In the context of "the Consultant's Materials are vested, and shall remain vested, in the Consultant." I can't guess the meaning!
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This sounds like a quote from a contract, so really this is a legal question. However the term "vested" here means " fully and unconditionally guaranteed as a legal right, benefit, or privilege" (Merriam-Webster). In other words the "Consultant" owns, and will continue to own, all rights to the "Consultant's Materials". This makes it clear that the client isn't buying any rights to these materials.
"Vested" often turns up where rights are transferred gradually. For instance a senior manager in a company may be given shares, but those shares aren't fully "vested" for some years. This might mean, for instance, that the manager gets dividend income but can't sell them and will lose them if he leaves the company.