In the book the octal group is used as an ultrafilter. Or rather, a set of symmetries of positive properties that themselves form an ultrafilter. Every subgroup of the octal is a closed group: but how does this relate to the notion of an ultrafilter? It must be shown (which was not done so very clearly in the book), that every application of virtue or of a positive property indiscernible from some virtue, operates in a triple that is likewise a closed set. Yet this is not immediately apparent!

The Problem With Closure
The greatest problem with applying the octal as an ultrafilter is the local closure of triples that are represented by subgroups. Each triple appears to not be closed on its constituent three symbols, and each triple always references the properties outside of themselves. Can I remedy this?

In The Case Of Virtue
It is a simple argument that closes each subgroup of the octal given that they include a virtue p. Those three subgroups together form the "K4 form" from the same application of virtue. But the group of all positive properties is somewhat larger and so given the case in terms of virtue, I may hopefully complete the argument on mere positive properties also.

The Debt Of Love
The metaphysics of the book has a straitforward symmetry, that both sides of a disjunction in virtue follow from the very same virtue. I examine that here, as it sounds counter-intuitive.

Given All The Facts
Every subgroup in the octal obeys the same similar pattern. Each triple as a K4 subgroup is closed in positive properties. There is a simple argument that shows such, for which I may only apologise as it was not included in the book!

Local Closure In The Octal
The octal ultrafilter is actually completely "unconstrained" in all its positive properties. That, is a consequence of the construction used! Though some positive properties appear missing they are most certainly not so. God is able to act in every case or not: there is only the sense of a logically impossible act that can not be made given virtue: It is not a hard thing to grasp.

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