Determining genetic fatherhood is very important in my worldbuilding experiment for lots of reasons, one being succession, but regardless it's the genetic component that's important But it's possible that people don't know who the father is of their child as well. Note that this is *not a medieval European world*. I specified tech for a reason as that's the only concept that carries over. The concepts of European royalty or culture really don't apply at all.

Given the idealized restrictions below, is there anything that can be done with medieval level technology that can determine with near 0% false positive who is the father given a set known amount of candidates, *or* if there is a false positive, you know there's a false positive and the test is still useful?

* Medieval level tech, is anything that was already available or could reasonably be available given aristocratic resources and modern knowledge *quickly* as in not 200 years from then (so you couldn't easily create microelectronics, you could however manufacture a spinning device with spinning glass tubes full of liquids with cork caps, even if it had to be manually turned by a person, as an example).
* A false positive is when a *single* candidate is tested positive for being the father, but it wasn't them. When more than one candidate tests positive, it's not a false positive, that's inconclusive.
* This is not a magic world.
* This genetic fatherhood.
* Assume neither candidates nor mother lie in this scenario.
* All candidates and mother are cooperative.
* The father can be guaranteed to be in the list of candidates.
* Man-power is not much of an issue, though I don't know if it's relevant.
* Test has to use medieval level technology, so assume you can make glass, work iron etc...
* Despite having medieval tech, this does not take place in medieval Europe, there's no need to tie customs there to an answer.
* A long the same lines, the test can be used along side *reasonable* original societal customs to aid testing. Maybe there's an invariant in society that means X can't be the father etc... But this can't prohibit non-monogamous relationships in general, otherwise the problem basically solves itself.
* The test has to be *useful*. So maybe you could use things like phenotypical traits, but if it mostly always boils down to a few people anyway, and barely ever narrows it down, it's not useful. Useful would be something like 30-50% of the time you can make a pretty strong assertion of the father under some average bounds of number of candidates (like 5 or less).
* Test has to have a low false positive rate. If it only works 50%-ish the time, then fine, but those times it better be pretty accurate.
* Alternatively, as long as it's still useful, as long as you *know* there's a false positive, so you can call the test "failed", that also works.
* The test can take long periods of time, but *ideally* (not a hard requirement) no more than 5 years.
* The test can assume some medical knowledge we know today, such as blood types.
* If it were possible to do limited genetic testing with said medieval technology, that would be a reasonable solution.
* Assume everyone are historically from the same place, so there's no "obvious" physical markers with respect to people from two completely different ethnic backgrounds making this work some significant amount of time.
* near 0% does not mean 0%, so 2% false positive may be acceptable, I didn't want to pin point a specific number.

EDIT:

* Phenotypical Book-keeping is possible, as in physically recording physical phenotypes that can be observed for each individual, from multiple generations prior (so you know a set number of specific phenotypes from parents of each individual, and parents of parents, and so on and so forth for several generations, and those people don't need to be alive for you to know this information, it's recorded).

EDIT2:

I have a new point that may help that didn't violate the above (which means this isn't really a "new rule", rather it serves to help answerers), but people are getting real stuck on way to euro centric world view of things, to the point where the top voted question earlier quite blatantly violates my entire question and doesn't answer it at all, plus assuming a marriage structure and civilization wide monogamy, or assuming that if there isn't monogamy it must be happening the same day constantly.

* Both the mother and men know who and when they had relations with and **when** (assume this knowledge is always accessible, as if they recorded it).
* When the father is ambiguous, it's not very common to have had relations with multiple different partners within the same day, and also not that common within the same week. The chance of different partners increases with time period when the father is ambiguous. If time can be used, there are accurate records to help.