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Chametz that was owned by a Jew during Pesach:
Complete Chametz that was owned by a Jew over Pesach, or that a Jew had liability over during Pesach, is forbidden in benefit for all Jews after Pesach. This applies whether it was owned throughout the entire Pesach, or was only owned through part of Pesach [such as he acquired it from a gentile on Pesach], and applies even if it was only owned for the last part of the last day of Pesach in the Diaspora. This applies even if the owner of the Chametz was not at fault for owning it, being that he was not able to destroy it, or did not know at all that this Chametz was in existence until after Pesach, nevertheless the Chametz is forbidden in benefit [for all]. [Thus, any Jew who did not perform Mechiras Chametz, all Chametz that was in his possession anytime during Pesach is forbidden in benefit for all Jews even after Pesach.]
Chametz of a non-frum Jew and Heretic: Chametz that was owned by a Jewish apostate is forbidden in benefit after Pesach. 
Chametz had joint ownership of Jew and Gentile: Chametz that was jointly owned by a Jew and gentile is forbidden in benefit until the Chametz is split up between the partners. Once it is split after Pesach, the portion of the gentile is permitted to be eaten, while the portion of the Jew is forbidden even in benefit.
Chametz of a gentile that was bought from a Jew: Chametz that was owned by a Jew during Pesach remains prohibited in benefit even if a gentile acquired the Chametz from him. Thus, if one buys such Chametz from a gentile it must be destroyed.
Feeding the Chametz to animals after Pesach? Any Chametz that is forbidden in benefit due to being owned by a Jew during Pesach is prohibited to be fed to any animal, whether the animal is owned by oneself or by another Jew or by a gentile, or is wild and ownerless.
Taaruvos Chametz: Taaruvos Chametz [foods which contained Chametz ingredients] is forbidden to be benefited from after Pesach if it was owned over Pesach. However if one redeems the contents Chametz value, then the food is permitted in benefit. This is done by assessing the worth of the Chametz ingredient [i.e. 10 cents] and throwing the money in an area that it will be lost from mankind. Nevertheless, even after redeeming the Chametz, the Chametz remains forbidden to be eaten, being that the taste of the Chametz is still found in the food.
Chametz Nuksha: Chametz Nuksha that was owned by a Jew during Pesach is permitted in benefit after Pesach, although is forbidden to be eaten.
Chametz that is questionable if it was owned by a Jew:
Any Chametz that is Halachically questionable if it forbidden after Pesach, such as it is questionable if it was owned by a Jew, or if it was disowned by a Jew before Pesach, is permitted in benefit, althou8gh is disputed amongst the Poskim as to whether it may be eaten. Some Poskim rule the Chametz is permitted even to be eaten. However, other Poskim rule that any Chametz in which there is question as to whether it was owned by a Jew during Pesach or disowned by a Jew before Pesach, is forbidden to be eaten, although is permitted in benefit. Practically, one is to be stringent like the latter opinion not to eat the Chametz, unless it’s a case of great loss [and the Chametz cannot be sold]. According to all, one may benefit from the Chametz.
Chametz found in public area after Pesach: All Chametz that is found in an area which has a majority gentile population and passersby, can be assumed to have been owned by a gentile and is permitted to be eaten and benefited from after Pesach. If, however, it was found in an area with a majority Jewish population or passersby, then the Chametz is forbidden in benefit if it was found immediately after Pesach. If, however, it was found some time after Pesach, or the area contains an equal Jewish and gentile population, then the status of the Chametz is subject to the above-mentioned dispute, and the final ruling is that the Chametz is not to be eaten, although may be benefited from.
One was walking outside two days after Pesach and found a closed Kosher cake sitting in the bushes. If the area contains a majority gentile population, the cake may be taken and eaten. If, however, the area contains a majority Jewish population, then the cake may not be eaten.
 Admur 448:1; Michaber 448:3; Mishneh Pesachim 28a
 Literally “complete grain”
 Chametz of a child: It is questionable whether Chametz that was owned by a Jewish child over Pesach is forbidden in benefit. [See Piskeiy Teshuvos 448:3]
 Admur 440:17; See chapter 2 Halacha 5C!
 The reason it is forbidden for all Jews: As the sages fined the owner of the Chametz due to the fact that he transgressed Baal Yiraeh and Baal Yimatzeh, and the sages did not want to differentiate in their decree between the owner and others and thus forbade it for everyone. [Admur ibid; M”A 448:3; Rava Pesachim 29a]
 Admur ibid; M”A 448:1; Chok Yaakov 448:7; Tur; Baal Haittur 2:126
Conservative Rabbi: Thus, if a conservative/reform Rabbi sold Chametz for his congregation, then even if he did all the sale laws properly the Chametz is forbidden on Pesach if he buys it back on the last day of Pesach in the Diaspora.
 Admur ibid; Michaber ibid; Rambam 1:4
The reason: They forbade it in benefit even in such a case in order to prevent people from leaving over their Chametz until after Pesach and then claiming that they were unable to destroy it. [ibid]
 Admur 448:26
Other opinions: Some Poskim rule that the Chametz is permitted after Pesach. [See opinions in Piskeiy Teshuvah 447:9]
 The reason: As even though he has sinned, nevertheless, he still remains a Jew, and thus [by him owning the Chametz] the Chametz becomes prohibited in benefit after Pesach, just like any Chametz that is owned by a Jew on Pesach. [Admur ibid]
 Admur 448:27
 Admur 443:13
 Admur 448:31
 Admur 442:3 and 6
 Admur 442:20
 The reason: Since he did not transgress a Biblical prohibition by owning it, the sages therefore did not fine him against benefiting from it. However, it may not be eaten being that it still has a “Shem Chametz Sheavar alav Hapesach.” [Admur ibid]
 Admur 448:30
 1st opinion in Admur ibid; Taz 449:1; Chok Yaakov 449:1
 The reason: As by every Rabbinical prohibition, when there is doubt we are lenient. [Admur ibid]
 2nd opinion in Admur ibid; M”A 449:2; Bach 449
 So is evident from M”A ibid and Bach ibid based on Eiruvin 64b as explained in Machatzis Hashekel, and so is implied from Admur ibid as even Chametz that was disowned before Pesach he defines as “Sheavar Alav Hapesach”
 Admur 448:30; Eiruvin 64b; See M”A and Machatzis Hashekel ibid; Upashut that this case is not considered a Safek at all, and the continuation of Admur regarding Safek and the dissenting opinion is only going in a case of majority Jewish, or equal Jewish population. Vetzaruch Iyun today that all products are kept in their wrapping, perhaps we would assume a Kosher product was bought and discarded by a Jew even in a majority gentile area. Vetzaruch Iyun!
 The reason: As in such a case we can assume that a Jew threw it out before Pesach. [Admur ibid]
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