In certain cases, the Torah prescribes the bringing of a pair of birds as an offering, one as a chatat and the other as an olah. See Lev 12:8, Lev 14:22, Lev 15:14, Lev 15:29, and Num 6:10.
The governing principle of mix-ups of birds is that we do not allow birds to be offered (a priori) if there is a possibility that we might accidentally offer both birds of a pair as olot or both as chataot, and if a group of X pairs is entirely mixed with a group of Y pairs, then we can bring min{X, Y} pairs from the X+Y birds. See Kinnim 1:3 (1:4).
The mishna Kinnim 2:2 describes what happens if two sets of four birds get mixed up (translation edited from Sefaria):
שתי נשים, לזו שתי קינים ולזו שתי קינים--פרח מזו לזו, פוסל אחד בהליכתו; חזר, פוסל אחד בחזירתו. פרח וחזר, פרח וחזר--לא הפסיד כלום: שאפילו הן מעורבות, אין פחות משתיים.
If there are two women, this one has two pairs and this one has two pairs and one bird flew from this woman's [pair] to the other woman's [pair], it invalidates one bird by its leaving. If it then returned [to its original pair] it invalidates one by returning. If it flew back and forth multiple times it does not cause further invalidation since even if they are mixed up, there are at least two [valid] birds remaining.
Birds are flying back and forth (one at a time) between the two groups of four, ending with four and four. The mishna allows each owner to offer two of the birds in front of them. I glean from this mishna that the minimum number of pairs we are able to offer in this case (even after much mix-up) is equal to the the number we could offer if all birds were mixed into one group.
The next mishna (2:3) speaks of owners each with different numbers of birds (translation edited from Sefaria):
לזו אחת, ולזו שתיים, ולזו שלוש, ולזו ארבע, ולזו חמש, ולזו שש, ולזו שבע--פרח מן הראשונה לשנייה, לשלישית, לרביעית, לחמישית, לשישית, לשביעית, וחזר--פוסל אחד בהליכתו, ואחד בחזירתו: הראשונה והשנייה, אין להן כלום; השלישית, יש לה אחת; הרביעית, יש לה שתיים; החמישית, יש לה שלוש; השישית, יש לה ארבע; והשביעית, יש לה שש. פרח וחזר--פוסל אחד בהליכתו, ואחד בחזירתו: השלישית והרביעית, אין להן כלום; החמישית, יש לה אחת; והשישית, יש לה שתיים; השביעית, יש לה חמש. פרח וחזר--פוסל אחד בהליכתו, ואחד בחזירתו: החמישית והשישית, אין להן כלום; והשביעית, יש לה ארבע. ויש אומרין, השביעית לא הפסידה כלום. ואם פרח מבין המתות לכולם, הרי כולם ימותו.
If this one has one [pair of birds], and this one has two [pairs of birds], this one has three [pairs], this one has four [pairs], this one five [pairs], this one six [pairs] and this one seven [pairs], and one bird flew from the first [group] to the second [group] and one [from the second] into the third, and one [from the third] into the fourth, and one [from the fourth] into the fifth, and one [from the fifth] into the sixth, and one [from the sixth] into the seventh, and then one returned [to its previous group], it invalidates one pair [for each] when it leaves and one pair [for each] upon its return. The first and second [groups] have no valid birds left, the third has one [valid pair], the fourth has two, the fifth has three, the sixth has four, and the seventh has six. If it [= a bird] flew back and forth [again], it invalidates one set when it leaves and one set upon its return. The third and fourth [group] have nothing, the fifth has one [valid pair], the sixth has two and the seventh has five. If a bird flew back and forth [again], it invalidates one pair when it leaves and one upon its return. The fifth and sixth [groups] have no valid birds and the seventh [group] has four... If a bird that is left to die flew into their group they all must be left to die.
More simply (as far as I can tell), say we have two groups, where group A has two birds (= one pair) and group B has four birds (= two pairs), and a bird flies from A to B and then a bird flies from B to A. For this case, based on this this mishna, no bird that is currently in group A can be brought, and one pair from those currently in group B can be brought. If another bird then flies from A to B, then no bird can be brought (by the last clause of the mishna). From here we see that there is a case where we bring no birds, instead of the one pair we would be allowed to bring if all six birds were mixed together.
How do I resolve this contradiction? Am I incorrectly inferring rules from the specific cases? Do Kinnim 2:2–3 not hold by Kinnim 1:3?
