Gemara

(Rav): The Halachah follows the four Tana'im according to R. Eliezer, who says that Shabbos and Yom Tov adjacent to each other are two Kedushos [regarding Eruv].

Contradiction - Rav Chisda (Rav): If Yom Tov is adjacent to Shabbos, an egg laid on the first day is forbidden on the second.

Answer (Rabah): That is [not because they are one Kedushah, rather,] due to Hachanah:

"V'Hayah ba'Yom ha'Shishi v'Hechinu" - a weekday can prepare for Shabbos or Yom Tov, but Yom Tov cannot prepare for Shabbos, or vice-versa.

Beitzah 2a (Mishnah - Beis Shamai): One may eat an egg laid on Yom Tov;

Beis Hillel forbid.

2b (Rabah): Beis Hillel forbid due to Hachanah.

4a (R. Yochanan): If Yom Tov is adjacent to Shabbos, an egg laid on the first day is permitted on the second.

(Rav Papa citing Rava): The Halachah follows Rav, whether this is a leniency or a stringency.

Pesachim 47b - Question (Abaye - Mishnah): One who cooks Gid ha'Nasheh in milk on Yom Tov and eats it is lashed five times - for cooking a Gid [ha'Nasheh] on Yom Tov, for eating a Gid, for cooking meat and milk, for eating meat and milk, and for burning on Yom Tov.

If we say Ho'il, he should be exempt for burning on Yom Tov!

Answer: Rather, it should omit burning, and it should say that the wood was Muktzeh.

Question: Is Muktzeh mid'Oraisa?! (Why is he lashed for it?)

i. Answer (Rabah): Yes! It says "... v'Hechinu Asher Yavi'u";

The Lav against using Muktzeh is "Lo Sa'aseh Chol Melachah."

Objection (Abaye): You contradict yourself!

Question (Rabah, of Rav Chisda or Rav Huna): If one slaughtered a [Muktzeh] sheep for the Tamid on Yom Tov, what is the law?

Answer (Beraisa): One may not offer something intrinsically forbidden, like Tevel;

Muktzeh is not intrinsically forbidden. It is Asur due to something else (Shabbos).

If Muktzeh is mid'Oraisa, is makes no difference whether or not the Isur is intrinsic!

Rishonim

Rif (Beitzah 1b): The Halachah follows Rav, whether this is a leniency or a stringency.

Ran (Beitzah 1a DH v'Ukma): It is considered Hachanah only because the egg was finished the day before. We are not concerned for growth on Shabbos. Otherwise, we could not slaughter an animal on Yom Tov after Shabbos!

Ran (1a DH Rabah): "V'Hechinu" teaches about verbal declaration, for it already says that physical preparation must be before Shabbos - "Es Asher Tofu Efu."

Ran (1b DH d'Rav): If an egg was laid on Yom Tov Rishon, Rav forbids it on Yom Tov Sheni mid'Rabanan. On Yom Tov Rishon itself the egg is forbidden only because of a decree lest people permit an egg laid on Yom Tov after Shabbos. Likewise, Hachanah applies mid'Rabanan whenever something is Muktzah on the first day, e.g. fruits that fell off a tree.

Rambam (Hilchos Yom Tov 1:19): A weekday can prepare for Shabbos or Yom Tov, but Yom Tov cannot prepare for Shabbos or vice-versa.

Rambam (24): The two days of Yom Tov of Chutz la'Aretz are two Kedushos. Therefore, if something was Muktzah or Nolad on Yom Tov Rishon, if one was Mechin it for Yom Tov Sheni it is permitted. For example, if an egg was laid or an animal was trapped or vegetation was detached on Yom Tov Rishon, it is permitted on Yom Tov Sheni.

Beis Yosef (OC 513 DH Kosav ha'Rambam): The Rambam permits on Yom Tov Sheni only if he was Mechin on Yom Tov Rishon. It is not clear what his source is. Perhaps he refers only to Muktzah, but not to Nolad.

Rambam (ibid.): An egg laid on the first day of Rosh Hashanah is forbidden on the second. The same applies to other Hachanos. The same applies to Yom Tov on Erev Shabbos or Motza'ei Shabbos, even if Yom Tov Sheni is on Erev Shabbos.

Rosh (Beitzah 1:1): R. Yochanan holds like R. Yitzchak, who decrees to forbid the egg lest people permit juice that flowed. This is farfetched, therefore the Rif rules like Rabah. Also, we hold that an egg is finished one day before it is laid. Also, Rabah holds that Hachanah is mid'Oraisa, so we are stringent about a Safek mid'Oraisa.

Question: If R. Yochanan considers it to be a reasonable decree, how can the Rosh decide that it is not?!

Answer (Korban Nesan'el 5): R. Yochanan does not forbid Hachanah, therefore he was forced to explain the Mishnah differently. We hold that Hachanah is forbidden, so we are not forced to find other explanations.

Note: Even though we forbid Hachanah, it is not obvious that this is the reason for the Mishnah. Perhaps we do not forbid Hachanah bi'Yedei Shomayim (like an egg), or we do not hold that the egg is finished the day before it is laid, or we do not forbid every Yom Tov because of Yom Tov after Shabbos. If we considered R. Yochanan's decree to be reasonable, perhaps we would establish the Mishnah like him! It is difficult to say that Hachanah is forbidden, yet there is no source in the Mishnayos for this.

Question: The Yerushalmi says that Rav forbids to light on Yom Tov a wick that extinguished on Shabbos (Erev Yom Tov), and R. Yochanan permits. Why do we permit this?

Answer (Rosh ibid.): This is not Hachanah (see the Mordechai below). The Yerushalmi concludes that they argue about whether the days are one Kedushah or two.

Rosh (1:7): A calf born on Yom Tov is permitted, for it is Muchan along with its mother. This is if the mother was ready to be eaten. If the owner wants the mother for offspring, the calf is forbidden according to the opinion that forbids Muktzah. Hachanah does not apply here. It applies regarding a laid egg, for that enables it to develop a chick. However, a calf found inside a slaughtered cow is as good to eat as a born calf.

Hagahos Ashri (1:1 DH Aval) and Tosfos (2b DH v'Hayah): Weekday meals are not important, so Hachanah does not apply to them. Hachanah applies only to something that was not around at all, such as an egg. It does not apply to food that merely needed to be fixed (i.e. cooked. Sefer ha'Chinuch (298) says that mid'Oraisa Hachanah applies only to an egg.)

Rashi (2b DH l'Ta'amei) Rabah forbids Muktzah mid'Oraisa.

Tosfos (38b DH d'Amar): Rabah retracted from this (Pesachim 47b).

Pnei Yehoshua (2b DH Sham bi'Gmara): Rabah retracted from saying that one is lashed for Muktzah, but he still holds that it is mid'Oraisa. Also, Rashi means that Rabah forbids the most severe Muktzah, such as Nolad, mid'Oraisa.

Mordechai (Beitzah 644): The Ri says that the Bavli holds that Hachanah does not apply to something that was already in the world but merely needed to be fixed.

Rebuttal (R. Baruch, brought in Rosh 1:7): Rav Chisda concluded that they argue about one or two Kedushos, because Rav Chisda permits Hachanah (Eruvin 38b). We forbid Hachanah, so we hold that they argue about Hachanah. Therefore, if a wick extinguished on Shabbos [that is Erev Yom Tov] or the first day of Rosh Hashanah, one may not light it on the next day unless it had been previously lit and extinguished. (Then it is not Hachanah, for the wick was not improved on Shabbos or Yom Tov). Alternatively, one may be Mevatel it in a majority of Heter.

Rebuttal (Beis Yosef ibid.): According to R. Baruch, if a wick extinguished on Yom Tov it should be forbidden to relight it that day, whether the Isur on Yom Tov Sheni is because both days are one Kedushah, or because of Hachanah!

Defense (Taz 501:7): Surely the Isur is due to Hachanah. Yom Tov prepares for itself! Hachanah applies to the second day of Rosh Hashanah only according to the opinion that it is a second Kedushah. We decree to forbid an egg laid on Yom Tov, for it is finished the previous day. If Erev Yom Tov was Shabbos it is forbidden mid'Oraisa. There is no reason to decree not to relight a wick the same day it extinguished, even if it is Motza'ei Shabbos. Regarding the second day of Rosh Hashanah or Yom Tov after Shabbos one should be stringent not to use a wick that extinguished the day before, but one may rely on the lenient opinions to move such a wick.

Poskim

Rema (OC 501:7): If a log burned on Yom Tov Rishon, one may burn what remains on Yom Tov Sheni, even if it is Motza'ei Shabbos or Yom Tov Sheni of Rosh Hashanah.

Shulchan Aruch (514:4): If a Ner extinguished on Yom Tov and one wants to relight it, he may cut the wick to make it light easier.

Rema: Leftover oil and wicks are permitted even on Yom Tov after Shabbos or on the second day of Rosh Hashanah.

Source: This is from the Mordechai (ibid.)

Magen Avraham (16): Regarding lighting we need not decree lest one permit when Yom Tov is after Shabbos, for people would not confuse Shabbos and Yom Tov!

Question: Avi ha'Ezri (brought in Mordechai ibid.) forbids even moving such a wick because of Hachanah, just like an egg. It should be permitted to light the side that did not burn, for the extinguishing did not improve it!

Answer (Terumas ha'Deshen 81): Avi ha'Ezri discusses a short wick that cannot be lit from the other side. Alternatively, if Hachanah is mid'Oraisa, we should decree not to, lest he use the side that burned, just like we decree about a double-edged Shechitah knife if one of the blades has a nick (Chulin 15b). However, it appears that Hachanah of a wick is only mid'Rabanan since it was already in the world. The Rosh holds that it is mid'Rabanan, for he permits to be Mevatel it l'Chatchilah. However, perhaps the Rosh says so only regarding the second day of Yom Tov or Rosh Hashanah, for mid'Oraisa it is not Kodesh. Even though the wick is a Kli she'Melachto l'Isur, one may not move it even if he needs to use it or its place because he was Docheh b'Yadayim (did an action showing that he will not use it).

Rebuttal (Yam Shel Shlomo 4): The Yerushalmi forbids leftover oil and wicks. Just like all the oil is forbidden, also both sides of the wicks! If a short wick can be lit, it can be lit from either side. It appears that extinguishing improves the entire wick, not just the place that was burning. The Isur to reuse the wick is not a decree lest one reuse the side that burned before. People normally reuse that side, so if it were permitted surely people would transgress! Hachanah is a mere stringency. Primarily, the wick is forbidden due to Nolad and Muktzah. If the other side were permitted, it would not be Muktzah, for it would still be good for its original use! Lighting before Shabbos is not Docheh b'Yadayim with respect to Yom Tov!

Mishnah Berurah (29): It is better to be stringent like the Poskim that forbid lighting on Yom Tov a wick extinguished on the previous day (Shabbos or Rosh Hashanah), or at least to light the other side.

Kaf ha'Chayim (63): Hachanah also applies to leftover oil, for it is clearer than oil that did not burn. It is good to be stringent l'Chatchilah, but if one adds other oil one may rely on the Beis Yosef and others who permit.