Since we study the theological system expounded in TriTrac in the commentary, following the systematic layout of the treatise itself, only a few words are necessary on the subject here.
Briefly summarized, TriTrac explains how the Father, who is One and who existed alone, desired to be known.
By this act of will the divine substance was externalized so as to become a congregation of autonomously existing personal entities with cognitive faculties.
However, knowledge and perfect existence are not granted the aeons from the beginning; these are goals to be attained through a process of education and formation.
This inherent imperfection gives rise to positive deficiency through the presumptuous anticipation of the goal by the last and least advanced aeon, called "the logos."
A rupture takes place within the logos: his presumptuous part is cut off and remains outside the world of the Pleroma, while his perfect part reascends there.
From the "thought of presumption" originate demonic powers of passions and vices, essentially material in character.
The logos himself, cut off from the Pleroma together with the offspring of his presumption, condemns his previous desire, is converted and remembers the Pleroma, praying to the aeons for assistance.

This second disposition, and the prayer, become another order of powers, which is psychic and which combats the material one. Then the Saviour is sent forth from the Pleroma as an answer to his prayer, and manifests himself to him. Through this vision the logos is illuminated and formed, and becomes capable of spiritual offspring, brought forth as a thanksgiving prayer of the logos after the image of the Saviour and his angelic retinue, who themselves manifest the forms of the Pleroma.
The logos proceeds through the medium of a Ruler of all the psichic powers, to shape the world, which becomes a structure composed of the material and the psychic powers and substances previously emitted, while the logos and his spiritual offspring form an aeon in the "Middle" between the cosmos and the Pleroma.
Man likewise is created as a mixture of the material and the psychic and with a third element deriving from the logos himself.
In the world there exist different categories of men professing varying opinions about the nature of the cosmos, in accordance with and inspired by the powers, the Greeks and barbarians belonging to the material powers and the Hebrews to the psychic ones.
Finally the Saviour is sent down to earth, assuming as his body The spiritual offspring of the logos, who thereby become incarnated as a spiritual Church in the world.
The purpose of their incarnation is that they shall be trained through living here below and receive the redemption through the ritual of baptism. So as to be reunited, together with the logos with the Pleroma, where the final unification now takes place. There does not exist a singular key to the understanding of the system of TriTrac. On the contrary it is essential to realize that this system, as indeed Valentinian thinking as a whole, combines several modes of thought deriving from disparate religious and philosophical backgrounds.
From one point of view TriTrac represents systematized salvation history in the Jewish-Christian sense. It provides an account of a process which unfolds itself in the medium of time and which encompasses the entire history of the world as well as an elaborate "prologue in heaven."
The telos of this process is the education and successive formation of the children of the Father towards their perfect Being and their complete knowledge of him.
A central concept in this context is that of the Father's will; it is his will to be known, but it is also his will that this take place through a process of gradual training and growth.
Thereby the Father has also willed the condition which made the fall possible. Moreover, the actual occurrence of the fall was in accordance with his will as well; it was necessary in a sense, although the text does not
explicitly define this necessity (which belongs to the level of philosophical interpretation: see below).
The creation of the world was also in accordance with the Father's plan: it is an instrument for the education of the spiritual seed, who receive in it the preparation for their acceptance into the Pleroma.
Closely allied to the concept of the will is that of providence, which indicates that the events of the salvation history take place according to a preconceived plan of the Father. In this context belongs also the term oikonomia, which in TriTrac, as in Valentinianism in general, has the specialized meaning of "the world" in its restricted spatio-temporal totality as a precalculated phase in the realization of the Father's plan for salvation, administered by lower powers who are themselves ignorant that they act only as instruments of a greater design.
From a different point of view TriTrac contains a system of physics in the philosophical sense.
The conceptual framework of the treatise is constituted by! the opposition of oneness and plurality. While the Father is One, emanation, although willed by the Father, implies plurality, and unlimited plurality at that.
This unlimitedness is epitomized in the presumptuous thought of the logos, who as a singular aeon attempts to grasp the Father, whose onenes is also an infinitude transcending the particularity of the individual aeon. But the fall fulfils a necessary function in the process of emanation, for through it unlimitedness is cut off from the Pleroma and a Limit is imposed upon it, which makes possible the conversion of the Pleroma towards the Father which is effected by the Son.
The evil aspect of plurality which now has been removed from the Pleroma, represented by the presumptuous thought, now expresses itself as a multitude of powers constantly struggling among themselves, their mutual strife and discord constituting the essence of matter.
For the logos this state of affairs implies a condition of passions and sufferings, from which he attempts to liberate himself through his conversion and prayer.
The vision of the Saviour brings about this liberation, and the state to which he then attains is characterized as rest and oneness of mind. Similarly the spiritual offspring which he now brings forth have an aspect of unity, but nevertheless do not possess the oneness of the Pleroma.
In the world of men the thinkers among the Greeks and the barbarians reflect the disruption and strife of the material powers, who inspire their thoughts, whereas the Hebrew prophets, who like good psychics are attentive upwards, all proclaim the same message, which derives from the spiritual region of the logos.
When the Saviour descends together with the spirituals he effects the final unification by being a single person in whom all the spirituals may participate at their redemption; the apokatastasis being a return to the initial oneness. This pervasive thinking in terms of the opposition of oneness and plurality is attributable to the influence of the Old Academic opposition of Monad and Dyad, as transmitted through Neopythagoreanism, where these two principles were first conceived in such a way as to form a monistic theory of emanation.

In my commentary I have attempted to show that such concepts as "extension," "Limit," "cutting off," "presumption" etc. belong within this tradition.
The logos of TriTrac, and Sophia in other Valentinian systems, in many ways correspond to the Dyad as the principle of unlimitedness inherent in emanation, and as the origin of matter. However, the logos (and Sophia) also possesses essential traits of the Platonic Soul (with no fundamental distinction being made by the Valentinians between the World Soul and the particular soul), in particular in the account of the fall and with regard to demiurgic function.
In terms of a hierarchic arrangement there is: 
(1) a supreme god, qualified as One as well as Good, and a
transcendent world, which does not constitute a level of its own, as in Plotinus, but which is the Father's thoughts, as in Middle Platonism although in a! dynamically conceived way which combines the Pythagorean notion of the Monad as potentially containing all numbers with the Stoic theory of the double logos;
(2) the region of the Middle, the aeon of the spiritual logos, corresponding to the Ogdoad where Sophia dwells according to other Valentinia' sources; and 
(3) the cosmos, which is composed of matter and soul and ruled by powers of either material or psychic nature, the demons of the philosophers arranged a hierarchic scale, one Ruler being placed over all the others.

The Saviour, who separates the material and psychic substances, and also manifests the forms of the Pleroma.

The Logos, who brings about the actual cosmic arrangement, in accordance with the model manifested by the Saviour.

The Ruler, corresponding to "the Demiurge" in other Valentinian systems, who is the instrument used in creation by the logos, but who in addition creates on his own account as well.

From a third point of view, which is also essential for the understanding of Valentinian thinking, the system of TriTrac is what may be called mysteriosophy, or mythology with a sacramental basis. Valentinianism is of course a religion, promising salvation through ritual acts, and the meaning of these acts is expressed conceptually through the system.
Therefore the Pleroma, although philosophically akin to the intelligible world of the Platonists, is called "Church": it is also the ideal, mythologically hypostasized community of the Elect.
This is also the background on which it becomes understandable how the most superior form of cognition of the aeons is the singing of hymns, and how the metaphysical concept of oneness is realized through the harmony, or consent, of the communal psalmody.
Furthermore, the emanation process itself, conceived as a generation from within the Father, is to be interpreted not exclusively in terms of current philosophical emanation theories, but also as reflecting sacramentally realized regeneration.
In this context it should be noted that such a term as "formation", in addition to the significance it has within the salvation historical outlook on the one hand, and Platonist physics on the other, also possesses sacramental connotations, being semantically closely related to "illumination".
It should also be pointed out that the myth of fall and restoration, while constituting. on the macrocosmic level. A cosmogonic theory, also provides the paradigm for the condition of the individual, and for his way to salvation.

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