I'm not sure what you mean. As far as I know, there is only one original genus in which edwardsi Van Duzee, 1930 was introduced, and whether the genus is paraphyletic is not relevant to the spelling of the specific name. Also, the nomenclatural rules do not require "authority"; they can be applied by anyone who has an interest. In this case the rule is Article 32.2 of the International Code of Zoological Nomenclature: "The original spelling of a name is the 'correct original spelling', unless it is demonstrably incorrect as provided in Article 32.5." There is no doubt that Chrysotus edwardsi is the correct spelling.