Judge Ralph Adam Fine, quoting from Brown v. Hartlage, asserts that "demonstrable falsehoods" are not protected by the First Amendment, but elsewhere in his opinion he states:
Certainly, it is not a true representation to imply through crafty sculpting of words that because Justice "Butler found a loophole[,] Mitchell went on to molest another child."(Brown v. Hartlage pits "demonstrable falsehoods" against "erroneous statement[s]," i.e., mistakes. Yet as Judge Fine himself points out in his footnote 4, this was no mere mistake, a finding of fact that all parties including ["apparently"] Michael Gableman acknowledge.)
Nevertheless, Judge Fine concludes that Gableman's speech is protected by the First Amendment. With all due respect to the learned judge, the difference between "demonstrable falsehood" and "certainly false representation" eludes me at the moment.
If it's certain, then it must somehow be demonstrable. And, indeed, Judge Fine did adequately (IMO) make that demonstration.
"Falsehood" and "not a true representation" are synonymous.