Gemara

(Beraisa): If Reuven keeps Chametz during Pesach, immediately after Pesach, others may benefit from his Chametz, because he trades (his Chametz with that of Nochrim).

Assumption: The Beraisa is like R. Yehudah, who holds that if a Yisrael keeps Chametz during Pesach, it is forbidden mid'Oraisa after Pesach.

Rejection: No. The Beraisa is like R. Shimon, who forbids only mid'Rabanan after Pesach. We are lenient to assume that a Mumar minimizes his sins only regarding (what could be) an Isur mid'Rabanan, but not for a Torah Isur.

Rejection of rejection: Even if the Beraisa is like R. Shimon, one may rely on this for Torah Isurim! It does not say 'we assume that he trades', rather 'he trades', i.e. definitely!

Pesachim 28a (Mishnah): Chametz she'Avar Alav ha'Pesach (a Yisrael owned it during Pesach) is Asur b'Hana'ah (forbidden to benefit from it), for it says "v'Lo Yera'eh Lecha."

29a (Rava): Our Mishnah is R. Shimon. He fines (to forbid Chametz of a Yisrael after Pesach) because he transgressed Bal Yera'eh and Bal Yimatzei.

31a (Beraisa): If a Nochri lent to a Yisrael on the Yisrael's Chametz (it was security), after Pesach all agree that one transgresses.

31b: The case is, the borrower put the Chametz in the lender's Reshus.

(Mishnah): If a Nochri lent to a Yisrael on the Yisrael's Chametz, it is permitted after Pesach.

Answer: In the Mishnah, when he put the Chametz in the lender's Reshus, he said '(If I do not pay you by the due date, it is yours) from now.' In the Beraisa, he did not say 'from now'.

Bava Kama 98b (Mishnah): If Reuven stole Shimon's Chametz, and Avar Alav ha'Pesach, he can return the Chametz and say 'here is your Chametz.'

Our Tana allows saying 'here is your object' even if it is Asur b'Hana'ah.

Rishonim

Rambam (Hilchos Chametz 1:4): If a Yisrael owned Chametz during Pesach, it is Asur b'Hana'ah forver. This is a fine mid'Rabanan. Because he transgressed Bal Yera'eh and Bal Yimatzei, they forbade it. This is even if he left it Shogeg or Ones, lest one leave Chametz in his Reshus during Pesach in order to benefit from it after Pesach.

Magid Mishneh: It is forbidden not only if he actually transgressed Bal Yera'eh and Bal Yimatzei, rather, even if one could have transgressed for it. This excludes a Nochri's Chametz.

Rosh (Pesachim 2:4): Our Mishnah is like R. Shimon, who forbids Chametz after Pesach only due to a fine. This is only for Chametz itself, but not for a mixture of Chametz. If a Yisrael found Chametz in his house after Pesach, letter of the law it is permitted. Since he did Bitul, he did not transgress Bal Yera'eh and Bal Yimatzei, so we should not fine him. If one made Chametz Hefker on Erev Pesach, in the Yerushalmi, R. Yochanan forbids, lest he scheme. I.e. he will not make it Hefker, but he will say that he did. Reish Lakish permits.

Poskim

Shulchan Aruch (OC 448:3): If a Yisrael owned Chametz during Pesach, it is Asur b'Hana'ah, even if he was Shogeg or Ones.

Gra: This teaches not to say that R. Shimon would not fine (Shogeg or Ones). A proof is from a Yisrael who put his Chametz in a Nochri's house for collateral. If he did not say 'from now (it is yours if I will not pay in time))', it is forbidden.

Mishnah Berurah (8): Shogeg is when he did not know about the Chametz.

Mishnah Berurah (9): Ones is when he knew about the Chametz, but he was not able to destroy it. Even though he did not transgress Bal Yera'eh and Bal Yimatzei, we fine Heter due to the case of Isur. If we permit this, people will come to leave over Chametz l'Chatchilah, and transgress. Beis Hillel (YD 132) discusses a Yisrael who had grain to grind in a Nochri's mill. It was in the mill until Chol ha'Mo'ed. On the seventh day of Pesach the Nochri ground it and made bread, and brought it to the Yisrael immediately after Pesach. I answered that he may take money from the Nochri for the bread, and the Nochri will eat it or sell it to another Nochri. Chak Yakov permitted even to eat it, for perhaps the Nochri switched the grain (with grain of a Nochri), so it is a Safek whether this is Chametz she'Avar Alav ha'Pesach. Beis Meir permits eating it for a different reason. Even though the Mechaber forbids Shogeg and Ones, not all Onsim are the same. This is an Ones that the Yisrael could not have anticipated, that the Nochri will bake his flour into Chametz. Surely Chachamim did not fine him. A Yisrael left wheat in a mill, and a Nochri told a Yisrael to be Loses (wet it just before grinding it) it and grind it. (If done properly, this does not make Chametz, but we are not experts about this - Mishnah Berurah 453:27.) Chak Yakov says that there are many Sefekos to be lenient. The Pri Megadim is skeptical where people normally do Letisah. He says that the Nochri should switch it with other wheat.

Bi'ur Halachah (DH Chametz): If Reuven's Chametz was stolen and Avar Alav ha'Pesach, the Noda bi'Yehudah (OC 20) permits to Reuven. The Acharonim disagree. It is clear from Ba'al ha'Itur, the Ramban, Ritva and Maharam Chalavah that it is forbidden to everyone. They learn from Bava Kama 98b. If a Nochri stole Reuven's Chametz and Avar Alav ha'Pesach, the Eliyahu Rabah, Pri Megadim and Maharit Algazi permit.

Kaf ha'Chayim (38):Maharam of Sarkasta says that neither the thief or owner transgressed Bal Yera'eh and Bal Yimatzei, but even so it is forbidden after Pesach. Mekor Chayim says that some forbid after Pesach even if a Nochri stole it.

Bi'ur Halachah (DH Afilu): The Mechaber did not specify. Perhaps he means that due to Shegagah or Ones he did not destroy or be Mevatel the Chametz, and therefore he forbids, but if he was Mevatel, but due to Ones he did not check, it is permitted, for he fulfilled his primary Torah obligation. Or, perhaps even if he was Mevatel, but due to Ones he did not destroy it, it is forbidden. Acharonim argue about this. The stringent opinion in Sa'if 5 forbids even if he checked and was Mevatel and found Chametz after Pesach, and all the more so here, when he did not check at all, even though this was due to Ones. According to the opinion that permits there if he checked and was Mevatel, we can say the same here even though he did not check. Since this was due to Ones, it is no worse than checking and finding after Pesach. Toras ha'Shelamim and Eliyahu Rabah say so. There, to avoid a big loss, one may rely on the lenient opinion that permits Hana'ah. The same applies here.

Bi'ur Halachah (ibid.): A case occurred in which Nochri servants were Loses a Yisrael's wheat to prepare it for grinding after Pesach. The Radvaz (1015) permitted to mix it with other wheat and grind all of it after Pesach and sell it to a Yisrael. The Pri Chodosh was astounded. Even if the wheat did not split, the Mechaber (467) forbids eating it itself. If so, we cannot permit to grind it and rely on Bitul, for one may not be Mevatel an Isur l'Chatchilah! Mateh Yehudah answered that since the Isur on the eighth day is only mid'Rabanan, and it is only a custom to forbid wheat due to Letisah, but letter of the law it is permitted (453:5), this is not called Mevatel Isur l'Chatchilah. He says that the Radvaz permitted only to sell to another, but he himself may not eat it, for he was Mevatel an Isur l'Chatchilah. It seems that neither of them saw the Teshuvah. The Radvaz said 'I saw the wheat on Motza'ei Yom Tov, and it was dry and had not become Chametz or split. Also, it is an Isur mid'Rabanan at a time (after Pesach) when Chametz is Asur only mid'Rabanan. The Isur is only due to a fine. Here, there is no reason to fine him. He did not transgress Bal Yera'eh and Bal Yimatzei, for on Yom Tov he could not destroy it. Do not say that he should have been Mevatel before it became Chametz, for he did not think that it is Chametz. Also, after Pesach the Isur is only mid'Rabanan, so l'Chatchilah one must be Mevatel it in a majority, like light Isurim such as Terumah of Chutz la'Aretz. Therefore I permitted to mix it little by little with other wheat, and to sell it even to a Yisrael.' Clearly, the Radvaz permits even to the owner. Mateh Yehudah erred, for his text said 'sell it to a Yisrael.' The word 'even' was omitted.

Kaf ha'Chayim (30): If others do not know that it is forbidden, one must tell them.

Kaf ha'Chayim (31): If he died, it is forbidden to his children. We do not say that we fined only him, but not his children.

Kaf ha'Chayim (34): If one saw on Erev Pesach that his wheat was soaked in water, and he did not have time to sell it to a Nochri, and he left it until after Pesach, if the wheat split, some forbid eating it, but permit selling it to Nochrim. Others totally forbid, for the Rambam forbids even if there was Ones.

Kaf ha'Chayim (35): Reuven gave money to Shimon in another city to buy merchandise. Shimon bought Chametz in the summer, and sent it on a boat to Reuven. Reuven received it only after Pesach. It is permitted, for he did not know about it at all. If he knew about it before Pesach, he must sell or give it to a Nochri. If he did not, it is Asur b'Hana'ah.

Kaf ha'Chayim (37): If one has Chametz on a boat or in another city, he may make it Hefker in front of three people for all of Pesach, and after Pesach he immediately acquires it, for he made it Hefker only for half of Erev Pesach and made it totally Hefker. The Radvaz says that one should not rely on this Heter.

Shulchan Aruch (5): If Chametz is found in a Yisrael's house after Pesach, it is forbidden, even though he was Mevatel.

Beis Yosef (DH v'Chametz): The Tur wrote Stam that the Yerushalmi forbids, for it is known that the Halachah follows R. Yochanan. He says that Ba'al ha'Itur forbids eating, but permits Hana'ah. Perhaps he holds that we are concerned for scheming only for a big Hana'ah, i.e. eating. Other Hana'os are small. The Tur rejected this.

Magen Avraham (8): The same applies to one who was Mafkir his Chametz. Rash ha'Levi (OC 12) said 'if one is on a boat, it seems that he may make all his Chametz Hefker, and acquire it again after Pesach and sell it, but not to eat it, for this is not common. (Chachamim decreed only about common matters - PF.) However, due to the severity of Chametz, I was not brazen to permit in practice.'

Mishnah Berurah (25): Several Acharonim say that even if one checked normally (on the night of the 14th) and found Chametz after Pesach, it is forbidden. Chachamim did not distinguish. Some are lenient if he checked and was Mevatel. He did what he was supposed to do. What else could he do? At least Hana'ah is permitted. One may rely on them to avoid a big loss. If one was on a boat and there was no Nochri to whom he could sell, and he was Mafkir in front of witnesses, Acharonim argue about whether he may acquire it after Pesach and benefit from it. To avoid a big loss, one may sell it to a Nochri or exchange with him. If one was on the road and remembered that he has Chametz in his house, and there was no one to sell to properly on Erev Pesach, and he was Mafkir the Chametz in his house in front of witnesses, he may rely on the lenient opinion and acquire it after Pesach and benefit from it. Since he did not have the Chametz with him, and he could not sell it or destroy it like Chachamim enacted, he was obligated to be Mafkir it, lest he transgress Bal Yera'eh and Bal Yimatzei. If so, we should not fine him.

Kaf ha'Chayim (95): If one knew that there is Chametz in his house, Bitul or Hefker does not help. If he did not know about it, or there was a total Ones, i.e. he thought that he sold it, and afterwards he found out that it was not sold, Chak Yakov permits Hana'ah after Pesach. Mekor Chayim disagrees. Perhaps one may be lenient to avoid a tremendous loss.