Gemara

43b (Mishnah): A one-day old exempts his mother from Yibum or Chalitzah... He is like a grown man with respect to his relatives.

44b (Rav Papa): This teaches that if he dies, they mourn over him.

Suggestion: This is unlike R. Shimon ben Gamliel, who says that a baby who lives 30 days is not a Nefel;

Inference: If he died within 30 days, he is a Safek Nefel!

Rejection: The case is, we know that he had a full term pregnancy. Even R. Shimon is not concerned for a Nefel in this case.

Shabbos 135b (Beraisa - R. Shimon ben Gamliel): Any child that lives 30 days is viable - "u'Fduyav mi'Ben Chodesh Tifdeh";

Inference: Before this time, it is a Safek.

(Shmuel): The Halachah follows R. Shimon ben Gamliel.

If he yawned and died, all assume that he was a Nefel. They argue about a baby who fell from the roof or was eaten by a lion. R. Shimon is concerned lest he was a Nefel. Chachamim assume that he was alive.

136a: The son of Rav Kahana died within 30 days. He was mourning over him.

Rav Ashi: Rav Yehudah taught that the Halachah follows R. Shimon!

Rav Kahana: I know that he was born after nine months.

A similar case occurred with Rav Dimi bar Yosef's son, and he gave the above answer to Rav Dimi.

(Ravina citing Rava): If Reuven died, and then his only child died within 30 days of its birth, and David (a stranger) was Mekadesh the widow:

If David is a Yisrael, she must do Chalitzah. If he is a Kohen, she does not need Chalitzah.

(R. Hoshaya brei d'Rav Idi): Chachamim assume that a baby was viable even if he died within 30 days. Since there is no solution for a Kohen to keep his wife if she does Chalitzah, we rely on Chachamim (to be lenient).

Rishonim

Rambam (Hilchos Yibum 1:5): If a man died when his wife was pregnant, and the baby was born alive, even if he died immediately, she is exempt from Chalitzah and Yibum. Mid'Rabanan, we must know that his months were complete.

Magid Mishneh: The Rambam holds that if we do not know whether a baby was born after seven months or eight months, and Simanim are not finished, if it did not live 30 days, it is a Safek. We are stringent for Torah laws. However, one may circumcise on day eight (for even if it is a Nefel, there is no Isur). The Rambam rules like Rebbi, who says that a full term pregnancy shows viability, and like R. Shimon ben Gamliel, that living 30 days shows viability, even if Simanim are not finished. This is Torah law. Chachamim were stringent about some things.

Nimukei Yosef (Yevamos 25a DH Bar, citing the Ritva): R. Shimon ben Gamliel is Machshir due to living 30 days alone, without finished Simanim. It seems that he agrees with Rebbi, for finished Simanim are better than merely living 30 days. Since we hold like R. Shimon, we hold also like Rebbi. The Ramban agrees. Therefore, the only Ben Shemoneh is when his Simanim are not finished and he did not live 30 days. Some explain that R. Shimon ben Gamliel requires both (living 30 days and finished Simanim), and he argues with Rebbi, who says that finished Simanim suffice. The Gemara rules like R. Shimon. He requires both only for a Safek Ben Tishah, Safek Ben Shemoneh. If it is a Vadai (nine - Bach) month baby, e.g. her husband had Bi'ah with her and separated, it suffices if he has finished Simanim or lived 30 days. Therefore, if we know that his months were complete, all agree that he is viable, even if his Simanim are not finished. If it is a Safek Ben Shemoneh Safek Ben Tishah, even if his Simanim were finished, R. Shimon holds that it is a Safek Nefel within 30 days.

Rivash (446, cited in Beis Yosef EH 156 DH uvi'Teshuvos ha'Rivash): The Rambam (Hilchos Milah 1:13) rules like Rebbi, that if his Simanim were finished he is a proper baby even on day 1. However, in Hilchos Yibum he wrote that this is mid'Oraisa, but Chachamim were stringent to obligate Chalitzah whenever he did not live 30 days. If so, here she needs Chalitzah even b'Di'eved, unless she became Mekudeshes to a Kohen. Alternatively, perhaps here she needs Chalitzah even if she became Mekudeshes to a Kohen, for even Rabanan hold that it is a Safek Nefel. Abaye said that if he yawned and died, all agree that he was a Nefel. Rashi, R. Yonah and the Rosh say that the same applies to if he fell sick and died. Rava said that we rely on Rabanan when he fell from the roof or was eaten by a lion. The Rif omitted Abaye's teaching. He holds that Abaye argues with Rava, for Rava taught Stam (in every case). The Ba'al ha'Ma'or says that Abaye holds that getting sick and dying is like falling from the roof. He and the Rif both hold that if the baby fell sick and died, Chachamim argue (and say that he was viable).

Poskim

Shulchan Aruch (EH 156:4): If a man died when his wife was pregnant, if she miscarried, she does (Chalitzah or) Yibum. If the baby was born alive, even if he died the moment (after) he was born, she is exempt from Chalitzah and Yibum.

Gra (11): If he was born dead, mid'Oraisa he is a Nefel (Bechoros 46a).

Shulchan Aruch (ibid.): Mid'Rabanan, we must know that his months were complete.

Beis Yosef (DH u'Mah she'Chasav b'Shem): R. Peretz wrote that when it is possible that his months were complete, even if he was born dead, she may not do Yibum. She does Chalitzah. I do not understand why he brings a proof from R. Shimon ben Gamliel. R. Shimon is a lone opinion. The Halachah does not follow him!

Beis Shmuel (3): The Magid Mishneh says that sometimes Chachamim rely totally on finished Simanim, even if he is Vadai a Ben Shemoneh, e.g. that he is not considered Muktzeh. Elsewhere, they did not rely on this. Mid'Oraisa and mid'Rabanan one may rely on complete months, even if he died within 30 days, and his Simanim were not finished. If he lived 30 days, even if Simanim are not finished and he is a Ben Shemoneh it is a proper child even mid'Rabanan, just we are lenient about Avelus in this case. Even though the Rambam discusses a Safek Ben Shemoneh Safek Ben Tishah here, the same applies to a Vadai Ben Shemoneh if it lived 30 days. If the Simanim were finished, mid'Rabanan we do not rely on them. Mid'Oraisa he exempts from Chalitzah or Yibum, even if he died immediately. The Tur did not discuss a Vadai Ben Shemoneh. Regarding a Safek, he said that if the Simanim were finished and he lived 30 days, he exempts. Regarding Avelus, he wrote (YD 374) that we mourn even over a Ben Shemoneh. This implies that likewise, he exempts, for we follow the lenient opinion in Avelus. However, Tosfos rules like R. Avahu, that a Ben Shemoneh does not exempt, even if his Simanim were finished, until he is 20 years old. Tosfos holds that R. Avahu (Yevamos 80a) discusses whether or not he is viable. The Nimukei Yosef holds that he discusses whether or not he is an adult. The Rivash says that such a Chalitzah is Pasul, and she must do Chalitzah with all of the brothers. This is difficult. If it is a viable baby, she does not need Chalitzah at all. If it is a Nefel, it is a Kosher Chalitzah!

Beis Shmuel (4): If her husband is here, we cannot know that the months are complete. If he is not here, perhaps we are concerned lest he came. Perhaps we are even concerned lest he came (a long distance in a moment) using Hash-m's name!

Gra (12): The Gemara rules like R. Shimon ben Gamliel. He discusses Stam children. If he had complete months, R. Shimon agrees. The Rambam holds that mid'Oraisa, R. Shimon agrees that we follow the majority. Most babies are viable. The Gemara asked that the law of Avelus is unlike R. Shimon. It did not ask about the entire Mishnah, for we could have said that the Mishnah teaches the Torah law, but Avelus is mid'Rabanan.

Shulchan Aruch (ibid.): If we do not know when he was born, if he lived 30 days, he is a proper child and he exempts his father's wives from Chalitzah or Yibum.

Rema: Some say that this is only if he has finished Simanim (Tur, and so it seems to me).

Beis Shmuel (6): This is when he lived 30 days. The Beis Yosef did not know the Tur's source for this. Several Poskim say so! The Nimukei Yosef says that R. Shimon ben Gamliel addresses Rebbi's words, and requires finished Simanim and living 30 days. Tosfos and the Rosh hold that R. Shimon ben Gamliel discusses both a Vadai Ben Shemoneh and a Safek Ben Shemoneh. This shows that even a Safek must live 30 days and have finished Simanim. The Taz says that if he lived 30 days and they did not check whether his Simanim were finished, we rely on the first opinion that he is viable, for mid'Oraisa he exempts if he was born alive.

Rebuttal (Gra 16): The Nimukei Yosef, Ramban and Rashba do not require finished Simanim for a Safek (Ben Shemoneh). Finished Simanim were mentioned only regarding a Vadai Ben Shemoneh. There is a printing mistake in the Nimukei Yosef. It should say oppositely. We do not find any Mefaresh who says that R. Shimon ben Gamliel requires finished Simanim.

Gra (OC 331:1): The Rambam wrote that if the baby was born alive, even if he died immediately, mid'Oraisa he exempts from Chalitzah or Yibum. Mid'Rabanan he exempts only if we know that his months were full, or he lived 30 days. If not, it is a Safek, and mid'Rabanan she needs Chalitzah. Regarding inheritance, we leave Torah law in place, for a leniency for one is a stringency for another. Elsewhere, the Rambam rules like R. Shimon ben Gamliel, except for Avelus, for which we follow the lenient opinion. The Rambam learned from a Kohen's wife, who does not do Chalitzah. If this were a Torah Isur, it would be better that she do Chalitzah, for then the Isur is only mid'Rabanan! The Shulchan Aruch rules like the Rambam. The Tur and Rema (EH 156:4) are difficult. They say that a Stam child exempts only after 30 days with finished Simanim. This is not like any opinion! Even Tosfos requires both only if we know that he was born in the eighth month!

Shulchan Aruch (ibid.): If he died within 30 days, even on day 30, whether he died due to illness, fell from the roof or a lion ate him, it is a Safek Nefel, Safek viable baby. She needs Chalitzah mid'Rabanan, but she may not do Yibum.

Gra (18): No matter how he died, mid'Oraisa it is a proper child. This is why a Kohen's wife need not do Chalitzah. The Gemara said Stam 'died', which includes through illness. The Poskim proved from here that the Halachah does not follow Abaye. Rabanan argue even when it died through illness, for the Gemara said 'since there is no alternative, we do like Rabanan.'

Rema: Some say that even if he was born dead, and even if there is a Safek that his months were complete, she does Chalitzah and not Yibum.

Beis Shmuel (7): This requires investigation. See Ben Lev 3 p.41.