
of the “__A__ddi__t__**ions**”
sub-category / level / scale of the _{C}__t__ category.

Likewise, each single “unit”
of the “**di**__V__**isio**__n__**s**” sub-category of category _{C}__n__, i.e., each typical individual “divisio__n__” operation, is a *negation*,
and also a *preservation*,
by way of also being an *elevation*
*to* *the* / *forming*
*the* “higher” / more inclusive “**di**__V__**isio**__n__**s**” sub-category / level / scale, of a *whole* [*sub**-*]*group** *of *unit*__s__
of the “__S__ubtrac__t__**ions**”
sub-category / level / scale of the _{C}__t__ category.

So, our full solution to the __s__tep **s ****=**** ****1** equation of our model is --

_{C}__)-|-(___{1}** ****=****
**_{C}__t___{ }**+ **_{C}__n__** ****=**** ****A**__ddi____t____ions & ____S____ubtrac____t____ion____s___{ }**+ **__Mu____L____tiplicatio____n____s____ & Di____V____isio____n____s__

**[---)**** **_{C}__q___{[}_{1}_{ + }_{1}_{i}_{]}_{ }**+ **_{ C}__q___{[}_{2}_{ + }_{2}_{i}_{]}.

If this model is working
right, __A____ddi____t____ion____s____ & ____S____ubtrac____t____ion____s__ is
the *simplest* category of
the domain of ‘basic arithmetical operations’; __Mu____L____tiplicatio____n____s ____& Di____V____isio____n____s__ is the *next more complex* category of
that domain.

__Stage ____2__. *¿*What additional ‘category-specifications’
do we generate in our next step, __s__tep **s ****=**** ****2**, that need “solving-for”*?*

Let’s find out:

_{C}__)-|-(___{2}** ****=**** **_{C}__t__^{2^2}** ****=****
**_{C}__t__^{4}** ****=****
****(** _{C}__t__^{2}^{ }**)**^{2}** ****=****
****(**_{ C}__t___{ }**+ **_{C}__n__^{ }**)**^{2}** ****=**** **

**(**_{ C}__t___{ }**+ **_{C}__n__^{ }**)** **x**** ****(**_{ C}__t___{ }**+ **_{C}__n__ **)**** ****=**** **

_{C}__t___{ }**+ **_{C}__n___{ }**+ ** _{C}__q___{n}_{t}**
****+ ** _{C}__q___{nn}.

This result arises by way of
two key rules of categorial algebra, plus the __gene__ral
rule for multiplication when one category-symbol is multiplied by a different
category-symbol [we used a __spec__ial
case of this __gene__ral
rule, for the case where the same category-symbol is multiplied by itself, in __s__tep **s ****=**** ****1**, above] --

**1**. __gene__ral case: _{C}__q___{Y} **x****
**_{C}__q___{X}** ****=**** **_{C}__q___{X }**+ ** _{C}__q___{YX}** ****=**** **_{C}__X___{ }**+ ** _{C}__q___{YX};

__speci__al case: _{C}__q___{X} **x**** **_{C}__q___{X}** ****=**** **_{C}__q___{X }**+ ** _{C}__q___{XX}** ****=**** **_{C}__X___{ }**+** _{C}__q___{XX}.

**2**. _{C}__q___{x} **+****
**_{C}__q___{x}** ****=**** **_{C}__q___{x}; the same category-symbol, added to itself, does not
make “two” of that category-symbol; one “copy” of each category is sufficient;
two or more copies of any category would be redundant, for the purposes of this dialectical-categorial algebra.

**3**. There
is no _{C}__q___{w}
such that _{C}__q___{x} **+**** **_{C}__q___{y}** ****=**** **_{C}__q___{w}; *different*
category-symbols, added together [as opposed to being ‘‘‘multiplied’’’], *do not
*__reduce__ to a single category-symbol, just like in the proverbial case of ‘__a____pples__ **+**** **__o____ranges__’, or __a__** + **__o__.

Well, we already know how to
“solve-for” _{C}__q___{nn}.

It describes a category
“containing” two sub-categories, the *first* sub-category being one of *‘***mu**__L__**tiplication**__s__ __OF__
**mu**__L__**tiplication**s’,
and the *second* sub-category being one of *‘***di**__V__**ision**__s__ __OF__
**di**__V__**ision**s’.

The *first* sub-category is one each of whose
units / operations is a *‘***mu**__L__**tiplication*** *__OF__
**mu**__L__**tiplication**s’,
i.e., each of which is a *‘***meta**-**mu**__L__**tiplication**’, such that each such *‘*__meta__-**mu**__L__**tiplication**’
operation is made up out of a multiplicity of **mu**__L__**tiplication**
operations.

Our experiences of / "in" the ‘basic
arithmetical operations’ domain suggest that such arithmetical operations do indeed
presently exist.

That sub-category-description
describes the sub-category of *‘**multi**-***mu**__L__**tiplication**’ operations -- i.e., of “__e____x____P____onentiations__”: “ex__P__onentiation” is a basic arithmetical operation which
is “made up out of multiple [repeated] **mu**__L__**tiplication** operations, viz. --

**2**^{3}** **__=__ **2 xx**** 3**** **__=__ **2 ****x**** 2 ****x**** 2 ****= ****8 **** ≠ ****9 ****= **** 3 ****x**** 3**** **__=__ **3 xx**** 3**** **__=__ **3**^{2}.

I.e., “two cubed”, or “two
raised to the exponent three”, is “made of” a product involving three twos,
that yields eight, whereas “three squared”, or “three raised to the exponent
two”, yields nine: in general, the
order of “base” and “exponent” cannot be reversed without changing the result
as well.

Generally, each order
will return a different result.

The *second* sub-category should be, per our
standard method, one each of whose units / operations is a *‘***di**__V__**ision*** *__OF__ **di**__V__**ision**s’, i.e.,
each of which is a *‘***meta**-**di**__V__**ision**’, such that each such *‘*__meta__-**di**__V__**ision**’ operation
is made up out of a multiplicity of
**di**__V__**ision** operations.

That is, the *second*
sub-category should be one of *‘**multi**-***di**__V__**ision**’ operations, “made up out of multiple [repeated] **di**__V__**ision**
operations.

We interpret this to
be the “inverse” operation of *‘*__de__-ex__P__onentiation’, or of “**n**th __R__oot extraction”.

The “log” operation, which returns exponents, not bases or roots, is
also a candidate for this “inverse” operation, but is not as fully this inverse
operation as is the __R__oot extraction operation.

Given our experience of / "in" the domain of ‘basic arithmetical operations’, this sub-category description may, at this point, give us
pause.

Many of us may be unfamiliar with the algorithms by which
the “**n**th” root(s) of a given number are “extracted”.

In what sense, if any, can an
exponentiation be reversed, the “root” “extracted” from its “power”, by
repeated division?

But let us consider the method of
extracting square roots that is perhaps the oldest such method still
known.

It is called “The
Babylonian Method”, and also “Heron’s Method”, because the storied Heron of
Ancient Alexandria is the most ancient source known to have written an explicit
account of this method.

This
method is, by the way, a __speci__al case of the more __gene__ral “Newton’s Method”, but predates the discovery of
“Newton’s Method” by many centuries.

The method involves guessing a “starting estimate” for the square root
sought, followed by repeated stages of, well, **di**__V__**ision** -- division of the square by the current best estimate of its
square root -- followed by, well, **di**__V__**ision** again --
this time division of the sum of the previous consecutive pair of estimates by two, thus
averaging them -- to obtain the next better estimate of the square root, all
leading to an improving estimate for the square root with each iteration of the
‘**double** **di**__V__**ision**’ just
described.

‘Formulaically’, the next better estimate of the square’s square
root, **x**_{n+1}, is derived from the previous best estimate, **x**_{n}, by **dividing**
the square, **S**, by the previous best estimate, **x**_{n}, summing **x**_{n} and
**S÷x**_{n},
then **dividing** that
sum by **2**:

**x**_{n+1} **=** **(** **x**_{n}** + S÷x**_{n} **)** **÷** **2**.

Let us apply this method to
“extracting” the “square root” from the “square”, **9**, with “starting estimate” of **x**_{1} **=** **2**:

__n__....__Current
Best Estimate__ ( **x**_{n })...__Di____V____ide____
Square by That Estimate__ ( **S÷x**_{n })..__Di____V____ide____
Their Sum by __**2 **for new est.

**1**....**2**............................................**9****÷****2** **=** **4.5**....................................................**(****2****+****4.5)****÷****2** **=** **3.25**

**2**....**3.25**.....................................**9****÷****3.25** **≈** **2.769**........................................**(****3.25****+****2.769)****÷****2** **≈** **3.01**

**3**....**3.01**.....................................**9****÷****3.01** **≈** **2.99**...........................................**(****3.01****+****2.99)****÷****2** **≈** **3.000**

**4**....**3.000**..................................**9****÷****3.000** **=** **3.000**......................................**(****3.000****+****3.000)****÷****2** **=** **3.000**

After **n** **=**
**3**, with "rounding" as shown above, the method reaches a “fixed point” / “equilibrium”
at **x**_{3+...} **=** **3**, which __is__
the positive square root of **9**.

Thus we see
in what sense, in this method at least, square root extraction is made up out
of repeated **di**__V__**ision**s.

We may thus “assert” our
solution as follows:

_{C}__q___{nn} **=
** _{C}__q___{e} __=__ _{C}__e__ __=__ _{C}__q___{[}_{P}_{ }_{+}_{ Ri}_{]}** ****=**** **_{C}__q___{[}_{LL}_{ }_{+}_{ VVi}_{] }** ****[---)**** ** _{C}__q___{[}_{4}_{ + }_{4}_{i}_{]}.

Our __s__tep **s ****=**** ****2**** **equation-model, as we have solved it so far, thus now
looks like this --

_{C}__)-|-(___{2}** ****=**** **_{C}__t__^{2^2}** ****=****
**_{C}__t__^{4}** ****=**** **_{C}__t___{ }**+ **_{C}__n___{ }**+ ** _{C}__q___{n}_{t}**
****+ **_{C}__e__

**[---)**** **_{C}__q___{[}_{1}_{ + }_{1}_{i}_{]}_{ }**+ **_{ C}__q___{[}_{2}_{ + }_{2}_{i}_{]}_{ }**+ ** _{C}__q___{[}_{3}_{ + }_{3}_{i}_{]}**
****+ **_{ C}__q___{[}_{4}_{ + }_{4}_{i}_{]}

-- since we have not yet
determined which actual category of the ‘basic arithmetic operations’ domain is
described by the algorithmically-generated symbol _{C}__q___{n}_{t} -- if any, i.e., if _{C}__q___{n}_{t} is not an “empty category”, “inoperative” for this
domain.

When, as a component of **(**_{ C}__t___{ }**+ **_{C}__n__^{ }**)** **x**** ****(**_{ C}__t___{ }**+ **_{C}__n__ **)**, the
“higher-complexity” category, _{C}__n__, operates upon / “multiplies” the “lower-complexity”
category, _{C}__t__ --

_{C}__n__ **x**** **_{C}__t___{ }**= **_{ C}__t__ **+**** **_{C}__q___{n}_{t}_{ }**= **_{ }** **_{C}__q___{[}_{A}_{ }_{+}_{ Si}_{]} **+****
**_{C}__q___{[}_{L}_{A}_{ }_{+}_{ VSi}_{]}_{}

-- __gene__rically speaking, the categorial
relationship to be called to the user’s attention by this operation, in this
‘categorial arithmetic’, is, again, a synchronic
«*aufheben*»
relationship, this time, that between _{C}__t__ and _{C}__q___{n}_{t}.

It calls the user to search
that user’s knowledge and memory of the domain in question -- in this __speci__fic case, the domain
of ‘basic arithmetical operations’ -- for a category which represents an
“uplift” of category _{C}__t__ entities to the level of the entities native to
category _{C}__n__, thereby “canceling” the _{C}__t__-type entities concerned, at their own native level, but, by the same
token, “preserving” those category _{C}__t__ entities at the _{C}__n__ level, combining _{C}__n__ and _{C}__t__ __q__ualities,
in the relationship of “elevation” of those category _{C}__t__ entities within the level typical of category _{C}__n__ entities.

Thus, the
additional category thereby presented, _{C}__q___{n}_{t}, signifies a category whose units are the *operational *__interactions__
of the _{C}__t__ operations with the _{C}__n__ operations, as codified in the axioms, and/or
theorems, and/or corollaries, and/or lemmas, and/or ‘‘‘rules’’’ of the “**R**eal Numbers” system of arithmetic.

The *first* sub-category of the category _{C}__q___{n}_{t}_{ }**= **_{C}__q___{[}_{L}_{A}_{ }_{+}_{ VSi}_{]} answers to a sub-category description which connotes
the way in which, or the ‘‘‘rules’’’ by which, the operation of **mu**__L__**tiplicatio**__n__ “subsumes” the operation of __A__ddi__t__**ion**, denoted herein by ‘**L**** | A**’.

To our lights, this
sub-category-description connotes the elementary arithmetical phenomenon often named '''__D__istribution''', or the “__D__istributive law”, e.g., of “**R**eal” arithmetic, an axiom of that system of
arithmetic, which codifies the interaction of the **addition** operation with the
**multiplication** operation -- the rule that the **multiplication** operation
“**distributes** over” the **addition** operation. This “law” involves two components, often called “**left**
**distributivity**” and “**right distributivity**”, respectively:

·
[“**left
distributivity**”]: For all elements **a**, **b**, **c** of **R**,

**c**** x (a + b)
= (c x a) +
(c x b)**.

·
[“**right
distributivity**”]: For
all elements **a**, **b**, **c** of **R**,

**(a + b) x c = (a
x c) + (b
x c)**.

The *second* sub-category of the category _{C}__q___{n}_{t}_{ }**= **_{C}__q___{[}_{L}_{A}_{ }_{+}_{ VSi}_{]} answers to a sub-category description which connotes
the ‘‘‘rules’’’ by which the operation of **di**__V__**isio**__n__ “subsumes” the
operation of __S__ubtrac__t__**ion**, denoted ‘**V**** | S**’.

To our lights, this
sub-category-description connotes a *“***non**-distributive rule” of “**R**eal” arithmetic for ‘**di**__V__**isio**__n__ / __S__ubtrac__t__**ion**’, although
this rule is, typically, not an explicit one in presentations and in
axiomatizations of “**R**eal”
arithmetic. It is learned
informally, as a joint consequence of other rules, i.e., as [partly] already
subsumed under, or included in, the “**distributive law**”, or is encountered as a
theorem, corollary, or lemma.
First of all, note that **di**__V__**isio**__n__ does __not__ fully “**distribute**” over [ ‘**|**’ ] __A__ddi__t__**ion**:

·
[‘‘‘left __non__*-***distributivity**’’’, ‘**V**** | A**’]: For some **a**, **b**, **c** of **R**, **(a + b) ≠ 0**,

**c ÷
(a + b) ≠ (c ÷ a) + (c ÷ b)**.

·
[‘‘‘right
**distributivity**’’’, ‘**V**** | A**’]: For
all **a**, **b**, **c** of **R**, **c ≠ 0**,

**(a + b) ÷ c = (a ÷ c) + (b ÷ c)**.

The **(a
+ b) ≠ 0** and **c ≠ 0**
proviso’s are necessary, in these assertions about ‘**V**** | A**’,
because **di**__V__**isio**__n__s by zero invoke a value that resides beyond the
‘‘‘number-space’’’of the set **R**.

But the *second* sub-category
of _{C}__q___{n}_{t}_{ }**= **_{C}__q___{[}_{L}_{A}_{ }_{+}_{ VSi}_{] }pertains directly to the interaction of the **di**__V__**isio**__n__ operation with the __S__ubtrac__t__**ion** operation, **not** with the __A__ddi__t__**ion** operation [although, given that the set **R** includes “signed numbers” with “negative” signs,
i.e., “additive inverses”, as well as the subtraction operation-sign, additions
can also express subtractions, i.e., if **b
= ****-****d**, then **a ****+**** b = a ****-**** d**], e.g.:

·
[‘‘‘left __non__*-***distributivity**’’’, ‘**V**** | S**’]: For some **a**, **b**, **c** of **R**,

** (a ****-**** b) ≠ 0**,
**c**** ÷ (a ****-**** b) ≠ (c
÷ a) ****-**** (c ÷ b)**,

e.g., **3 ÷ (1 ****-**** 2 ) = ****-****3 ≠ 1.5
= ( 3÷1) ****-**** (3÷2)**.

·
[‘‘‘right
**distributivity**’’’, ‘**V**** | S**’]: For
all **a**, **b**, **c** of **R**, **c ≠ 0**,

** (a ****-**** b) ÷ c = (a ÷ c) ****-**** (b ÷ c)**.

If we re-express subtractions
as additions, and divisions as multiplications -- which the “**R**eal” number arithmetic enables us to do, since it
includes ratios and “multiplicative inverses”, as well as “additive inverses”,
we see that the “**R**eal”
arithmetic’s '''rules''' for ‘**V**** | S**’ are *partly* implicit in the ‘**L**** | A**’ '''rules''',
e.g., if we set **b = ****-****d** and **c = 1÷e **__=__ 1/e, **e ≠ 0**:

·
[an aspect of ‘**L**** | A**’]: For **a**,
-**d**, **1/e** of **R**, **(a + d) ≠ 0**,

**1/e**** x 1/(a ****+**** d) = 1/((e
x a) ****+**** (e x b))**,

e.g., **1/(1/3) x 1/(1 ****+**** ****-****2 ) = ****-****3 =
1/( 1/3 x 1) ****+**** ( 1/3 x ****-****2) )**.

·
[‘‘‘right
distributivity’’’, ‘**L**** | A**’]: For
all **a**, **-d**, **1/e** of **R**, **e ≠ 0**,

**(a ****+**** d) x 1/e =
(a x 1/e) ****+**** (d x 1/e)**,

e.g., **(1 ****+ -****2) x (1/(1/3)) = ****-****3 = ****(
1 x (1/(1/3)) ) ****+**** ( ****-****2 x (1/(1/3)) )**.

The subscript ‘_{V}_{S}_{i}’ component of the subscript ‘_{[}_{L}_{A}_{ }_{+}_{ VSi}_{]}’ of category-symbol --

_{C}__q___{[}_{L}_{A}_{ }_{+}_{ VSi}_{]}_{ }**= **_{C}__q___{n}_{t}

-- can thus be interpreted as calling attention *systematically* and
explicitly, if *somewhat*
redundantly, to the __speci__fic
‘**V**** | S**’ rules, which differ from the __gene__ric ‘**L**** | A**’ rules,
in that the ‘**V**** | S**’ rules require the making explicit of __speci__al restrictions [e.g,
**0** denominators not allowed],
etc., as we have seen above.

A better **interpretation of **/ **solution for** the meaning of the _{C}__q___{[}_{VSi}_{]}_{ } sub-category of the _{C}__q___{[}_{L}_{A}_{ }_{+}_{ VSi}_{]}_{ } category would be the [sub-]category of/for '''**Di**__V__ided Differences''', which can form a portal to the **d**ifferential calculus, involving the Leibnizian *'***infinitesimal ****d****ifference**' operator, **d** -- and, which did, in part, [psycho]historically, actually serve as such, for Isaac Newton, in his pathway to that discovery:

**(Y - y) / (X - x) = **

**((y + delta(y)) - y) / ( (x + delta(x)) - x ) = **

**( (f(x + delta(x)) - f(x) ) / ( (delta(x) ) =**

**delta( f(x) ) / delta(x)**,** such that **--

**d( f(x) ) / dx = **

**limit as delta(x) --> 0(delta( f(x) ) / delta(x)**.

We may therefore write our *full* solution for __s__tep **s ****=**** ****2**** **as --

_{C}__)-|-(___{2}** ****= **** **_{C}__t__^{2^2}** ****= **** **_{C}__t__^{4}** ****=**** **

_{}
_{C}__t___{ }**+ **_{C}__n___{ }**+ ** _{C}__q___{n}_{t}**
****+ **_{C}__e__** **** **** **

__addi____t____ions & subtrac____t____ions___{ }**+ **

__multiplicatio____n____s____ & divisio____n____s___{ }**+ **

__n____
& ____t____ interactions___{ }**+ ** ** **

__e____xponentiations & d____e____-____e____xponentiations__.

Our categorial progression so
far can be summarized textually as below.

The ‘qualo-fractal’
content-structure of this psychohistorical dialectic to __s__tep **2** can be summarized as follows --

__e____xponentiations & d____e____-____e____xponentiations__ “contain” __multiplicatio____ns____ & divisio____ns__,

which, in turn, “contain” __addi____t____ions & subtrac____t____ions__.

The “five symbolic-elements
expression” for this model is thus _{C}__t__^{2^}^{2} [five if we count the underscore under the __t__ as a separate “symbolic-element”].

The systematic dialectic of
the basic operations of arithmetic ‘‘‘presently’’’ and “standardly” ends here,
at __s__tep **s ****=**** ****2**,
with the category named __e____xponentiations & d____e____-____e____xponentiations__
as its ‘meta-meristemal’ category, or
‘‘‘vanguard’’’ category.

We like the compactness of
the _{C}__Q___ representation
of this systematic dialectic, whose “final __s__tep” can be modeled via a single equation --

_{C}__)-|-(___{2}** ****=**** **_{C}__t___{ }**+ **_{C}__n___{ }**+ ** _{C}__q___{n}_{t}**
****+ **_{C}__e__.

However, there is also the
alternative of expressing each __s__tep of this
dialectic by *two *__separate__ equations,
using the _{N}__Q___ dialectical algebra instead, e.g. --

_{verse}__)-|-(___{2}** ****=**** **_{C}__A___{ }**+ **_{C}__L___{ }**+ ** _{C}__q___{L}_{A}**
****+ **_{C}__P__, for the *‘‘‘verse’’’* operations;

_{inverse}__)-|-(___{2}** ****=**** **_{C}__S___{ }**+ **_{C}__V___{ }**+ ** _{C}__q___{V}_{S}**
****+ **_{C}__R__, for the *‘‘‘*__in__verse’’’, or *‘‘‘*__re__verse’’’,

operations.
To isolate the **first triads** of **categories** from these paired **dialectical** **categorial**-**combinatoric progressions**, we can use twin __s__tep **s ****=**** ****1** **Triadic Seldon Function*** *'**[****meta**-]**model **[**meta**-]**equations**' as follows, using the modified notation also employed in the two images below --

_{verse}__)-|-(___{1}** ****= **_{O+}__)-|-(___{1}** ****=**** **_{}_{O+}__A__^{3^1}** ****= **** ****O+**__A__^{3}** ****=**

_{O+}__A___{ }**+ **_{O+}__M___{ }**+ **_{O+}__q___{M}_{A}** ****= **** **_{O+}__A___{ }**+ **_{O+}__M___{ }**+ **_{O+}__D__,

for *‘‘‘verse’’’* operations;

_{inverse}__)-|-(___{1}** **** ****= **_{O-}__)-|-(___{1}** **= ** **_{O-}__S__^{3^1}** ****= **** **_{O-}__S__^{3}** ****=**** **_{}
_{O-}__S___{ }**+ **_{O-}__D___{ }**+ **_{O-}_{}__q___{D}_{S}** ****= **** **_{O-}__S___{ }**+ **_{O-}__D___{ }**+ **_{O-}__V__,
for *‘‘‘*__in__verse’’’/*‘‘‘*__re__verse’’’, operations

-- which can then be depicted as follows --

__Stage ____3__. To
iterate our _{C}__Q___ ‘meta-equation’,
_{}
_{C}__)-|-(___{s}** ****=**** **_{C}__t__^{2^}^{s}** **__=__** ****(** _{C}__q___{[}_{A}_{ }_{+}_{ }_{S}_{i}_{]}_{ }**)**^{2^}^{s},

for __s__tep **s ****=**** ****3**, is to iterate the systematic presentation of the
domain of basic arithmetical operations beyond the “basic”, beyond the “present”,
beyond the conventional conclusion of that presentation, and beyond the
“systematic reconstruction” of this domain at present, to a somewhat
“preconstructive” -- somewhat “predictive” -- extrapolation of its possible
future. However, as we shall see,
we have already encountered units of the “vanguard” term of __s__tep **s ****=**** ****3**, in this very text.

Let’s see what are the
additional category-descriptions that this __s__tep **s ****=**** ****3** ‘self-iteration’ __gene__rates:

_{C}__)-|-(___{3}** ****=**** **_{C}__t__^{2^}^{3}**
****=**** **_{C}__t__^{8}** ****=****
****(** _{C}__t__^{4}^{ }**)**^{2}** ****=**** **

**(**_{C}__t___{ }**+ **_{C}__n___{ }**+ ** _{C}__q___{n}_{t}**
****+ **_{C}__e__^{ }**)**^{2}** ****=**** **

**(**_{C}__t___{ }**+ **_{C}__n___{ }**+ ** _{C}__q___{n}_{t}**
****+ **_{C}__e__^{ }**)** **x**** ****(**_{C}__t___{ }**+ **_{C}__n___{
}**+ ** _{C}__q___{n}_{t}**
****+ **_{C}__e__ **)**** ****=**** **

_{C}__t___{ }**+ **_{C}__n___{
}**+ **_{C}__q___{n}_{t}**
****+ **_{ C}__e___{ }**+ **_{C}__q___{e}_{t}**
****+ **_{C}__q___{e}_{n}**
****+ **_{C}__q___{e}_{n}_{t}**
****+ **_{C}__q___{ee }

**[---)**** **

_{C}__q___{[}_{1}_{ + }_{1}_{i}_{]}_{ }**+ **_{ C}__q___{[}_{2}_{ + }_{2}_{i}_{]}_{ }**+ ** _{C}__q___{[}_{3}_{ + }_{3}_{i}_{]}**
****+ **_{ C}__q___{[}_{4}_{ + }_{4}_{i}_{]}**
****+ **_{ }

_{C}__q___{[}_{5}_{ + }_{5}_{i}_{]}_{ }**+ **_{ C}__q___{[}_{6}_{ + }_{6i}_{]}_{ }**+ ** _{C}__q___{[}_{7}_{ + }_{7}_{i}_{]}**
****+ **_{ C}__q___{[}_{8}_{ + }_{8i}_{]}.** **

We know -- from past experience,
narrated above -- how to “solve-for”category

_{}
_{C}__q___{ee}**
****= **_{ C}__q___{[}_{PP}_{ }_{+}_{ RRi}_{]}.

It describes a category
“containing” two sub-categories, the *first* sub-category being one of *‘*__P__**owers*** *__OF__
__P__**owers**’,
and the *second* sub-category being one of *‘*__Root____-__**extractions*** *__OF__
__Root____-__**extractions**’.

The *first* sub-category is one each of whose
units / operations is an *‘***ex**__P__**onentiation*** *__OF__
**ex**__P__**onentiation**s’,
i.e., each of which is a *‘***meta**-**e****x**__P__onentiation’, such that each such *‘*__meta__-**ex**__P__**onentiation**’
operation is made up out of a multiplicity of **ex**__P__**onentiation**
operations.

But that is precisely the
new operation that we have encountered in this text, at the heart of the Seldon
Functions in __gene__ral,
and at the heart of our _{C}__Q___ ‘meta-equation’ --

_{}
_{C}__)-|-(___{s}** ****=**** **_{C}__t__^{2^}^{s}

-- __speci__fically.

A unit increment in the
‘meta-exponent’ of the ‘starting-category’ symbol of that ‘meta-equation’,
corresponding to a unit increment in its __s__tep-value, **s**, is equivalent to a two-fold exponentiation of that ‘starting-category’ symbol. e.g.
--

_{C}__t__^{2}**
= **_{C}__t__^{2^}^{1};** **

**(**_{C}__t__^{2}^{ }**)**^{2}** ****= ****(**_{C}__t__^{2^}^{1}^{ }**)**^{2}** **** = ****(**_{C}__t__^{2^}^{1}^{ }**)**^{2^}^{1}** ****= **_{C}__t__^{2^(}^{1}^{+1)}** **__=__ _{C}__t__^{2^}^{2}

-- because repeated exponents
mutually __multiply__,
and because *'meta-exponents' of* exponents __add__
together when those 'meta-exponents' have the same exponents as their bases, and
are multiplied together.

Let’s
call this sub-category __H__**yper**-**exponentiation**, or **H** for short.

The *second*
sub-category should be for operations which are *‘*__d____e____-____e____xponentiation__s* *__OF__
__d____e____-____e____xponentiation__s’, i.e., which are *‘***meta**-__d____e____-____e____xponentiation__s’, such that each *‘***meta**-__d____e____-____e____xponentiation__’ operation is made up out of a multiplicity of __d____e____-____e____xponentiation__ operations.

That is, the *second* sub-category
should be one of *‘*__multi__*-*__d____e____-____e____xponentiation__’ operations, “made up out of multiple [repeated] __d____e____-____e____xponentiation__ operations, each denoted by '**√**', the sign of the "square-root"-extracting operation.

We can use the ‘self-example’, exemplified in this very text, to
illustrate this process:

**√√√**_{C}__t___{ }**+ **_{C}__n___{ }**+**_{C}__q___{n}_{t}**
****+ **_{C}__e___{
}**+ **_{C}__q___{e}_{t}**
****+ **_{C}__q___{e}_{n}**
****+ **_{C}__q___{e}_{n}_{t}**
****+ **_{C}__q___{ee }**= **

**√√√**_{C}__t__^{2^}^{3}_{ }**=**** **

**√√**_{C}__t___{ }**+ **_{C}__n___{
}**+ **_{C}__q___{n}_{t}**
****+ **_{ C}__e___{ }**= **

**√√**_{C}__t__^{2^}^{2}_{ }**=**

** **** _____________**

**√**** **_{C}__t___{ }**+ **_{C}__n___{ }**= **

**√**_{C}__t__^{2^}^{1}_{ }**=**** **

_{}
_{C}__t__^{2^(}^{3}^{-3)}_{ }**= **** **

_{}
_{C}__t__^{2^}^{0}_{
}**=
**

_{}
_{C}__t__.

Let’s call this sub-category __D__**e**-**H****yper**-**exponentiation**, or **D**** **for short.

We may thus “assert” our solution
as follows:

_{}
_{C}__q___{ee} **=
** _{C}__q___{m} __=__ _{C}__m__ ** ****=**** **_{ C}__q___{[}_{H}_{ }_{+}_{ Di}_{]}_{ }**[---)**** ** _{C}__q___{[}_{8}_{ + }_{8i}_{]}.

Our __s__tep **s ****=**** ****3**** **equation-model, as we have solved it so far, thus now
looks like this --

_{C}__)-|-(___{3}** ****=**** **_{C}__t__^{2^}^{3}**
****=**** **_{C}__t__^{8}** ****=**** **

_{}
_{C}__t___{ }**+ **_{C}__n___{
}**+ **_{C}__q___{n}_{t}**
****+ **_{C}__e___{ }**+ **_{C}__q___{e}_{t}**
****+ **_{C}__q___{e}_{n}**
****+ **_{C}__q___{e}_{n}_{t}**
****+ **_{ C}__m__

-- since we have not yet
determined which actual categories of the ‘basic arithmetical operations’
domain are described by the algorithmically-generated ‘category-description’
symbols _{C}__q___{e}_{t}, _{C}__q___{e}_{n}, and _{C}__q___{e}_{n}_{t}, *if any*.

But we already know how to
characterize the *possible*
categories that these three category-symbols “call for”, viz.:

·
_{C}__q___{e}_{t}_{ } **[---)**** ** _{C}__q___{[}_{5}_{ + }_{5}_{i}_{]} “calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’, that
combines the __e__ and __t__ operations.

·
_{C}__q___{e}_{n}_{ } **[---)**** ** _{C}__q___{[}_{6}_{ + }_{6i}_{]} “calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’, that
combines the __e__ and __n__ operations.

·
_{C}__q___{e}_{n}_{t}_{ }**[---)**** ** _{C}__q___{[}_{7}_{ + }_{7}_{i}_{]} “calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’, that
combines the __e__ and the _{C}__q___{n}_{t}.

We may thus write our *full* solution for step **s ****=**** ****3**** **as --

_{C}__)-|-(___{3}** ****=**** **_{C}__t__^{2^}^{3}**
****=****
**

_{}
_{C}__t__^{8}** ****= **** **

_{}
_{C}__t___{ }**+ **_{C}__n___{
}**+ **_{C}__q___{n}_{t}**
****+ **_{C}__e___{ }**+ **_{C}__q___{e}_{t}**
****+ **_{C}__q___{e}_{n}**
****+ **_{C}__q___{e}_{n}_{t}**
****+ **_{ C}__m__** ****= **** **

__addi____t____ions & subtrac____t____ions___{ }**+ **

__multiplicatio____ns____ & divisio____ns___{ }**+ **

** **

__n____ with ____t____
interactions___{ }**+**

__e____xponentiations & __*d*__e____-____e____xponentiations___{ }**+ **

** **

__e____ with ____t____ interactions___{ }**+ **

__e____ with ____n____ interactions___{ }**+ **

** **

__e____ with ____n____ & ____t____
interactions___{ }**+**

__m____eta____-__**exponentiations**__ & ____de____-____m____eta____-__**exponentiations**.

Our categorial progression so
far can be summarized textually as below.

The ‘qualo-fractal’
content-structure of this **psychohistorical **__dialectic__ through __s__tep **3** can be summarized as follows --

__m____eta____-__**exponentiations**__ & ____de____-____m____eta____-__**exponentiations** “contain”

__e____xponentiations & d____e____-____e____xponentiations__, which “contain”

__multiplicatio____ns____ & divisio____ns__, which “contain”

__addi____t____ions & subtrac____t____ions__.

The “five symbolic-elements
expression” for this model, up to this __s__tep, is thus _{C}__t__^{2^}^{3}.

The meaning mnemonically
compressed into the **5**
symbolic-element expression _{C}__t__^{2^}^{3}^{ }can
be depicted as follows --