What happens to grasping at 'self' after termination of life?

My question is: in the described case of person X, will grasping stop after X's dead material body is put in the coffin and buried in the ground?

I sense this 1st question falls outside of the sphere of Buddhism because in the suttas I am not aware of any teachings about "grasping" & "rebirth". The teachings about "rebirth" are about good & bad kamma, which are mundane/worldly (refer to MN 117). Where as "grasping" falls into the supramundane/transcendent teachings. For example, in MN 144, it is said the monk that has "grasping" prior to the termination of life is "blameworthy" rather than is subject to kammic inheritance. The common idea in the suttas intepretated & translated as "rebirth" or "reappearance" is about kammic inheritance. These teachings assume there is grasping because these teachings refer to "a being" ("satta") that inherits kamma. The word "satta" literally means "grasping" because the word "satta" itself means "to cling" (refer to SN 23.2).

In summary, a person or mind that has comprehendeded "grasping", per the instructions on the 1st sermon, should abandoned the cause of that grasping (i.e, craving) to reach Nibbana. Where as the teachings about "kammic inheritance" (aka "rebirth") are for puthujjana who have not comprehended grasping.

Any mind that has truly comprehended the nature of grasping (the 1st noble truth) must truly comprehend the other noble truths. This shows the question here is merely intellectual.

For Z the incoming termination of life is liberating because of freedom from grasping.

The suttas list many types of liberation of mind (ceto-vimutti) that occurs to a living mind, such as:

The immeasurable (loving-kindness) mind-release, the (perception of) nothingness mind-release, the emptiness (sunnata) mind-release, the theme-less-mind-release... MN 43

The Z mind (not the 'Z person') was liberated before the termination of life. At the termination of life, there is no liberation because there is no mind, as follows:

'... after the termination of life, all that is sensed, not being relished, will grow cold right here.' MN 140

The term 'liberation' in Buddhism is liberation of mind (rather than liberation from mind). At the termination of life, there can be no liberation of mind because there is no mind anymore.

My question is: in the described case of person Z, will grasping re-arise after Z's dead material body is put in the coffin and buried in the ground?

The mind free from grasping views the termination of life as merely the ending of the aggregates. Please refer to SN 22.85 here, which answers the question conclusively.

Previously, my friend Sariputta, I did foolishly hold that evil supposition. But now, having heard your explanation of the Dhamma, I have abandoned that evil supposition, and have broken through to the Dhamma.

Then, friend Yamaka, how would you answer if you are thus asked: "A monk, a worthy one, with no more mental effluents: what is he on the break-up of the body, after death?"

Thus asked, I would answer, 'Form is inconstant... Feeling... Perception... Fabrications... Consciousness is inconstant. That which is inconstant is unsatisfactory. That which is unsatisfactory has ceased and gone to its end.

Very good, my friend Yamaka. Very good.

SN 22.85

The question is whether (in each case) grasping will stop or re-arise.

So far as I understand it:

• Aggregates arise and cease
• Aggregates may cease here (death) and re-arise there (birth)
• Generally clinging-aggregates are dukkha (says the first noble truth)
• View of 'self' (identity views) are dukkha (says the doctrine of anatta)

The above are (and remain) universal truths regardless of X and Z.

X and Z inherit their own kamma. One is and the other isn't ignorant of these truths, one does and the other doesn't suffer.

In case one, there is re-birth - hence he continues with grasping. In the second case (for an Arahant), there is no rebirth.