The brain is material form (rūpa dhamma) and cannot store memories, because memories are dharmic dust—formless and without appearance, having no intersection with the brain. Moreover, the brain arises and ceases; if it stored memories, after the brain ceases, nothing would remain, and future lifetimes would be unable to continue all dharmas from past lives. Even with psychic powers, one would not recall matters from past lives. All dharmas are recorded and stored by the Tathāgatagarbha (Buddha-nature), which provides them when needed. Dharmas originate from the Tathāgatagarbha, operate, and then return to the Tathāgatagarbha. When needed again, the Tathāgatagarbha can manifest them once more.
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