Our Father, Which Art In Heaven

When considering the K4 group, naturally I have always jumped to the assertion that it is primarily a subgroup of the octal group. However the K4 form of the ultrafilter is a group of K4 subgroups that spans the octal. It is the assertion "Sit thou at my right hand..." That would make the K4 group only a subgroup of the octal. Jesus Christ (In whom the Godhead dwells bodily) is not that simple to consider - there is in some sense a duality.

Given that an axiom of our construction was that the filter must contain itself as an element - (and that the filter is God's perspective on infinite regression) If the K4 form of the filter is composed of subgroups then the question of "unity" as a singleton element comes into consideration. Our axiom that "In Him we live move and have our being" was made to apply to God also - and Him foremost as not knowing anything that could exist aside from Himself, in that it would be impossible for God to know anything that He himself would not have created. (Indeed He may rest without knowing where every object is at any time, but it is implausible by construction that anything could exist without God Himself as the first creative cause. If there are other Gods, that is Gods concern and not ours - For God as "Most High" will never be able to know them, (or they He) without some form of union amongst them - thus the trinity is born in our construction.

However if the trinity in its K4 form "has itself" as its own "unity" (albeit with multiplication induced by the frobenius map in GF(8)) then we also know that there is the coset of that K4 group (being as unity) in the octal: And surprisingly, the coset contains four K4 groups of the seven. We simply assert that the K4 form has a direct association with a singleton (The intersection of its members) and although the four remaining K4 elements are associated with "The Earth" in Revelation, when constructing the K4 form this is not actually relevant: More relevant is the argument that Jesus Is God from the fact that Jesus "prevailed to open the seals" and the book of revelation asserts Jesus Is God from start to finish. (That Jesus, although "sat" as a subgroup of the octal, indeed is the K4 ultrafilter of the construction.)

So then, We deduce that outside of the requirement that the K4 form is indeed a K4 group of K4 subgroups - we must be able to assert that the K4 group is in some sense a steady state - that there is no infinite descent, and that unity must be as the Greeks asserted, that a Cartesian point is not necessarily a null vector, but a point is as unity commensurate everywhere. In order to understand the scripture we must understand the thought of the writer - it makes no sense to assume God talks merely to mathematicians and not those who would consider a point to have more than zero within it.

And again, when we perceive that the K4 filter has a coset, then we must also argue that there is a "heaven" and that the full octal contains the perspective of the viewer within and without - That the "earth" is closed and complete when heaven is included - (what is meant by within and without). Then we have a dichotomy to consider - is the K4 form true almost everywhere? Are we able to name our Father as Jesus Christ 's own? If Jesus, whilst sat at the right hand of the Father is no less diminished, that in some sense unity is commensurate with the perspective we have of creation, and in each several case the singleton "unity" in the octal (also the intersection of the K4 form's K4 subgroups) is both triune and also a state of rest, and in Christ such a state we find ourselves.

At college I was shown Hamilton’s quaternions - I always thought it strange to commensurate three different square roots of "-1". When I realised that multiplication is simply a map on an ordered pair of elements to another element, I put away my complaints. However I do complain that camouflage is given the K4 group that sits in the centre! We should realise without trouble that if we do our arithmetic modulo 2, then i*j = j*i = k. We then have a K4 group.

To put it plainly, if we start with a simple object and rotate it 180 degrees (a half turn) about two Cartesian axes, then we may "repeat or undo" our transformation with a third, bout the remaining axis. However, these order two transforms may be performed in any order and we always generate the elements of a K4 group, finding zero or the identity as our reference. To help you picture this I have prepared some images...

Given our starting position (zero),

We can apply any 180 degree rotation we choose;

And, if you can stretch your brain around it, recover the remaining one of the following two by application of either of them upon that first rotation. (about axes that are othonormal)

I have labelled the rotations with respect to the identity (zero): hopefully this is not a problem!

So, as to "unity", if the cartesian system has within it the embedding of a K4 group, we can discern that in all things these vectors, may be bent and twisted by gravity but they have not failed thus far - "unity" in the ultrafilter then (if it is a steady state) can be described as the realisation that "In Him we live, move and have our being". It is senseless to consider that there is a coset formed of mirror reflections of this system - or that space is somehow halfway there - rather time is a continually commensurate quantity and God provides the motive or continuing principle that the remainder of the octal is continually transformed into and from. "The Spirit giveth life" as it were.

Then we also now state that the K4 form of the ultrafilter begins to make more sense as a group of subgroups - That Christ also exists in a manner we may also be familiar with - and that He, humbled as a man is no less diminished in His divinity.

The perspective of the K4 group as unity in the K4 filter makes sense now to consider that in the life of Christ the Father was well pleased (in Him) as well as to consider that the perspective of Chrit was in "full knowledge" of the mind of His Father.

So, "Our Father, Which Art In Heaven"? We can only state that Heaven includes creation as well - that God the Father reigns from Heaven in full knowledge of the life He has given and placed in His children. Those that may be without such knowledge may come to it freely - wheras those for whom it is nonsense may only be lacking in faith.

So, space is commensurated with zero in the cartesian system - but time taken in terms of God in trinity appears to be comensurated with unity. Is there any evidence for that? Indeed there is. The (Minkowski) metric is used in General relativity to permute the tensors in their calculi - A metric may be thought of as a scalar that "normalises" the coordinate system. To put it plainly, the relativistic theory of gravity put forth by Einstein is subject to curvature: The "metric" is simply there to ensure that the local properties of space are "othonormal" and that things are locally flat. The system breaks down in a black hole because essentially, when dividing by zero, there is no longer a real number to multiply that infinity by in order to make the product equal to unity. A metric is literally, the scale of the system that enables the equations to hold as if time was - (you guessed it) commensurate as unity within space.

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