California, like many states, according to Professor Cahn, has conflicting statutes. One provides that any man can establish parentage if he “receives the child into his home and openly holds the child out as his natural child.” But another statute holds that a man who provides his sperm to a doctor for the purpose of inseminating an unmarried friend is “treated as if he were not the natural father” — unless there is a specific written agreement ahead of conception.

Mr. Patric and Ms. Schreiber had no such agreement. And her lawyers say there was nothing cavalier or last minute about it: “Danielle knew about the law before she chose to proceed with a known sperm donation,” Mr. Heather said. “She made a carefully considered judgment.”

Mr. Patric took her to court, holding up “intended parent” forms he signed at the sperm-donor clinic. Ms. Schreiber stood her ground, noting that Mr. Patric had asked that his name not be on the birth certificate. (“It would have thrust Gus into the limelight, and I wanted to protect him,” Mr. Patric said.) As for Gus calling Mr. Patric “Dada,” her lawyers say it doesn’t matter: Ms. Schreiber never intended to keep Mr. Patric’s identity a secret from Gus, but she did intend to prevent Mr. Patric from having any parental rights. (“The lies are stunning,” Mr. Patric said.)