Gemara

(Mishnah): If an animal gave birth for the first time, to two males, the owner gives a Seh to a Kohen.

9b - Suggestion: (This is like Chachamim, who say that Iy Efshar Letzamtzem, i.e. it is impossible for things to happen simultaneously. Therefore, one of them is a Bechor.) Chachamim must hold that even part of the (circumference of the) womb is Mekadesh (a Bechor);

If all of the womb were needed to be Mekadesh, even though Chachamim say that Iy Efshar Letzamtzem, if twins left almost at the same time, the second was a Chatzitzah (separation) between the Bechor and the womb!

Rejection (Rav): (Perhaps all of the womb is Mekadesh.) The second animal is the same species. Min b'Mino is never Chotzetz.

Chulin 70a - Questions (Rava): Is the child Kodesh if it left the womb wrapped (by people) in tendrils (vine sprigs), or in a fetal sac (of another fetus), or covered (by the hands of a woman helping birth it, or of a female twin)?

These questions are not resolved.

Gitin 43a - Question: If a half-slave was Mekadesh a Bas Yisrael, does it take effect?

Suggestion: We can learn from a Yisrael who told a Bas Yisrael 'half of you is Mekudeshes to me.' It does not take effect.

Rejection: There, he did not try to acquire all of her. Here, he was Mekadesh as much of her as possible.

43b (Rav Yosef bar Chama citing Rav Nachman): If a half-Shifchah was Mekudeshes to Reuven, she was freed, and David was Mekadesh her, Reuven's Kidushin is uprooted;

(R. Zeira citing Rav Nachman): Reuven's Kidushin becomes full Kidushin.

Rishonim

Rambam (Hilchos Bechoros 4:17): If a Bechor was wrapped in tendrils and left the womb without touching it, or it was wrapped in the fetal sac of another fetus, or its sister was wrapped around it, since it did not touch the womb, it is a Safek Bechor.

Ran (Chulin 45a DH ha'Mishtatef) Why does partnership of a Kohen exempt even the Yisrael's part from Matanos (foreleg, jaw and stomach)? Chachamim hold that even a partial Seh (e.g. the father was a deer) is (half) liable! There it is considered the entire Seh, for all of the Seh in it is liable. Here, part of the Seh (the Kohen's share) is exempt. The Torah did not obligate in this case.

Avnei Nezer (YD 393:13): I support this from Gitin 43a. (One opinion holds that Kidushin of a half-Shifchah takes effect, for he tried to acquire the entire woman. Here also, since the head touched all of the womb that remains, this is like the entire womb. Do not say that here it is not, for initially the entire womb was there. When one was Mekadesh a Chetzi Shifchah, and she was freed, the Kidushin either becomes complete or vanishes. This shows that it depends on her current status. Here also, what remains is considered the entire womb. If the majority of the womb touches, this suffices, for in the entire Torah, the majority is like the whole.

Poskim

Tur (YD 315): If a Bechor was wrapped in tendrils, or in the fetal sac of another fetus, or its sister was wrapped around it, since it did not touch the womb, it is a Safek Bechor.

Chelkas Yo'av (cited in Avnei Nezer YD 392:5): A Bechor was removed from the womb with cups under its ears. I saw that they surrounded less than the majority of the head, therefore the baby must be redeemed.

Avnei Nezer (394:3): Indeed, in Menachos (11a), the Gemara said that even a stick of Levonah (frankincense) disqualifies a Kometz (a handful taken from a Minchah), for the Kometz is incomplete. The Gemara asked that it should disqualify due to Chatzitzah, even though it is on the minority of the hand. We must say that the hand is different, for the hand determines the size of the Kometz, i.e. what fits in three fingers. Therefore, we require the entire hand. The majority does not suffice. In any case, I say that one should redeem the Bechor born through forceps without a Berachah.

Avnei Nezer (15,16): Even according to Chelkas Yo'av, the question of whether part of the womb is Mekadesh does not depend on the question of Chatzitzah on a minority. Even if part of the womb (i.e. all that remains) touches the entire fetus without any Chatzitzah, perhaps part is not enough. However, I explained why this is not a Sefek-Sefeka. If the forceps surround most of the head, even if part of the womb is Mekadesh, this is only when it touches most of the circumference of the head. Here it does not, so it depends on the question of Chatzitzah. I later found that the Shitah Mekubetzes in Bechoros explains like Chelkas Yo'av.

Igros Moshe (YD 3:125): The Avnei Nezer and Chelkas Yo'av hold that Chatzitzah applies to human Bechoros. If it is a Safek whether there is a Chatzitzah at all (Chulin 70a), so if it is a Safek whether the forceps covered the majority of the head, there is a Sefek-Sefeka. However, I say that Chatzitzah applies only to animals. The Sugyos discuss only animals. The Rambam, Rosh, Ramban, Tur, Shulchan Aruch, Poskim and Teshuvos of Rishonim did not mention that the same applies to people. Among the Acharonim, only the Avnei Nezer and Chelkas Yo'av said so, and some recent Acharonim began to discuss this. Most obligate, for there is not a Chatzitzah on the majority of the head. Presumably Chatzitzah applies only to animals, which are Chulin inside and become Kodesh when they leave. The Rambam and Ramban discuss the Sefekos only regarding animals.

Igros Moshe (ibid.): Regarding animals the question pertains to the baby itself, and this applies at the moment of birth. Regarding people, the Chiyuv comes only after 30 days. The womb does not obligate by itself. Presumably, it does not depend on touching the womb, just on leaving through the womb. Levushei Mordechai (YD 2:130) says that Chatzitzah applies only to animals, for it has Kedushah. This applies even to a firstborn donkey, for we rule that one may not benefit from it before Pidyon. He did not rely on this alone, only because also the Chatzitzah is not on the entire head. One must investigate whether we can obligate a father to pay due to a Sefek-Sefeka, and bless on the Pidyon. He also says that perhaps it is considered the way of birth. I disagree. It is done only for a difficult birth. Rather, my other reasons suffice. Likutei Pinchas says that it is not a Chatzitzah because the mother does not mind, due to the danger. This is wrong. She is not relevant to Kedushas Bechor. We do not care whether she minds.

Ateres Paz (1:2:17): According to Avnei Nezer and Chelkas Yo'av, the Safek is whether something between the Bechor and the womb is a Chatzitzah. Seemingly, the same applies to forceps used to hold a baby's head during birth. The father should be exempt from Pidyon due to Safek; ha'Motzi mi'Chavero Alav ha'Re'ayah. The Rambam (Hilchos Bikurim 11:19), Tur and Shulchan Aruch (YD 305:13) say so. Geza Yishai (Ma'areches Beis, 225) says that it is Isur (and one must be stringent about Safek Isur). Sha'ar ha'Melech and Beis Meir (YD 305:13) rejected this. It is Safek Mamon. She'alos u'Teshuvos ha'Ma'or says one does not bless on Pidyon ha'Ben if it was born through forceps, for it is a Safek.

Ha'Emek She'alah (9:6): The She'altos says that if we can hear the baby cry, we know that its head left the Pruzdor. We infer that 'outside the Pruzdor' means outside into the world, for before this the mouth is closed (Nidah 30b).

Ateres Paz (3): Even if we say that a Chatzitzah of the minority disqualifies, perhaps forceps do not, for they are put into the womb itself, rather, in Beis ha'Chitzon (the neck of the womb). Rashi (Nidah 45b) and other Rishonim explain that once the head is in Beis ha'Chitzon, even though it is still inside, it is considered born. Even though sometimes forceps are put in deeply, near the beginning of Beis ha'Chitzon, once the forehead leaves, this is birth. The Rogachover Gaon (7) and ha'Gaon R. Moshe Shternbuch, Shlita (Halachah b'Mispachah 9, Hagahah 4), concluded that one redeems with a Berachah, for doctors use forceps only after Halachic birth.

Ateres Paz (4): Indeed, Rishonim argue about this, and most Poskim say that birth is only when the head totally leaves, and this is primary. There is a Sefek-Sefeka (two doubts). Perhaps a minority is not a Chatzitzah, like the Avnei Nezer. And even if it is, perhaps forceps are used only after birth. Yechavah Da'as (5:108) says that Sefek-Sefeka applies even when both Sefekos are about whom the Halachah follows (as opposed to the facts of the case - PF). We do not say that it is like one Safek. Divrei Yisachar (111) and Dovev Meisharim (2:32) obligate due to the Sefek-Sefeka. Levushei Mordechai (2:130) and Chelkas Yo'av Briyash (YD 177) similarly obligate. They say that one of the Sefekos is whether Chatzitzah applies at all to Bechor, for in Chulin we did not resolve whether tendrils are a Chatzitzah.

Ateres Paz (7): R. Moshe Shternbuch said that initially he thought like the Rogachover, but later he saw that Acharonim argue about when is birth. However, some say that Chulin discussed only a Chatzitzah for an animal, but not for a person. The body of an animal is Kodesh, therefore Chatzitzah inhibits the Kedushah. There is no Kedushah in the body of a human Bechor. Therefore, Chatzitzah does not apply. It seems that all the other Acharonim hold that Chatzitzah applies even to humans. However, the Sugyos and the Rambam (4:17) discussed only animals. The Chazon Ish (YD 14) says that if the Gemara or Rishonim should have explained that a law applies also elsewhere, and they did not, we infer that it does not apply elsewhere. I conclude that there is a total obligation to redeem the baby, with a Berachah.