To The Left Or Right (Freedom)

We used the four elements of the coset of a (static under frobenius) K4 subgroup in the additive group of the finite field of eight elements as placeholders for the dialectic process. In using the symbols {d,e,f,g} we tacitly assumed that (0,a,b,c) is a K4 subgroup of the octal.

Likewise we noted the octal could be spanned by a K4 group composed of the subgroups (0,a,b,c),(0,a,d,e),(0,a,f,g). and using either the symbol '0' for the additive identity or the octal itself with the group under the operation (A v B)^c. (The complement in the octal of the symmetric difference of A and B.)

In the latter case we can happily assume a = 1 and frobenius then induces or forms a cyclic group of order three upon these three subgroups. Likewise in the octal if a = 1 then there is some subgroup held static that must not contain unity, say (0,b,d,f).

In the sense that a = 1 = (b,d,f) We have the association that to be found "In Christ" is to be a member of the subgroup (b,d,f) (dropping the zero in the notation for now) Likewise there is a span (as such above) of the octal with a = 1: We may reduce to the K4 form of the trinity with the span thus:

a = 1 = (a,b,c)
b = (a,d,e)
c = (a,f,g)

Which is not valid in the structure of the field GF(8) as before but maintains its cyclic multiplication under frobenius induced by or from the octal's multiplication. However the following associations to the elements {c,e,g} in the coset of (b,d,f) are valid in the octal and GF(8):

c = (a,b,c)
e = (a,f,g)
g = (a,d,e)

Where a = 1. Then (b,d,f) is held static by frobenius in the case that {c,e,g} is a triple also held static but {c,e,g} is not a subgroup. The elements in our coset in this case are not (d,e,f,g) but are actually (a,c,e,g).

So what can we deduce? That to be in Christ is somehow to be in the K4 form as well as the octal and GF(8). In both cases one would find one's self as a member of (0,a,b,c). To "have the Son as well as the Father" requires that one be spiritually placed in the intersection of GF(4) and GF(8) - which is either 0 or unity (1) or both together.

Movement then in Christ is from unity to one of a pair of inverses: ie from a = 1 to consider 'b' or 'c' where b*c = a = 1.

We will simply call one "movement to the right" and the other "movement to the left". In the octal the equivalence is to a product in the octal whether additive or induced under frobenius that the outcome of movement in Christ is done "separate to the fixed unity" and does not require it to shift.

In any case there is an arrangement under a relabelling or under a valid multiplication (of a possible eight compatible C7 groups for GF(8)*) that will transform the movement to the right or to the left back to the intersection of GF(4) and GF(8).

In the K4 form

a = 1 = (a,b,c)
b = (a,d,e)
c = (a,f,g)

'b' and 'c' are opposites, we could denote them as "true or false" - but both must find agreement in the freedom to be in either position, denoted by the common element of 'a' which is unity. In view are the elements of the coset {d,e,f,g} that are rightly and disjointly divided. Under every compatible multiplication, of the eight possible C7 groups for multiplication on each octal - subgroups are preserved.

We find that this disjoint division is unique to the additive group and these seven cycles, and whilst there are 30 separate octals in seven symbols, within each octal we may consider that "true" or "false" are compatible with the freedom of choice of either, as long as that choice finds its correctness in Christ as the unity element.

That there be a map from the multiplication on GF(8) or from the frobenius map inducing C3 on the K4 form, allows the transform of 'b' or 'c' back to 'a' in terms of singleton elements, or of the group (a,b,c) that is static; the concepts of "true and false" are better described by freedom of movement to the left or right: a true action may seem false in Christ: To be greatest is to be a servant etc: We find this often in the gospels. The Lord turned the world upside down. We will see this in the principle of "overlay" to follow.

Likewise the case in the octal of;

a = 1 = (b,d,f) and {c,e,g}
c = (a,b,c)
e = (a,f,g)
g = (a,d,e)

Is such that movement in the K4 groups actual members is closed also in the correspondence to the coset {c,e,g} of (b,d,f) - and also excluding unity 'a'. The members we used to construct the dialectic (here {a,c,e,g}) are associated with the elements of the K4 form, but the static subgroup of Christ that is sat at the right hand is (b,d,f).

There is a strange twist that equivocates the group (b,d,f) to ((a,b,c),(a,d,e),(a,f,g)) although only the coset (c,e,g) with a = 1 does so in the octal structure. Why?

One can not ask "Why?" but one can ask what is the effect of this.

Turning to the left or right is become the choice between alternatives rather than opposites. The disjoint choice 'b' or 'c' divides the option of {d,e} or {f,g} and is directly related to pairs of elements not unity. The elements in transition from (b,d,f) to (c,e,g) (respectively) as per above. Then we note that nothing can shift unity from unity's set place in Christ, but the freedom to choose whether one makes the transition into Christ from c=>b, or e=>d or g=>f is done in "overlay"; that the choice of distinction between b and c etc is up to the exampled freedom of alternatives available in Christ: And transitioning from one form to the other is what is truthfully "facilitated" from the world ({a,c,e,g} to Christ (0,b,d,f)) in the gospel.

If one holds fixed unity then the "love of the truth" is present in resisting the dialectic and if the "present truth" is compatible with the freedom of alternatives expressed in Christ, then the transition from the elements of a static bow {c,e,g} to those of the static subgroup (b,d,f) is freedom to choose not betwen right and left, but of God in choosing whom He will permit to be in approval as found in His Son.

"Right or wrong" then is not a matter of moving to the left or right, but is defined by the example in Christ that provides the freedom of alternatives that are in accordance with His laws and grace. Such movement to the left and right in all aspects of life, are essentially the freedoms present within the gospel.

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