Is enlightened cognition more than its parts?

Mahayana-Nagarjuna philosophy states everything is conditioned & comprised of parts. Original Buddhism does not. The impression is your question arises because of illogical philosophy, which makes the philosophy sound contradictory to you. The original Buddha was fully enlightened thus explained reality without flawed description. Enlightenment leads to the permanent end of suffering. Since enlightenment 'uproots' the causes of suffering, there is no longer spiritual work to do. So the 'parts' either end (in redundancy) or, otherwise, are permanently established.

From the flow of your words, I think that you may be confusing enlightenment with emptiness. Both can be confusing, but I will try to clarify each. According to Khenchen Konchog Gyaltsen Rinpoche, enlightenment is the ultimate achievement of buddhahood; and emptiness is the lack of inherent reality of a phenomenon or person. I have often heard lamas teach emptiness as the breaking down of phenomenon in their parts in order to establish that it does not exist as and of itself. In other words, we see a mundane table and we believe that it has an inherent existence, but the table is made up of legs,top,nails,screws,glue...not fixed, but constantly changing from moment to moment from the atomic level up to the fact that it ages and rots. All is impermanent. Also, in the mundane world one would look at the collective parts as having a practical value as they appear to make the table. However, within Buddhism we see all things as dependently arising; all is depend for its existence upon something else, cause and effect, which is karma. Ergo, the table is dependent upon its parts to make a table. A person is dependent upon karma to make a life. these principles can be applied to all that we perceive in this world.

When it comes to explaining enlightenment, I do not feel that I can explain it any better than Lama Surya Das: "Enlightenment is not about becoming divine. Instead, it is about becoming more fully human.In examining the archetypical experience of the Buddha, we see that his enlightenment represents the direct realization of reality---how things are and how they work. Enlightenment is the end of ignorance. when we talk about walking the path to enlightenment, we are talking about walking the path of an enlightened being."

If I have misinterpreted your question, please kindly rephrase the question.

I am going to break this down for clarity in my own mind, if not in the minds of others. By definition "enlightened cognition" means that one is "freed from ignorance and misinformation" in order to pursue "the mental action or process of acquiring knowledge and understanding thought experiences and the senses" (quotes from a couple of dictionaries). So to paraphrase, is the acquired knowledge and understanding of wisdom, in any Buddhist tradition, more than its parts? Wisdom, is insightful understanding. Simple put the question seems to be, is wisdom more than its parts?

Alright, now let's add the comment's reference to "direct realization of reality", which from the Buddhist view means Ultimate Reality, the direct realization of emptiness which leads to the end of suffering completely which is enlightenment, or buddhahood.

Both Theravada and Mahayana Buddhism translate Prajna and Bodhi as Wisdom. In the deeper sense, Prajna is insightful understanding; and Bodhi is the perfection of wisdom, or the teaching that all phenomenon are empty of self essence, which ties back into the direct realization of reality. This also takes care of the "in any Buddhist tradition" part of the question.

Is wisdom more than its parts? In the mundane world it is not, since one accumulates pieces and parts of wisdom, conventional wisdom, without sensing a higher pursuit. However, upon the realization of Ultimate Reality, the parts cease to play a role as they too become impermanent through emptiness.