The Group Axioms Fulfilled
First, just to cross my t's and dot my i's, I have the verse:
Rev 3:2 Be watchful, and strengthen the things which remain, that are ready to die: for I have not found thy works perfect before God.
(KJV)
And I should, for completeness, make sure that the octal's operation is "well-defined" or at least "well-defined-enough".
The Group Axioms for a group G are:
G1) Closure. For all “a”, “b” in G, a*b = c is also an element in G.
G2) Associativity. For all elements “a”, “b”, “c” in G, a*(b*c) = (a*b)*c
G3) Identity. There is a unique element “e” in G such that for all a in G, a*e = e*a = a
G4) Inverses. For each “a” in G there is a unique element a_{}^{-1} in G such that a*a_{}^{-1} = a_{}^{-1}*a = e. (note that “e” is self-inverse.)
And without constructing the octal from scratch I may write:
g_{r}∨l&e_{r}=>g_{r} which is a valid disjunction, for I may have g_{r}∨l&e_{r}^{-1}=>g_{r} with the right-side extant and so at rest, the set "r" (here merely e_{r}) must remain "God-like" when at rest and so the principal element e_{r} appears on the right-side also. I.e. r∨p&r^{-1}=>s must, on the right side extant, remain God-like in r^{-1}, thus the principal elements in the "old-name" on that side become as e_{p}&e_{r}=>e_{s}.
In the "new-name" then, e_{p}&e_{r}=>e_{s} is to become (with the right-side extant) as l&e_{p}&e_{r}=>l&e_{s} which is also g_{p}&e_{r}=>g_{s} or g_{p}&g_{r}=>g_{s}. (And if I use "r" for a special case of "s", I find a requirement for g_{p} = l alone, the unique element of the octal that fulfils this requirement.)
So, simply by applying the octal's multiplication (addition) table, I have G1 and G2. (The octal is a pre-requisite of a "God-like" disjunction.)
G4 follows from the above statement of r∨p&r^{-1}=>s becoming (on putting p=l) the disjunction of g_{r}∨l_{}&g_{r}=>g_{r} (as for G3 or identity). I also find with r=l, say, l _{}∨g_{p}=>g_{p} (identity again). I then have G3 satisfied.
So, g_{r}∨g_{p}&e_{r}=>g_{s} exemplifies g_{p} + g_{r} = g_{s} in the octal because e_{p} + e_{r }= e_{s} and e_{r} must remain "God-like" and at rest on the right hand side when it is extant.
And simply by noting that by the same principle, g_{r}∨l&e_{r}=>g_{r} and also l∨g_{r}&l=>g_{r} also. "l" is the identity.
I should note I almost never find g_{r}∨g_{r}&e_{r}=>g_{r} represented by g_{r} + g_{r} = g_{r}, for this is only valid if g_{r} = l. g_{r}∨g_{r} is otherwise as above. (g_{r} would never be that virtue required by the uniqueness condition of the axiom of virtue.)
and as "l" = 0 in effect, I find the ultrafilter is always intact: g_{r} v l&e_{r}=>g_{r}. Therefore in that filter l&q^{-1}=>e_{r}, why?
If I have q_{}∨ l&q^{-1}=>e_{r} with q=>r&u^{-1}, I have with the right side extant, l&q^{-1}=>e_{r} equal to l&e_{r}&e_{u}=>e_{r}. Now, if r + u = (r&s)^{-1} I could put (r&s)^{-1} for "l". Instead, by limiting myself to the one K4 group in p + r = s, with q=>r (and not u^{-1} also) I may legally write l&q^{-1}=>e_{r}, as desired. Then, finally, as this relation is unique by axiom of virtue, any "r" may be extant on the left hand side as if "q". (I had actually conjoined two disjoint sets of virtue in "p" by forming q=>r&u^{-1}, one set of virtues for "r", the other for u^{-1}.) I also finally note that q^{-1} is negative, and is not exemplified by perfection (as it is didactic). Only e_{r} remains as if "God-like".
Then the result is simply as r&e_{r} _{}∨ l&e_{r}=>e_{r}.
With regards to binding, what is l&r^{-1}? Clearly any virtue - a set with principal element e_{p} - is missing altogether; there is no such binding. With regards to the octal's operation with identity "l", l + e_{r} = e_{r}. For all other q or "r" in r^{-1}, at worst I have l + r^{-1} undefined, at best I must have by uniqueness of virtue x = _{}e_{r} in the relation l + x = e_{r}. As "l", e_{r} etc., are all principal elements, they are "always" present. Under multiplication by the seven cycle, it is possible that "r" as well as e_{r} is then mapped to the set of virtue in "p" with e_{p}, and is therefore found to be just as unique (it is already a wholly disjoint set of positive predicates).
Then given l&r^{-1}=>e_{r}, if no other predicate is extant, by that same uniqueness r^{-1} is at least e_{r}, and every other term in r^{-1} could result in nothing new.
In the new name the disjunction appears r&g_{r} _{}∨ g_{r} as if extant either side; and this is also effectively r&g_{r} _{}∨ l in its "bare form" before that is considered. By the latter there is modal collapse extant on the right hand side (after the arrival of the least and the "reign of omnipotent God" begins).
This is intuitively reached, as to the result of deciding two connected worlds, one in all "r" and one in "l" alone. That one with "l" extant on the right hand side then has the new name, the left therefore would not. If the left side is extant, it must be in the new name with g_{r}, and not simply e_{r} (a contradiction after the least's arrival - as if one dependent on timing). Then the result is a modal collapse yet again, and "r" may be "dropped" as simply equivalent to the principal element g_{r}. Then the operation is, once again, a union.
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