Is there a word for this? For instance, a medical test where a positive result would be conclusive but any other result would not be.
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You might use go/no-go test, which refers to a test that either certifies an item is satisfactory or does not so certify. In that regard, a go/no-go test is much like the vast majority of tests used in statistical hypothesis testing or in confirmatory data analysis, i.e. tests which "either fail to reject the null hypothesis or reject it in favor of the alternative." I suggest reviewing the vocabulary in hypothesis testing and statistical inference references.
A neologism to consider: "witness test", as inspired by Miller-Rabin testing, which either provides a witness (a certificate) that a number is composite, or fails to do so.
A test "where a positive result would be conclusive but any other result would not be" (i.e. a test with very few false positives) has high specificity. A test with a small number of false negatives has high sensitivity.