The best koans are those that give us direct insight into SupraRational wisdom. There are two limits to the Divided line of Plato. One is Contradiction, Paradox & absurdity which are differentiated into levels of striation of the intensity of illogicality. And then there is the single limit of the supra-rational which is unstriated. SupraRatonality is wisdom about the non-dual nature of existence. Basically all the exquisite Koans point to this non-conceptual non-experiential condition based on the interpenetration of all things which is possible due to their emptiness. The sublime Koans are those that take you directly to this state with the proper preparation as a transmission from mind to mine of enlightenment. The precious Koans are those that occur naturally, are not fabricated, are not elaborated, but arise of themselves from the context in which the transmission is made between enlightened beings (one knowing the nature of enlightenment and the other ignorant of it, until the indication is made, and recognized, and embodied utterly. These Koans are sui generis events of the happening of enlightenment as transmission. They are not thought up, and they have no interpretation. They merely exist, nonconceptually, nonexperientially, and unconfined by the limits of language. They are indicators within the total context of the enlightenment situation pointing directly at the suprarational nature of our existence as human beings within the context of interpenetration of everything with everything else based on the differences between things.
The gateless gate describes them perfectly, because they are indications that there is no gate. They are not a gate, but the recognition of this limit of experience and reason which is impossible to transgress called nonduality. Nonduality meaning Not One! Not Two! Not Many! but something else outside of the combinatorics of our logical possibilities as defined by the tetralemma. But because the tetralemma with its logical connectives are themselves empty, the tetralemma itself exemplifies this emptiness via the discontinuities between AND, OR, NAND, and NOR. Nonduality is an impasse. And the good Koans point out that Impasse. The better Koans ARE that Impasse. The best Koans show you that you are that impasse. In other words the truth of enlightenment is that there is no enlightenment other than ordinary mind which is already enlightened utterly. Koans facilitate that natural state of perfection to surface naturally within the total context of he transmission situation. The impeccable Koan is the one that is a gesture of recognition of the embodiment of enlightenment not in the self (because there is no self) but of the total aggregate that is entwined by dependent origination. As such nothing arises and nothing ceases. There is only the adamantine state of utter emptiness of all phenomena. And it through the Koan that we express our comprehension of that spontaneously as a rare event of realization, which is the manifestation of enlightenment in all sentient creatures including ourselves. The Bodhisattva includes himself last. But because of that he becomes the leader pointing the way to the pristine state of things just as they are.
The perfect Koan expresses our insight into the fact that enlightenment itself is a ruse. There is no enlightenment, and it is the realization of that which gives us enlightenment. That is if we devote every shred of ourselves to the pursuit of enlightenment, then when we realize that it is something non-existent to be we are brought to an impasse where we realize the nihilistic nature of our worldview, that we embody that nihilism, and by pursuing enlightenment we have so identified with the path of the bodhisattva that when we realize it was a ruse we played on ourselves, the self like a ship sinks in the sea of wisdom (prajna). That is why emptiness is in fact empty. but the emptiness of emptiness takes us back to the beginning to see ourselves and the world anew, as interpenetrating empty existences that are non-dual and supra-rational from the beginning. That never changed, and never will change even though existence itself is in flux of aggregates dependently arising throughout our experience. The magnificent Koans are those that speak to us across time and take us into the situation of transmission, and by that we ourselves become part of that transmission, and we gain wisdom via the embodiment of the supranational in our language as an indicator of our inner insight and realization of the nature of all things including ourselves.