Is it permissible for the husband to agree with his wife to register two houses in both their names equally, without an accurate calculation of the amounts paid by each, in order to guarantee the wife's right to the money she paid?

There is no sin upon you for writing the two couplets between you and your wife who shared with you in the building costs, whether what your wife paid was more or less than what you paid in the construction, as long as you both reconciled upon that willingly.