Are the positive qualities of enlightenment ultimate or conventional realities?
permanence, bliss, personality, purity.
If they are ultimately the case, can they be directly seen?
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Enlightened qualities are ultimate. One's mind is innately ultimate ie. enlightened.
A highly achieved buddhist practitioner often likens the enlightened mind with the clear blue sky.
Often clouds (defilements) obscure the clear blue sky and the clear sky cannot be seen. In other words, without the clouds, the clear blue sky (the ultimate) can be seen.
'Ultimate realities' (paramattha dhamma) in original Pali Buddhism means things that are always true. For example, that for suffering to occur, there must be craving & attachment as causes, is an ultimate reality. Suffering cannot occur without craving & attachment.
Conversely, the general truth that good karma leads to good results & bad karma leads to bad results is not an ultimate reality because this is not always the case. For example, good karma (eg. loving one's family) had lead to suffering (eg. if your family die) & bad karma (eg. hurting another person) can lead to enlightenment (since one learns to never perform that action again).
Convention realities can also refer to ideas such as 'man', 'woman', 'hot', 'cold', 'American', 'Chinese', etc, which are not always 100% or inherently true or fixed.
Therefore, full enlightenment (rather than partial enlightenment) is always permanent, blissful & pure for the entire lifetime of an enlightened being. This is an ultimate truth, which, naturally, can be directly seen by the fully enlightened being.
As for 'personality' or 'self', this is not an ultimate reality in Buddhism.
However, if 'personality' is taken to mean 'character' or 'mental qualities', yes, the virtuous & enlightened character of a fully enlightened being will have a permanent quality.
When the Buddha was old, it is reported he said:
I am now old, aged, burdened with years, advanced in life and come to the last stage: my years have turned eighty... Sariputta, even if you have to carry me about on a bed, still there will be no change in the lucidity of the Tathagata's wisdom.