"No need to tie customs there to an answer" -- making this irrelevant to the question.

I don't see how this answers the question, which asks for a test for biological parentage. Answering from a "legal" perspective seems useless.

@JordiVermeulen: No the question does not ask for biological parentage. Maybe that is what the OP wanted to ask, maybe it isn't, but for sure the question does not ask that. The question asks how to "determine with near 0% false positive who is the father". This is how they determined who was the father with almost absolute certainty.

@AlexP paternity test = biological parentage.

@Krupip: As I have explained, in the Middle Ages the paternity test came down to a marriage test. It was absolutely 100% legally binding. It could not be contested, except in quite exceptional circumstances. You are thinking of the modern world, where a parent-child relationship can be acquired by means of some sort of physical test; but in the Middle Ages, it could not. No amount of physical resemblance mattered, not at all. One either was or was not the father. As they say, the past is a different country: they do things differently there.

To drive home @AlexP's point: The idea of fatherhood as being tied to the transmission of genetic material is an entirely modern conception of paternity, and would be alien to one living in medieval times. In other words, OP uses the word "fatherhood" to refer to a concept which simply doesn't exist in the medieval mind, in whose mouths the idea of "fatherhood" refers to something different -- something which is testable via the marriage test.

You're playing a semantic game, and I'm telling you this doesn't answer my question.

"The past is a different country: they do things differently there." -- And this question is about another different country than our own past. Medieval technology is not 13th Century Europe with all the implied influences of Christianity, the Roman Empire, Helenic Greece, etc.

@Ceph: your claim is clearly untrue. People in medieval times very much understood and cared about biological fatherhood. That is the reason why adultery was such a big taboo, why adultery of a king's wife was often legally considered treason - not because it hurt the king's feelings, but because it cast doubt on the legitimacy of his children and thus the dynasty. Legally assuming that the husband of a woman was the father of her children was a practical simplification and done exactly because society so strongly enforced the sexual fidelity of women.

And by the way, these principles are still today in many places the default legal assumptions that determine fatherhood, simply because they are practical and usually true.

@MichaelBorgwardt, all the same, this answer is extremely useful for anyone who is interested in how a medieval paternity test might be received/instituted/reacted to/etc.

Since nobody asked so far: Got any sources?

were there really no incidents of higher-ranking men having affairs with lower-ranking married women and then acknowledging the children as their bastards (and not in fact the children of their mother's husband)?

@Tristan: Natural children with unmarried women, sure, plenty; it was a common occurrence in the first half millennium of the Middle Ages. The best known case is William the Bastard, son of Robert I of Normany and his acknowledged concubine Herleva; he was acknowledged as heir to the Duchy of Normandy and then went on to change his nickname into the Conqueror. With married women, I don't know any example, and I don't even know how that would be even possible.

Father=sire is a very modern, very Western concept. Check out The Life of Dad for some alternative concepts of fatherhood: goodreads.com/book/show/38357915-the-life-of-dad In some settings, it is quite clear that a person who is the only father that matters is not biologically related to his child.

-1 --- this isn't History SE but Worldbuilding

With married women isn't it just a matter of the high ranking man making them a widow and they are now unmarried?

@MikeM History seems like a great place to start when asking for a question that is built on what could happen based on history.