Gemara

(Mishnah): If dough is in cracks in a kneading trough:

If there is a k'Zayis in one place, he must do Bi'ur;

If not, it is Batel because it is so small.

Version #1 (Rav Yehudah): This (the first law) is only in a place where it does not strengthen [the trough]. Where it strengthens it, one need not do Bi'ur.

Inference: One need not destroy less than a k'Zayis even in a place where it does not strengthen it.

Version #2 (Mishnah): If not, it is Batel because it is so small.

(Rav Yehudah): This is only in a place where it strengthens it. Elsewhere, one must do Bi'ur.

Inference: One must destroy a k'Zayis even where it strengthens it.

(Beraisa #1): One is exempt for less than a k'Zayis, even in a place where it does not strengthen it.

Contradiction (Beraisa #2): One transgresses for less than a k'Zayis in a place that does not strengthen it.

Resolution #1 (Rav Huna): We delete the lenient [Beraisa] due to the stringent. (Beraisa #1 is erroneous.)

Resolution #2 (Abaye): Both Beraisos are like R. Shimon [who says that heavily leavened dough that was designated for sitting is Batel. They do not argue. They discuss different places];

Beraisa #1 discusses only where kneading is done. It calls the bottom a place where it strengthens.

Beraisa #2 discusses even where kneading is not done. It calls the walls a place where it strengthens.

(Rav Ashi): 'Where kneading is not done' refers [even] to on top of the rim.

Version #1 (Rav Nachman citing Rav): The Halachah follows R. Shimon.

Version #2 (Rav Nachman): Rav rules unlike R. Shimon. Rav Yitzchak bar Ashi said in the name of Rav that if one smeared the outside of the dough with mud, it is Batel.

Rishonim

Rif (13b): If there is not a k'Zayis in one place, it is Batel because it is so small. This is in a place that strengthens.

Rosh (3:2): The Rif brings only Version #2 of Rav Yehudah. Even though Abaye resolved the two Beraisos, the Rif disagrees. The Ba'al ha'Ma'or explained that this is because Abaye established both like R. Shimon, and the Halachah does not follow him. The Ba'al ha'Ma'or rules like Abaye, for he resolves also the two versions of Rav Yehudah, and Rav Ashi explains Abaye. I did not elaborate about this, for Yisrael are Kedoshim, and scrape and eradicate all Chametz, of any size, even if it is stuck to the wall of the house. Ra'avan found a support from the Yerushalmi to scrape walls and chairs that Chametz touched.

Gra (OC 442:7 DH Batzek): Rav Ashi comes to explain Abaye, but not to establish the Halachah.

Rambam (Hilchos Chametz 2:15): If dough is in cracks in a kneading trough, if there is a k'Zayis in one place, he must do Bi'ur. If not, if it is made to strengthen broken parts of the trough, or to seal a hole, it is Batel due to its small size. If not, one must do Bi'ur.

Hagahos Maimoniyos: The Re'em says that this is if it is somewhat proper to eat. If it is less than a k'Zayis and somewhat dirty, he is exempt from Bi'ur.

Magid Mishneh: The Rambam discusses in a place made to strengthen the trough. If not, one must eradicate even half a k'Zayis!

Kesef Mishneh: The Rosh says that the pieces need not stick to the walls. This is merely a typical case. I say that it is only when they stick to the walls, like the case of sealing a hole. The Rambam obligates Bi'ur on less than a k'Zayis. The Magid Mishneh and Ramach say so.

Sha'agas Aryeh (81 DH v'Achar): One who leaves over Chametz with intent to do Bi'ur transgresses. If so, why does the first Tana (Pesachim 29a) holds that Me'ilah applies to Chametz during Pesach? Even if it is a mixture of Chametz, Bal Yera'eh applies mid'Rabanan, so one cannot redeem it.? Tosfos (29b DH Rav) says that the Isur to redeem Kodshim in order to feed them to dogs is mid'Rabanan, and even so it prevents Me'ilah. Even the Ramban and Ran (6a), who exempt for owned Chametz not in his premises, obligate Bi'ur mid'Rabanan. Therefore, it has no redemption value. We must say that he redeems less than a k'Zayis at a time. Bal Yera'eh and Bal Yimatzei apply only to a k'Zayis (Beitzah 2a). Less than a k'Zayis there is not even an Isur mid'Rabanan. This is clear from Pesachim 45a. We must say so according to the Ri, who says that Bal Yera'eh is Nitak l'Aseh. If so, one is not lashed for it, even if he did an action. If one transgresses Bal Yera'eh for less than a k'Zayis, like R. Yochanan who forbids Chetzi Shi'ur (Yoma 73b), there is no difference between a Shi'ur and less than a Shi'ur! Why does the Gemara say that Beis Hillel and Beis Shamai argue about the Shi'ur for Bal Yera'eh (Beitzah 7b)? Rather, there is no Isur less than a k'Zayis. For other Isurim, the Torah forbids Chetzi Shi'ur because it can join. This applies only to eating, for foods join to a k'Zayis bi'Chdei Achilas Pras. If he eats Mashehu now, and finishes the Shi'ur of k'Zayis within Kdei Achilas Pras, retroactively the first eating was b'Isur, for it joins. Regarding Bal Yera'eh, even if later he will complete the Shi'ur, he transgresses only from then and onwards. There is no Isur even retroactively for having kept less than a k'Zayis. Even though we learn the Isur of Chetzi Shi'ur from "Kol Chelev", the reason is because it can join. Tosfos says so there (Yoma 74a DH Kivan).

Ohr Same'ach: It seems that the Rambam discusses when there is a k'Zayis, but not in one place, and it is in different cracks. Even so, [in Halachah 16] he is exempts for two half-k'Zeisim in the house only if they are stuck to the walls. If not, this is like in a trough not to strengthen, and one must do Bi'ur. We do not find a Chiyuv Bi'ur on less than a k'Zayis. It seems that the Rambam holds that there is no Chiyuv, unlike the Magid Mishneh and Kesef Mishneh. This is why Rav Huna said to delete the lenient due to the stringent. Even though there is no Isur for a half a k'Zayis, if a k'Zayis is split among two places, we must be stringent mid'Oraisa, and adopt the more stringent version. The Rambam clearly holds like this, for he holds that Bal Yera'eh is not Nitak l'Aseh, and one is lashed for it. On 29b, Rav Ashi says that the one who says that Me'ilah applies to Chametz during Pesach holds like R. Yosi ha'Glili, who permits benefit. How can one redeem it? He transgresses Bal Yera'eh! We must say that he redeems a Perutah's worth, and does not give to any one person a k'Zayis. This shows that there is no Chiyuv Bi'ur even mid'Rabanan for less than a k'Zayis.

Or Some'ach: Since Chetzi Shi'ur is forbidden to eat, perhaps also Chetzi Shi'ur of Bal Yera'eh is forbidden! R. Yosi ha'Glili holds that the Shi'ur [to be liable] for eating is k'Koseves, and the Shi'ur for Bi'ur is a k'Zayis. Since they have different Shi'urim, we cannot learn the Isur of Chetzi Shi'ur one from the other. We learn Bi'ur from eating, so we can learn also Chetzi Shi'ur, like Maharam Chalavah says.

Poskim

Shulchan Aruch (OC 442:7): If dough is in cracks in a kneading trough, if there is a k'Zayis in one place, he must do Bi'ur. If not, if it is made to strengthen broken parts of the trough, or to seal a whole, it is Batel due to its small size.

Beis Yosef (DH Batzek): The Gemara infers that less than a k'Zayis is not important, even if there are many half-k'Zeisim, since they are not in one place.

Mishnah Berurah (32): Since they are not in one place, they are Batel to the Kli.

Kaf ha'Chayim (76): They are Batel to the Kli only if he was or will be Mevatel them.

Beis Yosef (ibid. and DH Aval): Rashi holds that both Tana'im hold that less than a k'Zayis is Batel even in the wall, but only in a place of kneading, for there it helps strengthen the dough. Tosfos says that Chachamim did not decree about less than a k'Zayis, even in a place of kneading, if it strengthens. Since it is small, one can be careful not to knead there. We permit less than a k'Zayis not in a place of kneading, even if it does not strengthen. Whenever there are two reasons to be lenient (less than a k'Zayis, and it is not in a place of kneading or if it strengthens), we are lenient. When there are two reasons to be stringent (it is in a place of kneading, and does not strengthen), one must do Bi'ur.

Taz (5): In this Siman, whenever we exempt from Bi'ur of less than a k'Zayis, this is like the opinion that Bedikah is due to Bal Yera'eh, which is only for a k'Zayis. Tosfos holds that Bedikah is lest one eat it. One may not eat even a half a k'Zayis! Every "Chayav" [in this Siman] refers to lashes [but in any case, one must do Bi'ur]. Why did the Tur and Shulchan Aruch teach this? It is not relevant nowadays! The Rosh did not elaborate, for Yisre'elim are Kedoshim. Letter of the law one must do Bi'ur on any amount, lest he eat it! They cannot discuss when there is no concern lest he eat it. If so, why is Bedikah required? Also, the Beis Yosef brought from Re'em that it is somewhat proper to eat. I answer that it is in a place where people will trample it. Bal Yera'eh applies to a k'Zayis before it will be trampled.

Shulchan Aruch (ibid.): If not, one must do Bi'ur.

Beis Yosef (DH veha'Rif): The Rif brings only Version #2 of Rav Yehudah. It does not matter whether or not it is in place of kneading. The Rambam holds like this. I need not elaborate, for it is not relevant in practice. The Rosh says that Yisrael are Kedoshim, and scrape and eradicate all Chametz of any size.

Magen Avraham (10): The Kli joins everything inside it to a k'Zayis. If there is not a k'Zayis inside, one is exempt from Bi'ur, even if it is not to strengthen]. Sa'if 11 connotes like this, and Semak says so. However, many Poskim obligate Bi'ur even for less than a k'Zayis if it is not to strengthen. If it is not proper to eat, he is exempt from Bi'ur.

Mishnah Berurah (33): The Shulchan Aruch exempts less than a k'Zayis from Bi'ur. Many Poskim disagree when it is somewhat proper to eat. If it is slightly dirty, all exempt.