Finished mathematics is finished and published and accepted mathematical proof that has passed rigorous peer review and to which formal terms like "theorem" or "lemma" can apply. The term is inclusive of conjecture that attracts major attention and is deemed worthy of publication, but exclusive of minor speculations and conjectures studied only by one or two people, and exclusive of any "folk mathematics" in the sense of that term applied by mathematicians. It implies that high standards of mathematical practice have been, and continue to be, applied to these publications to find and correct errors.
Mainstream mathematics or accepted mathematics are equivalent terms.
