Is the contract that the brother concluded with the institution to prove his residency, in which he stated that he had been working for them for 6 months despite that not being true, considered a forgery, and is the wage resulting from it permissible or forbidden?

Forging and falsifying documents, and signing them with the intent to deceive and lie, as well as forging a fictitious contract, are not permissible except in cases of dire necessity. This is because it falls under false testimony (shahadat az-zur) and false speech (qawl az-zur), which the Prophet (peace be upon him) warned against. False speech is describing something contrary to its true nature. If the work is permissible and the worker performs it as required, then their wage for it is permissible; otherwise, it is not.