There are specific rules for introducing sources of fact in a normal (first-instance) trial, either through evidence or through witnesses, with constraints around hearsay, etcetera. A fact (or an alleged fact) introduced or referred to in a trial needs to be sourced following these rules, or be disregarded.
It would also seem logical to me (although I'm not sure about it) that since a suit where one party fails to attend entails a default judgement in favor of the other party, then a judge must only apply a law if it has been invoked by at least one party in the trial (when it comes to sentencing I mean, not on procedural grounds during the trial itself). Is that inference true ?
It seems to me (as said in the movie On the Basis of Sex) that a US court of appeals and the Supreme Court work like the french Court of Cassation, in that it only (re)judges questions of law, and not questions of facts. If so, does the same standard of only applying sources of law mentioned during the trial apply there ?
I'm asking the question after having read the infamously leaked Alito opinion from Dobbs v. Jackson, which ends by exhaustively listing old state abortion laws. Although I didn't read the transcript of the hearing that thoroughly, I doubt all these sources of law (or of fact, depending on how you look at it, since the opinion cites them as evidence and does not seek to apply them) were all cited by the party arguing against Roe.
If judges (and justices) are not bound by what law(s) and which facts have been cited by the parties, what is the limit on how judges can do the parties' job, inventing their legal arguments for them ? If they are not, how could a poorly-defended case be prevented from setting a terrible jurisprudence precedent ? Final question, is there a difference between how these rules bind judges and justices (i.e in Courts of Appeals vs in the Supreme Court) ?