Horses, Bows, Crowns, Swords And Balances

A "horse" in the text refers to the generalised form of an additive subgroup cycled under addition. The term is analogous to motion whilst the rider is "at rest". The static subgroup of the left handed octal is the "horse" in view in the seals whereas the army of 200,000,000 horsemen of the fifth and sixth trumpets refer to those multiplicative cosets of additive subgroups.

We then term the following as a "horse"; that if G is an additive subgroup of a field then placing every element of G under product with an element x from the multiplicative group of the field sends "one horse to another" That is, every such product xG = H where H is another additive subgroup. Sets of horses forming a superstructure with similar properties are also "horses" (if it is true G1,G2,..Gn form the group G under a cycle x, i.e. G is a subfield of F) if yG = H in F is also a "horse. However groups such as H are "teams of horses" more appropriately the "chariots" we see referenced in the text.

So if we have a field F and a subfield K, Then under a generator of K* our "x" we may generate a cycle of additive subgroups of K+ by acting upon them with x. So G1 = H, G2 = xH, G3 = xxH etc. Then it is the case that for elements of F* (our "y") we have cycles of both the G[i] and the H under action by xy and y repsectively. We would term the G[i] as horses and the H, yH, yyH etc, as "chariots."

A "bow" is simply a subgroup of the left handed octal that is held static under the frobenius map upon the right handed field. There is one such group in the octal under multiplication by GF(8)*. The term "bow" is taken most often to refer to the static subgroup of the left hand.

The "crown" (as given to the rider of the white horse) is equivocated with the "rest" or "static" property of the element, bow or field itself. Just as unity is held fixed under frobenius (as is the Lamb as unity and the twenty four elders as particular octals) a crown may also represent not just the 168 full automorphisms of the field GF(8) (with all its C7 groups and with floating unity) but also represents the static subgroup(s) of the left hand. It is this static group of the left hand that represents a "crown" as mentioned in the first seal.

The "sword" refers in the text as that which "cuts off" two parts and preserves one of three elements. Just as Christ "comes to fight against them with the sword of His mouth" as imaged in the two parts cut off of the Lamb as slain - (To whom He states, "I never knew you", and "I know not from whence ye are".) The presence of two bows allow for a similitude of interaction using cross substitution.

By allowing ambiguity over the unity element and the current bow to which it is associated we have two correspondences in the "sun" octals left hand - we use the right hand to display this here.

so, if a = 1 = [b,d,f] = {c,e,g} and then c = 1 = [b,d,f] = {a,e,g} Then cross substitution yields a = {a,e,g,e,g} or c = {c,e,g,e,g}. Then the two parts cut off are {e,g} or {{e,g},{e,g}}. This "great" sword represents not just one way of "cutting off" but is a "doubled" action. The result is that depending on the choice of one element as primary in the substituion, the other as under the sword is eliminated, or as the text puts it "that they shoulld kill one another."

In permitting further ambiguity to three bows we have the balances of the black horse.

We would have say a = 1 = [b,d,f] = {c,e,g} and then c = 1 = [b,d,f] = {a,e,g} and also e = 1 = [b,d,f] = {a,c,g}. Then there is one element common to all, the "wine" as at the table, served to all and by which the men are "made drunken", here that element is "g". Likewise each of the three shares one element with one of the other two besides "g".

We pick any order we wish of the three left hand bows and write: {c,e,g}=>{a,e,g}<={a,c,g}. Without loss of generality we assume that the opposing element of unity corresponding to the centre of the "balances" {c,e,g}=>{a,e,g}<={a,c,g}. is "c". We treat this as it were unity and by absorbtion we mimic the effect that we "lighten" the scales of the balances to {e,g}=>{a,e,g}<={a,g}. Then the "balances" find their equilibrium as if it were true that "thesis => synthesis <= antithesis." The middle of the balance is an unjustified union of the scales - a synthesis.

The element in view as unity is "the oil" from the verse. We usually equate oil with doctrine, but in this case or process it may also be thought of as "leaven". I.e we can "agree to disagree" with accepting the synthesis: it becomes agreeable to ignore the facts (unity) on the basis that it is our "stance" on them we disagree over rather than the facts themselves.

The "measure of wheat" for a penny refers to one singleton of the right hand octal (wheat) in balance with a scaled measure (here the weight of a penny) that is in balance with a "triple measure" (of barley) from the left hand. The "penny" or "unity" is the days wages, (a "crown" of reward when in the right hand but not so here as in the left. There is ambiguity over the wages or unity in view, as three are assumed rather than one.) Then in balancing the singletons (wheat) with the left handed bows (barley) with three such bows as here we have our balances.

"Hurt not the oil and the wine" informs us we should preserve the action of unity (we used "c" here) and likewise we should not alter the operation on "g". I.e. that which is left in the balances at the end! (NB a + e = g in the left hand with c=1.) We should not permit the structural inconsistency of switching oil and wine with use of the fourth bow, however the carnal man in the system (as in the image) will gravitate to the choice of the two that dispenses with the hard facts as oil - as if they were painful, and gravitate to the synthesis of the wine as if it were pleasurable, a continuation of the dopamine rush from being "part of the group". We are predisosed to approach pleasure and avoid pain.

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