This is how I understand.
After parinibbana:
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The Arahant or Tathagata is not a person or an object or something that we can identify. They have no name for us to call or identify. We cannot refer them to anything, so there is no way to identify them. We cannot refer them as ArahantX, TathagataY, or Arahant who is always happy or the Arahant who lives or used to live at such place, or the Arahant who was Sariputta…
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The Arahant or Tathagata is not the five aggregates, but they are also not something apart from them.
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The Arahant or Tathagata does not take anything as “This is mine, this I am, this is myself.”
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The Arahant or Tathagata is not a consciousness or stream of consciousness.
Therefore, to my understanding, Arahant or Tathagata is the remaining when nothing can be referred as “This is mine, this I am, this is myself.”
Just like the “I” that we often use to identify ourselves. The “I” does not exist by itself alone, but it exists when the mind takes something as “This is mine, this I am, this is myself.” We can think it in term of a pointer. When a pointer points to something, it becomes that object and has all the properties of that object. If it does not point to anything, it is meaningless.
Since we can see the existence of the “I”, we cannot say that the “I” does not exist. Since we can see the non-existence of the “I”, we also cannot say that the “I” exists. Same for Arahant and Tathagata.
After parinibbana, the purified mind-base of the six senses is released from the body (can be inferred from MN43.)
This purified mind-base is not a consciousness. It will need its objects for mind-consciousness to arise (it is a base just like an eye).
This purified mind-base can function as the six senses (See MN43), and it is not limited by a body, so it has no limitation.
Since it is purified, It does not take itself as “This is mine, this I am, this is myself.” Therefore, we cannot say it is the ArahantX or the Tathagata.
When this purified mind-base takes no object. That’s what we called Nibbana or the ultimate refuge where Mara cannot see or reach. When it takes its objects, it will take them with wisdom knowing the danger and the escape from them. When it takes its objects, six senses consciousness arise, feeling arises, perception arises, volitional formations arise. Therefore, it is not apart from the five aggregates. When it does not take any object, it is not the five aggregates. We can see it is acting like a pointer! (We can infer from examples in MN25.)
We cannot say that purified mind-base is the ArahantX or the mind-base of ArahantX because by doing so, we are trying to objectify something that cannot be objectified (or we take the object as the pointer while the pointer is not an object!)
We also cannot say that the ArahantX is that purified mind-base just like we cannot say the pointer is an integer number 5 when it does not point to anything.
I think this purified mind-base is some kind of pure energy which has no beginning and no end, and it is indestructible. The defiled mind-base is what carries a person from life to life. It is not consciousness, or stream of consciousness, and it is not permanent since it can be changed from a defiled mind-base to a purified one, and it is indestructible.