Is it valid for a half-sibling to claim his share of the inheritance after having relinquished his right and forgiven the remaining heirs in this world and the Hereafter?

If the adult, sane heir willingly relinquishes their share, their relinquishment is valid and is considered a gift (hiba). This gift becomes binding if the donee takes possession of the gift before the donor retracts it, unless the gift was already in the possession of the donee, in which case it becomes binding simply upon offer and acceptance (ijab wa qabul).

Therefore, if the brother relinquishes his share after possession has been established, even before the house is divided, his share becomes your property, and he has no right to retract. The same applies if he retracts before the division of the inheritance, and the house is already in your possession and control.

However, if he retracts before the division of the house, and the house was not in your possession, he has the right to take his share, because the gift has not yet become binding.

It must be noted that a father's writing some of his properties in the names of some heirs does not make them their ملك (property/ownership).