Ratios of Hameitz-to-Other for Owning

Question

For the prohibition of owning hameitz on passover, what is the ratio of hameitz-to-other that makes a mixture hameitz? The prohibition of owning hameitz is a separate mitzvat lo-ta'asei, as listed by the Rambam at the beginning of Hilkhot Hameitz u-Matzh, so I assume it might have different laws than the prohibitions of eating hameitz or of deriving benefit from hameitz, which is why I'm asking this as a separate question.

Answer 1

For the first answer, I was involved in this big Suggiah, Rambam, Ramban on Sefer Hamitsvot, and Rabanu Tam, others. It appears that explaining the halacha lead to high degree of complexity. To make a list of rules is more easy but is not a "limudic" approach. Rambam, Ramban in Sefer Hamitsvot and Megilat Ester on S.H. are an harmonious cascade. I am sparing the Megilat Ester here.

Rambam Lo Taase 198

We are prohibited to eat products that contain mixed Chamets, despite their are not bread, Murias, Kutach, Beer, and all that ressembles them. This rule comes from the verse ( "any 'Machmetset'"). Let me quote here from Mechilta (Derashbi): "Do not eat any Machmetset, this includes Kutach Habavli, Medean beer, Edomian vinegar. May be that one may be punishable by Karet , ... they are excluded from Karet because they are not fully fledged Chamets. The Gemara Pesachim 44a rules that despite the ban to eat mixture of Chamets, flagellation is only for eating a mixture with Kezait Bikde Achilat Pras, a mixture of Chamets with less that this proportion is prohibited to eat but do not lead to flagellation.

The objection of Ramban

His opinion is that when the proportion of Chamets is Kezait Bikde Achilat Pras, it is banned from Torah and from this ban (198) we learn flagellation but not Karet. And when the proportion is less than Kezait Bikde Achilat Pras, no flagellation because the ban is not from Torah but a Rabbinic ban, excepting Makat Mardut (if someone is rebelling against Rabbinic rules). He wrote the same in his great book (Mishne Torah).

However, the truth on this issue is totally contrary to his words.

Summary of the opinion of Ramban

1. A mixture with a proportion of Chamets of Kezayt Bikde Achilat Pras is equivalent to fully fledged Chamets, not mixed.
2. Each ban, when the proportion is KAP, has all his properties, if flagellation, flagellation, if Karet, Karet. This Is Called Taamo Umamasho and Gemara AZ 67b states flagellation concerning bans punishables by flagellation.
3. The rule regarding mixed Chamets involves only lowest proportion. And this rule has not been selected on Halacha. It is the view of Rabbi Eliezer and we follow Rabanan
4. The verse of Machmetset for Rabanan prohibits Chamets whom chimuts was catalyzed by something not chamets. This is not an additional mitsva but a modallity and is not counted (according to shoresh 9).

Regarding the prohibition of owning Chamets

1. According to Rambam, mixture of KAP is prohibited as mixture, according the Halacha (we follow Rabbi Eliezer). A lowest proportion is cancelled and not considered as a mixture of chamets from a Torah point of view.
2. According to Ramban, a mixture is not prohibited from a Torah Point of view. But when the proportion reaches KAP, this is fully fledged Chamets and is prohibited for owning and Karet for eating.
3. Concerning the owning prohibition, the result is the same for Rambam and Ramban, but the explanation is different: a mixture with KAP is prohipited for ownning on pessach.

But the practical Halacha involves different views concerning the texts of Rambam

and my first answer tried to explain this very confusing issue.

Answer 2

Halacha

(The following is not a comprehensive halachic discussion)

We are dealing only with Chamets mixed before Pesach

1. Min beMino (homogeneous mix), in a dry mixture (for example flour Chamets with flour Non Chamets): Cancelled by ratio of one against two; a liquid mixture too. No problem of owning (Mishna Berura 442, sk 1).
2. Min besheEino Mino (heterogeneous mix, the tastes are different) in a volumetric isur/heter ratio up to 1/59 (when the taste may be identified), if the Chamets mixed was initially an entire piece. According to the Mishna Brura there is a prohibition of owning (when we have a global amount of Kezait of Chamets into the mixture op. cit.).
3. Min besheEino Mino (the tastes are differents) in a isur/heter ratio less than or equal to 1/60, even if the Chamets mixed was initially an entire piece, no prohibition of owning Chamets.
4. If the taste of the Chamets is only exsudated from dishes, according to Mishna Brura, the prohibition of owning doe's not apply. But we can object that Torah uses the same measures for owning and eating (See Peri Megadim Pticha Lesiman 442)

Discussion

As mentioned in the OP, owning Chamets is a separate prohibition of the Tora, and it is admitted that Chamets owning may be prohibited even when the Chamets is only a "mixt partner" (we adopted the choice of this expression according to Shulchan Aruch, which reproducts the Rambam's words). We did not study other opinions which addressed some mixtures as if Chamets is alone and non-mixed).

1. The concept of mixture is obvious in common language. But halachic relevance of mixture is not easy to define. The "method for measuring the concentrations" is particular. Let me give you an example. A quantity of 1 olive (the name of the unit of volume) of Chamets in mixture, can be eaten within a unit of time called "half loaf". Some Rishonim would prohibit to owning it as "non mixed chamets", some others prohibit too but because they think that the prohibition of owning exist also for mixed Chamets. We can see relative convergence of practices despite a divergence of theoretical view.
2. • We use the concept of the Eating Rate Concentration, ERC of Kazait Bikde Achilat Pras KAP,
   • The fact that a prohibited component of a mixture may be detected and identified by the taste, is called Taam KeYkar TKY
3. The link between the owning and the fact that some aliments need to be destroyed because containing Chamets should be left out.
4. The existence of rabbinical rules derived from the prohibition of owning is left out.
5. There is a considerable overlapping between the owning prohibition and the Mitsvat Byur, from a Torah and from a Rabbinical point of view. We do not talk about it. Poskim inquired about the nature of taste of Chamets exudated from dishes. It seems that most poskim opinion ruled out this taste from problem of owning Chamets.

Mishna Pesachim Pessachim 3, 1

‏ ‏‏ אֵלּוּ עוֹבְרִין בְּפֶסַח,‏‏ כֻּתָּח הַבַּבְלִי, וְשֵׁכָר הַמָּדִי, וְחֹמֶץ הָאֲדוֹמִי, וְזֵתוֹם הַמִּצְרִי, וְזוֹמָן שֶׁל צַבָּעִים , וַעֲמִילָן שֶׁל טַבָּחִים, וְקוֹלָן שֶׁל סוֹפְרִים. רַבִּי אֱלִיעֶזֶר אוֹמֵר, אַף תַּכְשִׁיטֵי נָשִׁים. זֶה הַכְּלָל, כָּל שֶׁהוּא מִמִּין דָּגָן, הֲרֵי זֶה עוֹבֵר בְּפֶסַח. ‏‏הֲרֵי אֵלּוּ בְאַזְהָרָה‏, וְאֵין בָּהֶן מִשּׁוּם כָּרֵת: ‏
(Soncino in note, translation from the perspective of Rashi): (On account of) the following [things] you transgress the injunction (leaven) 'shall not be seen' and (leaven) 'shall not be found' (in the house) on Passover: Babylonian Kutah, Median Beer, Idumean Vinegar, Egyptian Zithom, the Dyer's Broth, Cook's Dough, and the Scribes' Paste. R' El(ie)['a]z(e)[a]r (according to Baal haMaor "El'azar") said: women's ornaments too. This is the general, rule: whatever is of the species of corn must be removed on passover 'these are subject to a warning (of flagelation, if someone eats them), but they do not involve kareth. {The first three items enumerated in our Mishna are food , mixtures containing Chamets. However, for some reason, the ERC is less than KAP}
The Gemara observes that in the list we have mixtures and hardened Chamets (not intended for feeding) ‏ מאן תנא דחמץ דגן גמור - על ידי תערובות {רש"י: כותח הבבלי ושכר המדי}, ונוקשה {רש"י: תכשיטי נשים} בעיניה בלאו? ‏ Which Tanna [holds] that real leaven of corn in a mixture (Rashi Kutach Habavli, Shechar Hamadi), and spoiled leaven(Rashi: Cosmetic products for women) in its natural condition, is subject to a negative injunction? (I.E. Witch is the Tana of our Mishna?). In conclusion this Tanna is Rabbi Eliezer according to the Beraita ‏ על חמץ דגן גמור - ענוש כרת, על עירובו בלאו ‏ For real leaven of corn there is the penalty of kareth; for a mixture of it [one is subject to] a negative injunction. This stam Mishna follows the opinion of Rabbi Eliezer.

Mishna 5

שְׂאֹר, יִשָּׂרֵף, וְהָאוֹכְלוֹ פָטוּר. סָדוּק, יִשָּׂרֵף, וְהָאוֹכְלוֹ חַיָּב כָּרֵת. אֵיזֶהוּ שְׂאֹר, כְּקַרְנֵי חֲגָבִים. סָדוּק, שֶׁנִּתְעָרְבוּ סְדָקָיו זֶה בָזֶה, דִּבְרֵי ‏‏רַבִּי יְהוּדָה ‏‏‏‏. וַחֲכָמִים ‏‏ אוֹמְרִים , זֶה וָזֶה, הָאוֹכְלוֹ חַיָּב כָּרֵת. וְאֵיזֶהוּ שְׂאֹר, כָּל שֶׁהִכְסִיפוּ פָנָיו כְּאָדָם שֶׁעָמְדוּ שַׂעֲרוֹתָיו:‏
Si'ur must be burnt, while he who eats it is not culpable {as Hardened Chamets, there is no Kareth, nor "subject of warning" (this Stam Mishna follows the opinion of Rabbi Yehuda and Eliezer about hardened Chamets, if we eat hardened Chamets, no flagellation, and the quote of Rabbi Eliezer above is in reality Rabbi El'azar Gemara 43a, Baal haMaor)}; sidduk must be burnt, while he who eats it [on passover] is liable to kareth. what is si'ur? [when there are lines on the surface] like locusts' horns; sidduk is when the cracks have intermingled with each other: this is the view of R`Judah. but the Sages {In Gemara 48b, they said that "Sages" are Rabbi Meir} maintain: regarding the one and the other, he who eats it is liable to kareth. and what is si'ur? when its surface is blanched, like [the face of] a man whose hair is standing [on end] {and Chachamim (=RM) think that who eats this kind of Si'ur is subject to 40 [flagellation] (Brayta) }.

Rambam, Zemanim, Chamets Umatsa, 4, 8

Looks as a paraphrase of the mishna following the comment of Rashi
‏ תערובת חמץ עוברין עליה משום בל יראה ובל ימצא. כגון המורייס וכותח הבבלי ושכר המדי שעושין אותו מן הקמח. וכל כיוצא באלו מדברים הנאכלים. ‏

The Magid Mishne

Magid Mishne diverted the text from its literal meaning of the Rambam (to avoid internal contradiction with Rambam in chapter 1, 6) and says that Bal yerae is only with Kazait bikede Achilat Pras (because KAP is as id the Chamets is alone and not mixed). Rambam do not follow R. Eliezer but Rabanan. So, mixed Chamets has no problem of owning. But KAP prohibited for owning.

Magid Mishne and kesef Mishne Ramach

(You can see a very lengthy pilpul of the Kesef Mishne chapter 1, halacha 6) Regarding Byur, Rambam follows Rabbi Eliezer (stam Mishna 3, 1) that mixture is on Bal Yerae only. But regarding eating, he follows Rabanan (stam Mishna 3, 5 concerning Hardened mixture), there is no negative injunction. So a mixture of 1/59 that contains an entire Kazait of small pieces of Hamets should be destroyed or annulled. Magen Avraham rules as the Magid Mishne and the Gra as the Ramach. And Mishna Brura seems to rule as the Gra.

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Conceptually, Kazait bikde Achilat Pras design a substantial presence of the element whiich allows us to eating it with hight rate as if we eat it alone. See Gemara A.Z. 67b "Taamo Umamasho".[. 1- the total amount 1 kezayt, 2.- the ERC is KAP. See Rabenu Tam and Rav Chayim Cohen (in Rosh Chulin, perek 7, siman 31). See the mishna of mikpa here and Tos. Nazir and Pesachim, One might carefully read the mishna in tevul yom (2, 3). See also Ran in Chidushim (different from Ran al harif for this topic) A.Z. Suggia of Noten Taam lifgam about the volumetric importance of the Issur beeyn. KAP is also a way to mesure the possibility to eat the issur as if it is alone.It is as if you can keep several little particles alone without mixture. 2 steps to build the Issur. 1 .- detected presence and not bitul (TKY). 2.- proximity or concentration and accessibility between particles that makes an effect as if you eat them alone (KAP)].