Siddhis, or perfections. These are supposedly magical or supernatural powers that confer extraordinary abilities on a yogi who achieves them. They are only manifest at the highest levels of development.
As I've explained in some detail (readers who want the details can check the enneagram resource for them) the enneagram represents a fractal structure, in which each note of an octave represents the "do" or first note of an octave below it. What this means is that the attributes or names of God — which corresponds to Gurdjieff's "centers" — each have a complete octave of development proper to themselves.
Each note within an octave affects the octave only to the extent to which it is developed in its own octave. So, for example, if we look at desire, it might develop only to the level of re, or 1, which represents urge, an instinctive or animalistic desire, a material desire. In this case, that is the highest influence the note mi could exert on the octave above it; whereas, if it developed to the note sol, it would have what is called a real wish, or aim, which is associated with Being, or real "I". In this case, desire would exert a much higher influence and have a proportionately greater effect of expression in a man's life and work.
Unfortunately, it's a very rare for desire to develop to this level. Under ordinary circumstances, even though we may think we have a wish, it is always attached to one of the lower manifestations or attributes, that is, urge, lust, or passion. Each one, under the right circumstances, may be appropriate to a particular situation; and yet in a well ordered Being, the appropriate level of desire is always applied to the appropriate circumstance, and man as he is is not generally arranged in that way. Al 'Arabi explains this quite specifically in his work The Divine Governance of the Human Kingdom, even giving a clear indication of what Gurdjieff would have called right work of centers.
The point is that while we need every iteration of every note both in the superior and subordinate octaves to function properly, they need to function according to right action. In the state we are currently in, we are constantly trying to use an urge, lust, or passion in the place of what Gurdjieff called wish, which is a conscious influence. Real Wish can, logically enough, only arrive with Real Being, which is why they are so closely associated in the prayer for the first conscious shock: I am, I wish to Be.
It's interesting that although it's vaguely understood, in inner work, we must surrender our wish, it is not understood that there is a specific reason for it, which you can see quite easily when it is detailed in the enneagram in this manner. This is why Gurdjieff said the diagram was so useful for men who understood how to use it. Of course we don't have real wish! The situation is lawful. Once we admit that to ourselves, perhaps we can begin to let go of some of these precious wishes we have, all of which are attached to lower influences — very useful and quite necessary under the right circumstances, but otherwise, completely and absolutely an obstacle.
It is in the action of discrimination — that is, understanding what is necessary in these matters — that a man orders his inner kingdom.
The given iteration of desire is but one example. If desire was developed to the highest levels — which are very difficult to describe in a diagram, let alone words, so I have approximated as best I can — what one would get would be a Siddhi, or perfection. That is, a note in an octave developed to the level of what al 'Arabi called Knowledge. But there are other principal attributes. Thus, the six absolute perfections available to the Siddhas are related to the six principal attributes of God, that is, perfect material, perfect desire, perfect power, perfect being, perfect purification, or perfect knowledge. Attaining any one of these levels would represent an absolutely extraordinary individual with what we would consider to be magical powers; yet only a yogi who completed all of the notes in all of the octaves would reach to the highest possible level; and that is almost unknown.
This phenomenon only takes place when the energy from the absolute is sent backwards through the diagram, that is to say, in our own words, God becomes incarnate. Action that begins from this direction can perfect all the Siddhis automatically, because the energy that influences the development has an unerring understanding of every level, beginning from the top. So, one sees, the diagram also explains exactly why Christ and Buddha were able to achieve what they did. Again, their action was not mysterious or supernatural, but entirely lawful.
One should remember, over and over, that the hierarchy depicted by the enneagram does not indicate that one note is "more important" or "better" than another. In the system, although there is a hierarchy, and influences beget and affect one another, the essential unity of the system itself cannot be contradicted simply by favoring one element over another.
The post would not be complete without a mention of the fact that different systems propose different Siddhis. (See the Wikipedia link above.) While one could have lengthy arguments about this point, because of the basic principles embodied in the enneagram, we can understand that there can only be at most eight principal Siddhis: six related to the notes in the octave, one related to conscious labor, and one related to intentional suffering. The six which are related to the actual notes, I must insist, must be in direct relationship to the six principal attributes. Some of the traditional Siddhis from the Hindu tradition do indeed appear to reflect that; but sources that attempt to analyze or understand this without a fundamental grasp of the cosmological principles will definitely end up with a confusing array of definitions – which is exactly what we see, overall. Once again, the enneagram provides us with a succinct organizing tool. We might call the fundamental perfections "objective" perfections.
How these objective Siddhis might manifest as a matter of conjecture, because their actual presence is so rare. Everything that we read about them is hearsay and probably not second or third hand, but passed down through one hundred sources before it reaches us. One thing that is accurate is that, of course, each principal siddhi has subsidiary siddhis under it, which represent varying constituent aspects of its nature.
One peculiar consequence of this is that there are perfections of this kind at every level of the universe. In other words, some perfections are quite ordinary and even familiar to us, because they are appropriate to this level. And this — for the astute reader who contemplates it — will bear a great deal of fruit to ponder in relationship to the last post, because the entire nature of the manifestation of what we call reality is folded into this question, which is not actually separate from any other question.
I respectfully hope you will take good care.