> deliberate and often also reckless, harmful falsehood is not protected speech

IANAL (I am not a lawyer; and IANAAJA—I am not an Alex Jones apologist), but I think deserves qualification. My understanding is that it’s not the falsehood which is restricted, it’s the damage caused by purveying the falsehood. If I’m not mistaken I can claim (anywhere under US jurisdiction, unless under oath or otherwise legally sworn to be truthful) that I have a cat (I don’t) without any legal risk to 1A protection whatsoever. Granted, that could raise other eyebrows which might gain legal force if I persisted, presumably around competency to care for myself (or my dog, who I assure you does exist).