Gemara

(Rav): [During Pesach, any amount of] Chametz forbids both b'Mino (if it is mixed with its own kind, i.e. grain) and she'Lo b'Mino;

Rav and Shmuel): All forbidden foods, if mixed b'Mino, forbid b'Mashehu. Lo b'Mino, they forbid b'Nosen Ta'am.

Rav decrees [to forbid] she'Lo b'Mino due to b'Mino.

(Shmuel): Chametz [b'Mashehu] is forbidden b'Mino, but is permitted she'Lo b'Mino.

He does not decree.

(R. Yochanan): Chametz forbids b'Nosen Ta'am, whether b'Mino or Lo b'Mino;

(R. Yochanan and Reish Lakish): All Isurim forbid b'Nosen Ta'am, whether b'Mino or Lo b'Mino.

(Rava): The Halachah is, Chametz forbids b'Mashehu, whether b'Mino or Lo b'Mino, like Rav;

After Pesach, it is permitted even b'Mino. This is like R. Shimon.

Question: Rava said that R. Shimon fines [to forbid Chametz after Pesach] because one transgressed Bal Yera'eh and Bal Yimatzei!

Answer: He fines intact Chametz, but not a mixture.

42a (Mishnah): The following are Ovrim during Pesach: Kutach of Bavel (a dip made with stale bread and whey), beer of Madai, vinegar of Edom...

The general rule is, during Pesach, one transgresses for anything from grain. There is a Lav for it, but not Kares.

43a (Beraisa): "Kol Machmetzes Lo Sochelu" includes Kutach of Bavel, beer of Madai, and vinegar of Edom;

"Ki Kol Ochel Chametz v'Nichresah" obligates Kares for pure Chametz, but one transgresses [only] a Lav for mixtures.

Rishonim

Rif and Rosh (13a and 3:1): Rabanan exempt from lashes when there is not k'Zayis of Chametz bi'Chdei Achilas Pras (in half a loaf, i.e. three or four eggs, of the mixture). If there is k'Zayis bi'Chdei Achilas Pras, Chachamim agree that one is lashed.

Rambam (Hilchos Chametz 1:5): If Chametz became mixed with something else during Pesach, whether Min b'Mino or b'Eino Mino, any amount forbids. Chametz of a Yisrael she'Avar Alav ha'Pesach is Asur b'Hana'ah. If it became mixed, whether b'Mino or b'Eino Mino, one may eat it after Pesach. Chachamim forbade only the Chametz itself, but not the mixture.

Rambam (6): A mixture of Chametz such as Kutach of Bavel, beer of Madai and similar things, if one ate it during Pesach he is lashed, but there is no Kares - Kol Machmetzes Lo Sochelu. This is if he ate a k'Zayis of Chametz bi'Chdei Achilas three eggs. For less, he is lashed only mid'Rabanan.

Ra'avad: The Rif rules about these like Chachamim [who say that there is no Lav for mixtures], for there is not k'Zayis bi'Chdei Achilas Pras.

Rambam (4:8): One transgresses Bal Yera'eh and Bal Yimatzei for a mixture of Chametz such as Muryas (fish oil with flour), Kutach of Bavel, beer of Madai made from flour, and all such foods.

Magid Mishneh: The Rambam (1:6) said that there is a Lav for eating a mixture only if there is k'Zayis bi'Chdei Achilas Pras. Surely, one transgresses Bal Yera'eh only in such a case. Some Ge'onim say that Bal Yera'eh applies even to a sparser mixture. The Ramach says that the Rambam agrees. Even though he is exempt for eating it, he must do Bi'ur, for the Chametz is here. I am unsure.

Kesef Mishneh: I agree with the Ramach. Regarding eating, if there is less than k'Zayis bi'Chdei Achilas Pras, he does not eat a k'Zayis within the time. It is like eating half a k'Zayis today and half tomorrow. Bal Yera'eh applies to a mixture with a k'Zayis of Chametz. It does not matter whether it is mixed with more or less than a Pras!

Magid Mishneh (12): A mixture of Chametz is like other mixtures, just other Isurim cannot forbid 60 times as much Heter, but Mashehu of Chametz forbids. When other mixtures do not need 60 to permit, rather, it suffices to peel off a layer or cut off a small amount, the same applies to Chametz. I.e. if two dry hot foods touched, a layer suffices. Since the Rambam holds that one transgresses for a mixture of Chametz, surely he forbids after Pesach. R. Shimon fines only for intact Chametz, i.e. but not if it became mixed after Pesach, for no Isur was done with this mixture. If it became mixed beforehand, he was obligated to do Bi'ur. Since he transgressed, it is forbidden. Some Meforshim said so. This is only if there is k'Zayis bi'Chdei Achilas Pras.

Poskim

Shulchan Aruch (OC 442:1): One transgresses Bal Yera'eh and Bal Yimatzei for a mixture of Chametz such as Muryas, Kutach of Bavel, beer of Madai, and all such foods.

Taz (1, b'Sof): The Shulchan Aruch did not distinguish whether or not there is k'Zayis bi'Chdei Achilas Pras. This is from the Rambam.

Magen Avraham (1): If Min b'Mino became mixed, e.g. flour with flour, if there was Bitul b'Rov, one does not transgress, for mid'Oraisa one may eat it l'Chatchilah. All the more so one may keep it. Min b'Eino Mino, one is not lashed for eating it if there is not k'Zayis bi'Chdei Achilas Pras, but one must do Bi'ur, for it joins automatically. The Tur and Semak say that anything the Torah did not forbid to eat, l'Chatchilah one may mix it and keep it. Tosfos (2a DH Ohr) says that Chachamim did not distinguish.

Gra (DH Ovrim): The Ramach is primary. The Mishnah discussed Kutach and beer of Madai. The Gemara says that they do not have k'Zayis bi'Chdei Achilas Pras. Also, the Rambam distinguished regarding eating, but he did not distinguish regarding Bal Yera'eh. The Magid Mishneh equates this to eating. The Rambam does not. He says (4:11) that Bal Yera'eh does not apply to Nuksheh (inedible Chametz) used to launder or for glue, for it lacks the form of Chametz. This implies that if not for this, one would transgress, even though it is Nuksheh.

Kaf ha'Chayim (3): The Acharonim hold that Bal Yera'eh applies even if there is not k'Zayis bi'Chdei Achilas Pras.

Mishnah Berurah (1): The Mechaber discusses only Bal Yera'eh and Bal Yimatzei. He holds that one transgresses even without k'Zayis bi'Chdei Achilas Pras. Even though all exempt one who eats it from Kares, and some exempt even from a Lav, Bal Yera'eh and Bal Yimatzei apply. Since there is a k'Zayis in all, and it is all in one Kli, it joins. This is only for Min b'Eino Mino, which is not Batel in the majority according to the opinion that Ta'am k'Ikar is mid'Oraisa. Min b'Mino is Batel mid'Oraisa in the majority, and one may eat it, just mid'Rabanan one may not keep it, lest he come to eat it during Pesach. It is permitted after Pesach, for he did not transgress Bal Yera'eh and Bal Yimatzei. The same applies if there is 60 times as much as Heter as Chametz. Chametz forbids b'Mashehu only mid'Rabanan. The same applies to anything that for which Bi'ur is only mid'Rabanan. If one remembered during Pesach he must burn it then, but if he did not, it is permitted after Pesach. If the Chametz is not intact, rather, it is a mere taste, e.g. one cooked something with Chametz and removed the Chametz, and only the taste remains, for there was not 60 parts of food to be Mevatel it, the Torah forbids eating it due to Ta'am k'Ikar. Even so, Chak Yakov says that Bal Yera'eh does not apply, since there is no substance of Isur. Surely mid'Rabanan one may not keep it. He must do Bi'ur if he finds it during Pesach. If he left it, it is permitted after Pesach. Since he did not transgress mid'Oraisa for keeping it, there is no fine. Perhaps one should be stringent not to eat it. Some disagree. Since the Torah forbids eating it, also Bi'ur is mid'Oraisa, so after Pesach it is Asur b'Hana'ah. To avoid a big loss, one may be lenient like the first opinion. All this refers to a Tavshil (cooked food) that received taste of Chametz. If a Tavshil was cooked before Pesach in a pot in which Chametz was cooked, even if the Tavshil is not 60 times as much as the Chametz emitted from the pot, one may keep it until after Pesach. (One must hide it in a room where he does not normally go.) It may not be pickled during Pesach in a Chametz Kli, i.e. [rather] it is emptied into a Pesach Kli (i.e. one never used for Chametz). If it was cooked or pickled l'Chatchilah during Pesach in a Chametz Kli, even if it was not a Ben Yomo, one must do Bi'ur.

Kaf ha'Chayim (5): The Chasam Sofer (108) forbids keeping a food cooked in a Ben Yomo pot only if it is sharp.