Daf Yomi Beitza 16a -- Leftovers or initial eruv tavshilin?

The Mishna {from Beitza 15b}:

Mishna:
יום טוב שחל להיות ע"ש לא יבשל בתחלה מיו"ט לשבת אבל מבשל הוא ליו"ט ואם הותיר הותיר לשבת:
ועושה תבשיל מע"יוט וסומך עליו לשבת
בש"א שני תבשילין וב"ה אומרים תבשיל אחד
ומודים בדג וביצה שעליו שהן שני תבשילין
אכלו או שאבד לא יבשל עליו בתחלה
שייר ממנו כל שהוא סומך עליו לשבת

Yom Tov which occurred on erev Shabbat, one should not begin cooking from {/on} Yom Tov for Shabbat; however, he may cook {on Yom Tov} for Yom Tov and if he left over, he left over for Shabbat.

{Alternatively,} He may make a meal from erev Yom Tov {for Shabbat} and rely upon it for Shabbat {and thus when he continue cooking on Yom Tov for Shabbat, for now he is not beginning to cook but is merely continuing}.

Bet Shammai say: Two cooked foods.
And Bet Hillel say: One cooked food.

And they {=Bet Shammai} admit that a fish and the egg {smeared} upon it that they are {=count as} two cooked foods.

If he ate it {on Yom Tov} or lost it, he should not cook {=relying} upon it {to continue cooking on Yom Tov for Shabbat}.

If he left over from it any amount, he may rely upon it for Shabbat.

There is a surprising gemara on Beitza 16a:

תני ר' חייא עדשים שבשולי קדרה סומך עליהן משום ערובי תבשילין והני מילי דאית בהו כזית
אמר רב יצחק בריה דרב יהודה שמנוני' שעל גבי הסכין אדם גורדו וסומך עליו משום עירובי תבשילין והוא דאיכא כזית

Rabbi Chiyya taught {a brayta}: Lentils at the bottom of a bot, one may rely upon them for eruv tavshilin {to cook for Shabbat}.

{The gemara clarifies:} And these words are where they are of an olive's measure.

Rav Yitzchak son of Rav Yehuda said: One may scrape off the fat off a knife and rely upon it for eruv tavshilin.
{The gemara clarifies:} And this is where there is an olive's measure.

Rashi explicitly interprets both the lentils at the bottom of the pot, and the fat from off the knfe, as collecting them before Yom Tov and designating it as eruv tavshilin. This Rashi is surprising! Why should he bother to do this on erev Yom Tov, as opposed to simply making a cooked dish?

In fact, it is clear that both the lentils at the bottom of the pot and the fat residue on the knives are leftovers. Rather than designate them as initial primary eruv tavshilin, it makes a whole lot more sense to take them as leftovers of the initial eruv tavshilin.

That is, there are requirements of initially establishing an eruv tavshilin. If the eruv tavshilin is lost or eaten up, then one may not continue to rely upon it to cook from Yom Tov for Shabbat. However, if there are any leftovers, one may rely on the leftovers. Indeed, this is what the Mishna states explicitly:

אכלו או שאבד לא יבשל עליו בתחלה
שייר ממנו כל שהוא סומך עליו לשבת

If he ate it {on Yom Tov} or lost it, he should not cook {=relying} upon it {to continue cooking on Yom Tov for Shabbat}.

If he left over from it any amount, he may rely upon it for Shabbat.

This case of eating the eruv tavshilin appears to fit perfectly into the two cases in the gemara. Lentils at the bottom of a pot after having eaten the rest of the lentils or grease on a knife after having eaten the food are exactly a case of leftovers, so it makes a lot of sense that we are dealing with leftovers, as in the gemara.

Now, it is possible to take exception to the gemara's two clarifications -- that we are dealing with an olive's measure left over. Perhaps one can say this by the lentils at the bottom of the pot, but it is more difficult (though admittedly not impossible) to say this by residue on knives.

Indeed, it is the gemara and not the brayta that state that this is where the lentils are a kezayit. And it is the setama digmara and not Rav Yitzchak son of Rav Yehuda who states that the residue on the knives are a kezayit.

Pashut peshat would be that they are not a kezayit. And if so, it would be in accord with a simple reading of our Mishna. Recall that our Mishna stated:

שייר ממנו כל שהוא סומך עליו לשבת

kol shehu means any amount whatsoever, not specifically a kezayit.

Later, the gemara redefines the kol shehu in the Mishnato mean kezayit, but it does not yet do so at this point in the gemara. And there is no challenge on the requirement of kezayit from the Mishna. This is just the setama retrojecting its reinterpretation of the Mishna onto statements elsewhere in the gemara. But the fact that there is no challenge here from the Mishna, or discussion of kol shehu may well be what prompts Rashi to understand that this was done on erev Yom Tov, at the initial establishment of the eruv tavshilin.

Why does the gemara reinterpret kol shehu as kezayit? Because on Beitza 16b, Rabbi Abbi states that eruv tavshilin requires a kezayit. This means one kezayit, and even if 100 people are relying upon it, we do not require 100 olive measures.

The setama digmara, however, challenges this statement of Rabbi Abba on the basis of the Mishna where it states that as long as there is kol shehu, any amount. The setama digmara offers a pat answer, that by kol shehu the Mishna meant "so long as it is a kezayit."

To be frank, this seems a bit unlikely. A more likely answer is that there is a distinction between the requirement for initially setting up the eruv tavshilin and what still counts as eruv tavshilin once Yom Tov has come in and one ate from it. Indeed, that is the plain implication of the Mishna when it states that if one left over any amount he may continue to rely upon it. This is a chiddush in the Mishna that one may rely on it in any amount. The idea behind it was that while you need a basic shiur to initially set up an eruv tavshilin, once it is set up, so long as any amount of it is left, it is as if it is still there.

And, if the basic amount for initially setting up an eruv tavshilin is identical to the one for keeping one around after it was eaten, then what in the world is the Mishna coming to tell us in the seifa. This is obvious!

The brayta reinforces this reading of the Mishna, that kol shehu really means any amount. And the statement by Rav Yitzchak son of Rav Yehuda reinforces that interpretation and demonstrates that Amoraim held by this. Ravina and Rav Ashi are sof horaah. If so, the setama digmara should not be able to uproot this Tanaaitic and Amoraic law, and we should say that even if any amount is left, even less than a kezayit, the eruv tavshilin is still effective.

Not to be relied upon halacha lemaaseh.