"I think; therefore I am".
According to the trillema fallacies (Munchauseen, Fries, Agrippa, Albert) "I" is the point of awareness in which we assume reality.
It is a self referential statement where self referentiality necessitates creation of multiple contexts of "I". The self looking on the self observes multiple selves where the I progresses to another I.
This can be observed where "I" think is one context of self referencing:
And "I am" is another context of "I".
Each I, as a point of awareness (using Sextus Empiricus version of the trillema (all things change as there relations change)), effectively is the I viewing itself from different angles.
It is creating itself through the recurssion of contexts:
(I --> I) <--> ((T)-->(A))
In one respect "thinking" is a context of "modality" the "I" exists through.
In another respect "am" follows the same course.
Each modal state, where "think" and "am" are both very general modes, effectively is a sub context of I itself, thus is an extension of "I".
With each new context I exist in comes a new progressive I, with I being purely assumed as a point of assumption and the following contexts following the same course.
The repetition of these assumptions forms a tautology that can neither be proven or disproven. So yes you are right. But where Descartr fails is his ability to point out "I" is assumed, thus the contexts stemming from "I" are also assumed.
The pyrhonists (western variation of the buddhist, see wiki page) observe this.
Existence thus is grounded, through the I, as the recursion and isomorphism of assumptions.
Isomorphism is the inversion of one context:
Into a symmetrical opposite state (ie many contexts)
Which again inverts
Thus one assumption, "I" is an empty context which is defined by its continual progression to further contexts that as assumptions through the original assumption are inherently "void".
This inversion of one to many requires recursion and with recurssion necessitating isomorphism. It's a loop, showing the nature of assumption as fundamentally "void" negating itself into "form".
Wittgenstein "tautology of tautology" statement, most every philosopher is loosely aware of, is a variation of descartes because of these principles alone.
This recursion/isomorphism is also observed in Euclid's first postulate of a line as the distance between two points. One point, inverts to another point (many points) and the recursion is defined through the line as a Directional quality. One "I" is directed towards itself resulting in two "I".
This connection of the "I", as the most base of assumptions, is the first act of definition in thought. Thought is thus the projection and connection of assumptions that have a recursive nature expressed through "the line".
One assumption individuates (divides/multipies simultaneously) into another assumption and this always occurs through recurssion thus necessitating the basic "line" as the objective form or subjective understanding.
Thus self reflection is grounded in spatial axioms, where Platos Justified True Belief as an expression of the forms, is really the objectification of the subjective nature of space that kant observes.
In these respects Descartes claim of "I think therefore I am" is really Justified True Belief from a seperate angle of awareness. With belief being the assumption of I, "thinking" as the justification through definition observed by the first recursion of the I, and "Am" observing the "thought of thought" as a "dualism of dualism" of "I".
Fundamentally "I think therefore I am" observes 4 different points of view based upon the recursion argument presented above. This corresponds with the "axiom of maria" (Google this because I want you to research it) Jung promotes inherent within the process of individuation.
This process of individuation is what Descarte is actually philosophizing about and presents roots for young.
I(X)=I(Y) still necessitates as base I=I that is rooted in aristotelian identity properties. Thus descartes statement, while a tautology has aristotelian roots and the act of self reflection takes on a form of "self measurement".
This self measurement, with measurement being the foundational meaning for the Greek "arithmetica", is fundamentally mathematical in nature. Math being the quantification of reality through the individuation of reality, we all reality being assumed fundamentally through the "I" as void in it's own context.
The I is fundamentally assumed, and as assumed it is void in such a manner it is no different than a dot considering the most apriori observation that exists is the Dot as "self". It is a universal symbol among all cultures...why if not apriori?
How is this different than the "I" directed towards itself under a self assumption considering the subjective nature of space according to Kant?
Even time as subjective, is observed through Heideggers conception of Time, as universally expressed by two points and a line (thus necessitating Euclid's first postulate as the grounding of what we understand as finite).
Thus "thought" as the first context the "I" is created through, by descarte", is the creation of definition where the I as subjective is fundamentally finite and subject to it's own perception of time which stems from itself.
Self reflection thus requires an inherent subjective state of time and space as each I is a point of awareness that occurs by recurssion
Considering Kant's premise of space being primarily subjective, euclids first postulate is descartes axiom given a platonic form. But this platonic form of the line is merely the objective viewing of how the subjective operates.
Even Jung's process of individuation, expressed in Descartes "I...therefore I", necessitates a form of evolution that is protectively linear in nature. This is further reflected in aristotelian actuality and potentiality, where being takes form (ie "volume") through the dynamic movement of formless potentiality (ie mass).
Thus "I" actualizes through the "thought" descarte observes, where the "I" as previously "void" or "formless potential" exists through a basic recursive self measurement embodied under the "line" as a universal form for the subjective connection and seperation of assumptions.
Thus we can see how "I think therefore I am" can be viewed as the evolution of assumption through a multigenerational recurssion and isomorphisms of "I's" that stem from prior to Euclid all the way to the present.
"I" thus can be implied as a Jungian Zietgeist under certain respects.
So descarte, as the premise point you are observing, can actually be viewed as given definition to the nature of consciousness and how we assume through basic recursion and isomorphism.
Hofstastder's book: "Godel, Escher and Bach" and "I am a strange loop" explores this nature of isomorphism, recurssion from another angle as well.
So this should help in viewing Descarte from different perspective "I"'s by observing it through different philosophers.
I may have to clarify some points...and ignore the typos from the ipad.