Smoke, Locusts, Green Things

Firstly, we note that in the text a star falls from heaven "unto" the earth. The star in view is a fifth element added to the four earthly elements to form the full set of permutations in S5 upon them. The true "fifth element" in the octal is actually an element of the static K4 subgroup, so this star has fallen to take it's place. This has the result that no K4 subgroup is formed from the earthly elements with an element from the static group: rather this star has fallen to prevent that. In place then is the full group S5 with its normal subgroup A5 (a simple group) as opposed to the subgroup of S4 which has a K4 normal subgroup.

In being given the key to the bottomless pit we see that this group S5 is used to take the place of the octal (the sun) and interrupts the proper operation of the seven cycles (spirit or "air"). It is in this sense that the operation of any seven cycle is obscured and as "air" the general sense of the spirit as over the earth (or the four beasts) it is now completely obscured by the "smoke" of the pit. The "smoke" is basically poor air that can not be breathed (is not "life" to us as is the Spirit) and it has taken the place of the Holy Spirit.

For those whom live in spiritual ways, the octal underneath is a secondary concern - the spiritual requirement to hear God's word of life is primary. In interrupting the action of the Holy Spirit upon the earth with a "false" spirit (in the person of the fallen star) There is substituted in underneath a set from a different fiinite field than GF(8) (the sun). It is the elements of the multiplicative group of this substitute field that are "locusts".

However there is the mention in the text of a command that "they should not hurt the grass of the earth, neither any green thing, neither any tree" - we see then that the locusts are not to interrupt the singletons of the earthly elements - the operation on them must be preserved, neither should the operation affect anything with "new life" (a green thing) and should not affect anything that branches (trees). The correspondence of subgroups on the one hand to the singletons on the other must be preserved. (ie K4 subgroups)

The command to preserve these elements shows that from the octal the elements of the four beasts are intact: as is a singleton added to that set: (forming a K4 group with them) which is a "springing up" of new life as a K4 subgroup sharing a singleton with the static subgroup of the Lamb; Then "any tree" preserves the branching operation of the sun octal. We see that in these three things, we have in the earth and green things a complete definition for the octal, and the tree as any subgroup. The locusts are commanded to not redefine the GF(8) field in any way other than direct 1-1 correspondence with their counterparts in the extensions themselves. (I.e. a bijection of labels rather than simply an automorphism) However the definition is such as to leave any C7 group as GF(8)* with any choice of unity from the four beasts free to vary. In other words the full "24" elders are permitted to reign still.

The one thing that these locusts have authority to harm are those men without the seal of God in their foreheads. The seal here relates to that of the Father - as an element of the octal in one particular arrangement or another of the 168 possible floating unities. The presence "In God" of the believer shows that those without the Father (in fact without a rest state in Christ) are such as to be present in some field other than GF(8). In fact the only place that they could go elsewhere is either GF(2) or GF(4). (ie, at the "altar" or in Christ) But the effect is that they may spiritually dwell in S5 upon those five earthly elements with this star. It is by changing octals that these groups have a potential to act.

The "switch" of multiplication is such that the addition is required to be altered as well. Extension fields are chosen over GF(8) for this; then instead of "additive locusts" we reserve additive subgroups (horses) to be operated on by multiplication also. There is no additive operation for these groups but the action of the locusts upon crowds is such as to give the illusion of spiritual activity and "a form of fellowship".

Since the C7 groups are free to vary they are not affected by the locusts, and since the seven cycle is not given to those apart from Christ there is no need for GF(8) to be a subfield of the extension of locusts: In fact it simply requires the octal to be an additive subgroup of the extension that preserves the "earthly" elements along with a static subgroup excluded - and this is done since three cycles generate A5. (A5 contains every three cycle in S5, these elements are considered as "bows") In five symbols we see our four earthly elements with the fifth permit "green things" to be unharmed I.e. if the earth is {d,e,f,g} and [a,b,c] is static then [a,f,g] and [a,d,e] are excluded from S5 since "a" is not enjoined, only a "label" or "star", an "x" is.

Likewise the elements in the static K4 group [a,b,c] are fully excluded from S5 in this case - the intersection property requires every other K4 subgroup of the octal to have one element from [a,b,c], to have "a green thing". Thus every tree is excluded also.

The embedding of the earthly elements in A5 is a good point to know.

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