If a husband divorces his wife judicially, not religiously, and the wife does not take the matter to court, is their marital life sound?

The ruling of a judge does not alter the inner reality. If a judge rules based on false testimony or a fabricated confession, what he has ruled in favor of the plaintiff does not become lawful for them, whether it be property or a marital right (intercourse). This is the view of Malik, Shafi'i, Ahmad, and the majority of scholars, in contrast to Abu Hanifa, who held that marital rights become lawful but not property. So, if a judge rules to divorce a woman based on false testimony or a fabricated confession from the husband, the woman remains his wife in the inner reality, and it is permissible for him to enjoy her if she knows he is truthful or if she predominantly believes so.