Is it permissible to marry a woman who has been Islamically divorced without documenting her divorce, knowing that official papers will indicate she is a virgin due to the absence of a divorce certificate?

Firstly: Documentation is not a condition for marriage (nikah (definition)) and divorce (talaq (definition)) if their pillars are fulfilled.

Secondly: If the woman is still married and has not been divorced according to Islamic law, it is not permissible to propose to her or to corrupt her relationship with her husband. However, if she has been divorced and her waiting period (iddah (definition)) has ended, then it is permissible to marry her.

Thirdly: Falsely claiming virginity is a lie, and lying is fundamentally forbidden. There are several ways to avoid lying, including performing the contract without documentation, or convincing the notary that the first marriage and divorce were not documented. If it is impossible to avoid lying, and it does not result in the loss of a right or a mischief, then it is permissible if it is a means to achieve a recognized benefit, for "speech is a means to ends. So, every commendable end that can be achieved through both truthfulness and falsehood makes falsehood forbidden due to lack of necessity. However, if it can be achieved through falsehood and not through truthfulness, then falsehood is permissible for what is permissible, and obligatory for what is obligatory."