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leonos: The group leader has allowed the PostDoc to apply as PI for career reasons. PhD was not involved in that discussion. Anyon: There are no formal requirements but I am confident that a proposal from a PhD will not be taken seriously.
It seems you're not sorry you cheated, but you're sorry you got caught. was this the first time you ever cheated? How many times have you removed cheating cases from memory? I think your professor was kind enough to let you go, it won't be on the transcript.
Surface level foundation will contribute only to surface level knowledge. And you risk building on something that will eventually collapse. If your aims are getting a grant, that might work. If your aim is to advance human knowledge, that won't necessarily work.