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- It is important to buy an etrog that has a reliable hechsher (certification that it is not grafted) in order to be sure that is not grafted.
- An etrog must be at least the size of an egg. If the etrog is longer than an egg but not as wide as an egg, but its total volume is more than that of an egg, it is still kosher.
- If the pitom falls off or the stem on the bottom of the etrog is completely removed, the etrog is invalid.
- Some poskim invalidate an etrog that has black dots found on its chotam, while others deem it valid.
- One can inspect a Etrog with one's eyes and doesn't need to use a microscope or magnifying lens.
- An etrog that is completely green is invalid. If, however, if it started to become yellow, it is valid.
- See note for a list of other ideal qualities. Besides for the qualities that Chazal specified, the niceness of an etrog includes its subjective beauty.
Placing the Etrog in Wool
- Some hold that it is permitted to place an etrog on top of a tallit even if they will make the tallit smell nice if one doesn’t intend for that result, while others hold it is forbidden. However, putting the etrog back into the wool it was in before hand is permitted since that wool already smelled nice.
- A grafted Etrog is pasul whether it was grafted with lemon, promerance, or any other fruit. 
- The signs given by the Achronim to distinguish a grafted etrog from a real one are not to be relied upon nowadays when it’s possible to graft an etrog and make it look very much like an etrog. 
- Rather one should not buy an Etrog except from a trustworthy seller and can testify that the Etrog isn’t grafted. 
- Even if the Etrog is only in doubt one may not make a Bracha on the Etrog. 
- If one has an etrog that’s safek if it’s grafted and an etrog that’s not as hadar (“beautiful”), one should make the Bracha on the true etrog first and after shaking the minim with it, one should take the one that’s a safek grafted. 
From Israel, Morocco, and Chazon Ish
- Many say that the etrogim from Eretz Yisrael have a good mesorah. However, some claim that the etrogim from Eretz Yisrael have no mesora since religious Jews didn’t live there for many years before and after the Ramban. However, others uphold the mesora of the etrogim from Israel.
- Moroccan etrogim have a long tradition of being authentic.
- Chazon Ish etrogim are from trees of Shechem. Rev Chaim Braverman with the support of Rav Yehoshua Leib Diskin authenticated the trees in Shechem as authentic etrog trees. Years later the Chazon Ish went to Shechem and picked one that he felt looked authentic. Afterwards he distributed the seeds and one successfully grew in the yard of Rav Michel Yehuda Lefkowitz.
A Dry Etrog
- An Etrog which is so dry that it doesn't have any moisture is invalid. 
- It is possible to check if an Etrog is invalid by piercing it with a needle with a string attached and if there's moisture on the string the Etrog is not considered dry. In order that this procedure not make the Etrog invalid because of a hole, one should pierce the thick part of the Etrog without piercing the Etrog completely through and through, others suggest that one should pierce it along the length of the Etrog.
- An Etrog from the previous Sukkot is considered a dry Etrog which is invalid. Some say, however, that if an Etrog from a previous year was left closed in a moist and cool area it is possible to be moist and valid for the mitzvah.
A Punctured Etrog
- A hole of any size invalidates the etrog. Many poskim hold that it is not an issue if only a piece of the outermost peel is removed, while some say that it is acceptable even if a piece of the thin yellow peel is removed.
- An Etrog which is punctured through and through is invalid. Some say that it is valid as long as no part of the flesh of the Etrog is lacking. Only in extenuating circumstances may one rely on the second opinion.
- An Etrog which is punctured up until the inner chamber where the seeds are but doesn't go through completely, some say it is invalid and others argue.
- If any of the flesh of the Etrog is lacking, it is invalid. Others, however, say that an Etrog which is lacking is only invalid if it is punctured completely and is lacking, or it is lacking at least the area of an Isur (a coin at the time of the gemara). In extenuating circumstances one may rely upon the second opinion. If the outermost peel, which acts like a thin ice-frosting on the Etrog, is lacking the Etrog is still valid.
- If the flesh of the Etrog deteriorated but the inner chamber in which the seeds are is intact, the Etrog is valid.
Etrog without Pitom
- An etrog whose pitom falls off is not valid to be used for the mitzva of daled minim, see footnote. According to some poskim, this invalidation only applies for the first day of sukkot since that is the only time that it's diorayta. However, all this only applies if the etrog had a pitom from the outset. If it never had a pitom, then it is valid.
Trumot and Maaserot
- Trumot and Maaserot must be taken off for the Etrog be to be used for the mitzvah, otherwise the Etrog is pasul for all seven days of Sukkot.  Therefore, one must be careful in finding a reliable seller. 
- If one cut an etrog from a non-Jew’s field in Israel that etrog should not be used unless one forgot to take off Trumot and Maaserot and it’s already Yom Tov (when it’s forbidden to take off trumot and masserot). 
- Any of the four minim that are stolen are unfit all 7 days of Sukkot. 
- Any of the four minim that are borrowed are unfit for the first day of Sukkot. 
- One of the four minim that are borrowed is fit the shakings of hallel and hoshanot but not the first shaking of the lulav on the first day of Sukkot. 
- One shouldn’t cut any of the four minim from the field of a non-Jew (even with his permission) but rather one should allow the non-Jew to cut it himself and give it to you.  This is only preferable, however, after the fact, the minim are fit for the mitzvah and one may make a Bracha on it. 
Eating the Etrog
- It’s forbidden to eat an Etrog all the days of Sukkot and if one does take a bite out of the Etrog it’s unfit 
- In Israel, it's permissible to eat the Etrog on Shemini Aseret, and in Chutz LeAretz (outside Israel) one may eat it on the ninth day of Sukkot.
- Regarding biting off the Pitom as a segulah for an easy pregnancy, see Pregnancy and Labor
- The Rama (Responsa 226) writes that an etrog that is the product of an etrog tree grafted with a lemon tree is invalid because it no longer qualifies as an etrog. The Levush (649:4) writes likewise but for a different reason. He asserts that a grafted etrog is invalid since it was created in violation of kilayim (grafting trees of different species) even if a non-Jew did it. The Shevut Yaakov (1:36) disagrees with the Levush’s reasoning, since we hold that non-Jews are not obligated in the laws of kilayim. The Shevut Yaakov quotes a story in which the Shach permitted making a bracha upon a grafted etrog, but he adds that the Shach later retracted. In sum, almost all poskim consider a grafted etrog to be invalid, including the Magen Avraham (648:23), Taz (648:3), and Chazon Ovadia (p. 223). Rabbi Zvi Sobolofsky (The Laws of the Daled Minim” min. 3) advised getting an etrog with a reliable hechsher in order to make sure that it is not grafted.
- See Mishna (Sukkah 34b), Rambam (7:8), and Shulchan Aruch (648:22).
- Chazon Ovadia Sukkot pg. 275, Yalkut Yosef Moadim pg. 154
- The Mishna (34b) establishes that if the pitom fell off an etrog, the etrog is invalid, but if the oketz fell off, it is valid. The Gemara (35b) explains that the pitom is the rod-like protrusion (“buchanto”). Rashi (s.v. Tani) cites a dispute between his teachers about how to explain what the pitom and oketz are. Rabbeinu Yaakov explained pitom as the protrusion on top of the etrog and the oketz as the thick stem on the bottom. Rabbeinu Yitzchak, however, explained that oketz refers to the outer part of the thick stem on the bottom, while pitom refers to the inner part of that stem, meaning that the etrog would be invalid only if the stem on the bottom is totally removed, even the part that penetrates the etrog. Rashi sides with Rabbeinu Yaakov.
- The Rabbeinu Chananeil (36a) explains that the Gemara was not explaining the Mishna at all but rather adding another potential disqualification. According to his view, the pitom is on the top of the etrog and “buchanto” is on bottom. He adds that the oketz is the outer part of the stem on bottom, and if that is removed, the etrog remains valid. The Rif (17b) and Rambam (8:7) agree. The Rosh (3:16) writes that the minhag was to follow Rabbeinu Chananeil. Shulchan Aruch (648:7-8) concurs. Mishna Brurah (648:31) writes that if the bud on top of the pitom falls off, one should avoid using the etrog unless it is the nicest one available.
- The Rosh adds that if the etrog grew without a pitom in the first place, it is valid. Rama (648:7) codifies this. Mishna Brurah (648:32) explains that the reason an etrog without a pitom is invalid is either because its lacking or is not “haddar.” Accordingly, if an etrog grew that way initially, it is valid.
- The Mishna (34b) explains that an etrog upon which there is a protruding growth (“chazazit”) that covers the majority of the etrog is invalid. The Gemara (35b) adds that if the growth is found in two or three separate places, it invalidates the etrog, since it appears “spotted.” Additionally, the Gemara states, if the growth is found on the chotam, it invalidates the etrog regardless of its size. The Rosh (3:20) writes that a discoloration of white or black has the same status as a growth. Therefore, the Tur and Shulchan Aruch (648:12) write that a discoloration of any size on the chotam would invalidate the etrog.
- What is the chotam? Rashi (35b s.v. Ubechotmo) explains that the line at which the etrog begins to slope inwards is the chotam. See Halachos of the Four Minim (p. 10) for a diagram. The Rosh explains that the Rif holds that the chotam is the entire slope up to its pitom. The Beit Yosef (648:12) writes that the minhag was to be strict for both Rashi and the Rif. The Beiur Halacha (648 s.v. Mimakom) explains that the Rif, and by implication Shulchan Aruch, agree that the pitom itself is included in the chotam.
- The Pituchei Arbaat HaMinim (p. 266-7) discusses black dots that are caused by bug sprays that farmers use. He suggests two reasons to be lenient. First, he argues (based on the Terumat Hadeshen responsa 99) that since these black dots are part of the normal appearance of the etrog, they do not present a problem. Furthermore, the black dots are external to the etrog. Nonetheless, he concludes that one should be strict unless the dot can be removed. Similarly, Rav Dovid Miller (“Hilchot Arba Minim”) and Halachos of the Four Species (p. 22) say that a black dot on the upper part of the etrog invalidates it.
- Rabbi Zvi Sobolovsky (The Laws of the Daled Minim. min 4-5), however, explained that most of the etrogim we have do not have any issue with discoloration; the little black dots are just specks of dirt. Similarly, Rabbi Eliakim Koenigsberg (“A Practical Guide to Purchasing Daled Minim” min. 32-3) quotes the Chaim Ubracha (p. 33 note 87), who says that black dots invalidate the etrog only if they develop because of rotting, which is not usually the case. Mishna Brurah (648:46) writes that a discoloration disqualifies an etrog only if it is noticeable upon a normal glance without staring at it carefully.
- Chazon Ovadia p. 270. He proves it from the Aruch Hashulchan 84:36, Tiferet Yisrael Avoda Zara Boaz 2:6:3, and Shoel Vnishal YD 5:64 who say that it isn't necessary to use a magnifying glass for kashrut. He also cites that Rav Shlomo Zalman in Shalmei Moed p. 134 agrees that for an etrog you don't need a magnifying glass to check if it is punctured.
- The Mishna (34b) cites Rabbi Yehuda’s view that an etrog that is as green as grass is invalid. The Rosh (3:21) cites Tosfot’s assertion that an etrog that is green but will turn yellow over time is valid, since it must be a complete fruit in order for it to turn yellow. Shulchan Aruch (648:21) codifies this view. Mishna Brurah (648:65) writes that the Achronim decided that one should not rely on the fact that the etrog might potentially turn yellow later on unless it has begun to start doing so. Chazon Ovadia (p. 256) agrees. Rabbi Hershel Schachter (“The Halachos of the Daled Minim,” min. 33-5) cited the Mishkenot Yaakov’s opinion that the etrog is invalid even if it started to yellow.
- Bumpy: Rama (Responsa 126) writes that the differences between a grafted etrog and a real etrog include: 1)A real one is bumpy, while a grafted one is smooth. 2)A real one has an indented oketz, while a grafted one has an oketz that protrudes. 3)A real one has a thick peel with very little juice, while a grafted one has a thin peel and a lot of juice. The Tiferet Yisrael (Mishnayot Sukka 3:6) says that a person ideally should look for an etrog that is very bumpy and has an indented oketz. Nitei Gavriel (p. 140) as well as Kashrut Arbaat Haminim (p. 8) codify this view.
- Ball-like: The Gemara (36a) says that an etrog that is round like a ball is invalid. This is quoted by Tur and Shulchan Aruch (648:18). Mishna Brurah (648:59) explains that a round etrog is invalid since it is not a normal shape of an etrog. Beiur Halacha (648:18) elaborates that it is not necessary to be stringent for the opinion of Tosfot that a cylindrical etrog is invalid, since most authorities disagree. Kaf Hachaim (648:113), however, says that ideally, one should accommodate this view of Tosfot. Chazon Ovadia p. 279 cites the Mata Yerushalayim who permits a ball-like etrog if it has a pitom and oketz and disagrees because the poskim didn't mention this distinction. Interestingly, the Radvaz (1423, 5:50) writes that the Rambam doesn't quote the halacha of a ball-like etrog since it doesn't occur naturally. If it was crafted with a shaped bottle it is invalid in any event.
- Tower-like: Tiferet Yisrael (Yachin Sukkah 3:6) writes that ideally, the etrog should be like a tower, meaning thick at the bottom and thin on top. Nitei Gavriel (p. 140) and Arbaat Haminim Lamedharim (p. 252) agree.
- Symmetrical: Tiferet Yisrael (Yachin Sukkah 3:6) writes that ideally, the pitom should be lined up with the oketz. Kashrut Arbaat Haminim (p. 8) agrees. Arbaat Haminim Lamehadrin (p. 177) cites Rav Nissim Karelitz, who says that this criterion is met if the pitom and oketz are approximately lined up. Chazon Ovadia p. 279 cites the Ramban's comments to the Raavad's Hilchot Lulav that the invalidation of a bent lulav doesn't apply to a etrog; therefore, a bent etrog is valid.
- Aesthetic beauty: Chazon Ovadia (p. 278) quotes the Maamar Mordechai, who asserts that besides for the properties that Chazal specified, the beauty of an etrog depends on the subjective view of the individual. Accordingly, Rabbi Mordechai Willig (quoted by Rabbi Eliakim Koenigsberg “A Practical Guide to Purchasing Daled Minim” min. 44-6) would ask his wife to pick the nicest-looking etrog from amongst the valid etrogim.
- The Maharil (Etrog no. 15) held that it is forbidden to place an etrog on top of a clothing on Yom Tov since it will create a nice smell in the clothing which is molid. He explained though that it is permitted to put the etrog back into a cloth or wool that it was sitting in before Yom Tov since it already smelled nice beforehand. The Magen Avraham 511:11, Mishna Brurah 511:26, and Aruch Hashulchan 511:12 quote the second halacha. The Maharshal (Yam Shel Shlomo Beitzah 2:34) argues that there's no prohibition of molid in clothing if one's intention isn't to create the smell. A pesik reisha is only forbidden for a melacha but not for molid. The Ginat Veradim OC 3:16 agrees. Yabia Omer 4:34:30 quotes many who are lenient and Yalkut Yosef (Shabbat v. 3, Molid no. 3) holds that way. However, the Magen Avraham 658:2 agrees with the Maharil that it is forbidden if it is unintentional. Mishna Brurah 658:7 agrees with the Magen Avraham unless it is an extenuating circumstance.
- The overwhelming consensus of Achronim hold that a grafted Etrog is unfit all days of Sukkot and one may not make a Bracha on it. These Achronim include Magen Avraham 648:23, Taz 648:3, S”A HaRav 648:31, Mishna Brurah 648:65, Chazon Ovadyah Sukkot (pg 223).
- The Sh”t Rama 126 brings 3 signs that distinguish a real etrog from a grafted one: 1) The real etrog have many small bumps while grafted ones are smooth, 2)the real etrog has a stem (o’ketz; opposite the pitom) that’s indented, while grafted ones have stems that protrude from the fruit, and 3) the peel of an etrog is very thick and the fleshy-fruit part is small and dry, while grafted ones have a thin peel with a large and juicy flesh part. However, the Sh”t Chatom Sofer 207 questions the reliability of the signs. So writes the Sh”t Shenot Chaim 270, Mishna Brurah 648:65, Chazon Ovadyah Sukkot (pg 223). Nonetheless, Mishna Brurah 648:65 writes that one may rely on the signs for using the etrog on the second day.
- Mishna Brurah 648:65, Chazon Ovadyah Sukkot (pg 223)
- Chazon Ovadyah Sukkot (pg 228-232), Sh”t Mishneh Halachot 6:117
- The Achronim quote the story of when the Grach (Rav Chaim Soloveitchik) had an etrog that was safek grafted and one that’s surely non-grated but not as hadar. The made the Bracha on the safek grafted one so that he could fulfill his mitzvah with the more hadar etrog, rather than use the true etrog and once he fulfills the mitzvah he would no longer be able to fulfill the mitzvah of hadar. However, the Mikrei Kodesh (Sukkot 2:9) and Chazon Ovadyah Sukkot (pg 244) argue that shaking the safek kasher one first would be a safek hefsek between the bracha and shaking on the true etrog and so one should take the true etrog first.
- Avnei Nezer CM 415 the Etrogim from Eretz Yisrael are the best because there were religious Jews living there the entire time of the exile, but because they’re difficult to get, we use Korfu. Taharat Haetrogim p. 298 cites this.
- Taharat Haetrogim p. 299 quoting Sod Yesharim p. 356, Horat Heter p. 5. He also cites a testimony from R’ Yakov of Karaz that the minhag Yerushalayim was to use Korfu etrogim.
- Taharat Haetrogim p. 300 quoting Rav Meir Auerbach author of Imrei Binah (Nefesh Chaya OC 4). He also cites Rav Chaim Nissim Abulafia supporting those etrogim.
- Rav Meir Mazuz (in a letter printed in Pri Etz Hadar), Hegid Mordchai 13, Yishrei Lev, Simchat Kohen 1:13-14 cited by Taharat Haetrogim p. 342. Among Ashkenazim those who supported the Moroccan etrogim include: Rav Shlomo Kluger (Yalkut Pri etz Hadar p. 34) and Rav Yakov Etlinger (Tochacha Megula and Bikurei Yakov end of 648) cited by Taharat Haetrogim p. 344 where he disapproved of the Korfu etrogim in favor of the Moroccan etrogim. Aruch Hashulchan 648:28 writes that we can’t rely on the simanim since some of the Korfu etrogim have the simanim and many hold that they are grafted by those from Israel and Morocco don’t all have simanim. Brisker Rav (cited by Taharat Haetrogim p. 349) tried to get Moroccan etrogim. The Satmer Rav (cited by Tahraat Haetrogim p. 350) supported Moroccan etrogim.
- Mishpacha Magazine
- S”A 648:1 writes that a dry Etrog which has no moisture is invalid.
- The S”A 648:1 writes that one could check whether an Etrog is considered dry by piercing it with a needle that has a string attached and if the string comes out moist the Etrog is considered moist. The Mishna Brurah 648:3 quotes a dispute regarding how to do this procedure without invalidating the Etrog by having a hole in it. Some say that one should make a hole in the thick part of the Etrog so that its not a hole through and through. Others say that one should make a hole along the length of the Etrog, along the Uketz-Chotam axis.
- The Rama 648:1 writes that an Etrog from a previous Etrog is invalid as an dry Etrog. Mishna Brurah (Shaar HaTziyun 648:8) quotes the Raavad who holds that there's no time limit for an Etrog, rather it all depends on the moisture of the Etrog. He quotes the Bikkurei Yacov who saw an Etrog from a previous year which was moist since it was kept in a closed, moist, and cool place.
- The Mishna (34b), according to Rashi, states that an etrog that is punctured and lacking even a little bit is invalid. If, however, it is punctured but not lacking, it remains valid. The Gemara (36a) limits the leniency of the Mishna to where the hole does not penetrate all the way through the etrog and the hole is not the size of an issar coin. The Rosh (3:18) agrees with Rashi. Rabbeinu Chananeil, though, maintains that the Gemara’s limitation was regarding the stringency of the Mishna. Thus, in his view, the Gemara means that an etrog that is punctured and lacking is invalid only if the hole goes all the way through or is at least the size of an issar coin.
- The Rambam (8:7, as understood by the Beit Yosef 648:2) explains the Mishna as meaning if an etrog is punctured or lacking it is invalid, unlike Rashi. For all practical purposes, though, the Rambam agrees with Rashi, because he explains the Gemara as referring to the first case, like Rabbeinu Chananeil, thus limiting the stringency of the Mishna to where the hole goes all the way through or is the size of an issar coin. Though the Rif (17b) is not clear, the Beit Yosef suggests that he agrees with the Rambam.
- The Gemara (35b), according to the explanation of the Rabbeinu Chananeil and Rosh (3:17), establishes that an etrog that is peeled in its majority is invalid, while an etrog that has only a minority peeled is valid. The Ran (17a s.v. Niklaf) explains that the Gemara is discussing a case where the outermost peel (which is thin like frost) is removed. If, however, the thin yellow peel is removed, the etrog would be invalid, as it is an etrog that is lacking. The Rambam (8:7, as understood by the Bach) agrees that it is valid only if none of the thin yellow peel was removed. The Rashba (Responsa 1:58), however, says that it is invalid only if part of the thick white section is removed (see the Machon Yerushalayim edition of the Tur 648 note 13). Chazon Ovadia p. 269 adopts the Rashba's view.
- The Bach concludes that the view of the Ran should be normative. Shaar Hatziyun (648:27) elaborates upon the Bach (see also Beiur Halacha 645:2). The Chazon Ish (147:1), however, claims that there never was a dispute and everyone really subscribes to the view of the Rashba. See Mishna Brurah (648:26), who discusses the status of an etrog if the area where the outermost peel was removed became discolored.
- The Terumat Hadeshen (Responsa 99) writes that if a hole was made while the etrog was growing but the flesh and peel subsequently grew over it, it is valid. Rama (648:2) agrees. Chazon Ovadia p. 269 cites the Rashash Sukkah 36a who permits a hole that isn't noticeable.
- S”A 648:2 writes that an Etrog which is punctured completely is invalid, however, some say that it is valid as long as it isn't lacking. Mishna Brurah 648:9 writes that we hold like the second opinion in extenuating circumstances.
- S”A 648:3. Mishna Brurah 648:23 writes that one should be strict not to use an Etrog which is punctured up until the inner chamber, unless it isn't possible to find another Etrog, in which case one may be lenient especially if the Etrog isn't lacking.
- Shulchan Aruch O.C. 648:2. The Rama rules that in extenuating circumstances one may rely upon the second opinion.
- Beiur Halacha 648:2 s.v. VeIm, Chazon Ovadia p. 269
- Shulchan Aruch O.C. 648:4
- Mishna Sukkah 34b. for the following discussion see Diagram of Etrog. Shulchan Aruch Orach Chaim 648:7 rules like the Rabbeinu Tam Tosafot Sukkah 35a and Rambam Hilchot Lulav 8:7 that as long as the dad is intact, the etrog is still valid. However, the Rif Sukkah 17b according to the Ran there d"h "Gemara" says that if the shoshanta falls off, the etrog is invalid. Rama 638:7 says that one should preferably be stringent and follow the opinion of the Rif, that if the shoshanta falls off the etrog is invalid. However, Mishna Brurah 648:31 explains that the Rama only refers to a case where one has a choice of purchasing two equivalent etrogim, but one has a broken shoshanta but if the etrog with the broken shoshanta is a nicer etrog, one should purchase that one. The Levush 648:8 disagrees and says that the etrog is invalidated if any part of the dad breaks off. Mishna Brurah 648:30 quotes the opinions of Taz 648:11 that the etrog is invalidated only if the entire dad is broken off and the Magen Avraham 648:9 that if any part of the dad remains protruding from the etrog it is still valid, but does not rule conclusively on the matter. Rashi Sukkah 35b "Nitla" quotes an opinion that the pitom refers to the bottom of the etrog and therefore if what we usually call the oketz falls off the etrog would be invalid.
- Tosafot Sukkah 29b "Bainan" writes that an incomplete etrog is valid from the second day onward but an etrog which lacks hadar is invalid throughout Sukkot. Rambam Hilchot Lulav 8:9 disagrees saying that an etrog that lacks even hadar is valid from the second day onward. Shulchan Aruch Orach Chaim 649:5 rules like the Rambam but the Rama there rules like Tosafot. Therefore, according to Shulchan Aruch an etrog which lost its pitom is kosher after the first day. However, according to the Rama it would be a problem if it's a problem of hadar. Rabbeinu Yerucham Sefer Adam 8:3 says that an etrog whose pitom broke is similar to an incomplete etrog and is valid starting on the second day. However, Rav Avraham of Prague cited in Darkei Moshe 649:5 says that an etrog whose pitom broke lacks hadar and is therefore invalid throughout Sukkot. Rama rules like Rabbeinu Yerucham, and therefore according to him it would be mutar to use an etrog with a broken pitom after the first day. However, the Magen Avraham 649:17 rules like the Rav Avraham of Prague since the Maggid Mishneh Hilchot Lulav 8:7 rules that way and therefore an etrog with a broken pitom would be invalid even after the first day. Mishna Brurah 649:36 concludes based on the Elia Rabba 649:15 that even if Rav Avraham of Prague and the Magen Avraham are correct, you can rely on the aforementioned Rambam that lacking hadar is still permitted for after the first day, but he adds that one shouldn't recite a beracha on such an etrog unless it's something that would be permitted to recite a beracha on the first day as well.
- Rosh Sukkah 3:16, Rama, OC 648:7. Mishna Brurah 648:32, explains that since it grew without the pitom, it isn't in the category of lacking hadar or incomplete.
- Rambam (Hilchot Sukkat 8:9), Kolbo (Siman 72), Eliyah Rabba 649:4, Pri Megadim A”A 649:20, Bikkurei Yacov 649:29, S”A HaRav 649:15, Mishna Brurah 649:45, Natai Gavriel 36:2, Chazon Ovadyah Sukkot (pg 248)
- Kaf HaChaim 649:41, Natai Gavriel 36:2
- Chazon Ovadyah (pg 246)
- S”A 649:1, Nitei Gavriel 44:1
- S”A 658:3
- Mikrei Kodesh pg 78, Natai Gavriel 44:3 in name of Rabbi Moshe Soloveitchik.
- Rama 649:1 based on the Rashba; even though, the Kaf HaChaim 649:21 quotes the Smag and Radvaz who say that nowadays there’s no need to be strict in this matter, nonetheless, the Kaf HaChaim concludes like other achronim that it’s preferable to be strict for the Rashba.
- Mishna Brurah 649:10, Kaf HaChaim 649:23 quotes the Get Mekushar who says that one shouldn’t make a Bracha in such a case and concludes that one may make a Bracha as times have changed (see there).
- Rama 649:5
- Chazon Ovadyah (Sukkot pg 444)