What do one-thought ignorance and beginningless ignorance mean?

One-thought ignorance (āvidyā), as the name implies, means that one's thoughts and views are ignorant and defiled. Why are they ignorant and defiled? Because there is a "self" in the mind, taking the five aggregates (pañca-skandha) and the eighteen realms as self and what belongs to self, not knowing that this so-called self is suffering, empty, impermanent, and non-self. All of one's thoughts and views revolve entirely around self and what belongs to self, firmly grasping this so-called self; thus birth-and-death is also firm. When one's bodily, verbal, and mental actions are guided by the view of self, of course body, speech, and mind are all defiled karma, and from this defiled karma, the cycle of birth-and-death inevitably turns on without cease. Therefore, one-thought ignorance can continue all birth-and-death, and liberation cannot be attained.

However, one-thought ignorance is also a kind of beginningless ignorance (anādi-avidyā); it has existed throughout beginningless kalpas, with no point of beginning. If there were a point of beginning, then before that time there would have been no one-thought ignorance, and sentient beings would be arhats, abiding in Nirvāṇa without remainder; in nirvāṇa there are no sentient beings, nor any arhats. All ignorance has existed innately since before beginningless kalpas; it is all beginningless ignorance, and it is only through the conditioning of later circumstances that beginningless ignorance increases or decreases.

After one-thought ignorance has been completely eradicated, there is still beginningless ignorance that must be completely eradicated before one can become a Buddha. Beginningless ignorance means that sentient beings, throughout beginningless kalpas, have always taken the world of the five aggregates to be real, not knowing that it is an illusory appearance conjured forth by the tathāgatagarbha, not knowing that it is empty. From this arise the grasping of self and the grasping of dharmas, and thus segmented birth-and-death (pariccheda-maraṇa) and transformational birth-and-death (parināma-maraṇa), with no liberation. To eradicate beginningless ignorance, one needs to investigate Chan (cān chán), personally realize the tathāgatagarbha, personally realize the true suchness nature (tathatā) of all dharmas, and step by step realize the nature of the one true dharmadhātu. When all dharmas have been realized as possessing the true suchness dharmatā, one will completely transform consciousness into wisdom and accomplish the Buddha Way.