The law of a Pesel-The status of the Sechach that extends past the walls of a Sukkah:[1]
Contains two extending walls from back wall [Illustration 1]:[2] If a Sukkah contains Sechach that extends past the middle[3] wall of the Sukkah[4], and two [parallel] walls extend with this Sechach [past the middle wall], then if that area contains the minimum dimension of a Kosher Sukkah, which is at least 7x 7 Tefachim, and the Sechach provides majority shade in that 7 x 7 area, then it is Kosher, being that it has three walls. The following are its three walls; the middle wall of the Sukkah, plus all the long parallel walls that extend past the middle wall. This area is thus considered a complete Sukkah and is valid.[5]
Contains one extending wall from back wall [Illustration 2]: If a Sukkah contains Sechach that extends past the back wall of the Sukkah, and there is one wall that extends together with the Sechach, then seemingly even if that area is 7Ă—7, that area is not considered part of the Sukkah and it is forbidden to eat under it being it contains only one extending wall.
Contains one wall extending from open side [Illustration 3]:[6] [If a Sukkah contains three walls and] there is Sechach extending past the fourth direction which is an open side where there is no wall, and one of the walls of the Sukkah extends together with this Sechach which contains Hechsher Sukkah[7], then that area is also considered part of the Sukkah and it is permitted to dwell there [and eat and sleep under it] even though it only contains one extending wall. This applies even if the Sukkah owner had expressed his intent to make the two long walls extend throughout the entire length of the Sechach[8], nevertheless that area [which currently contains only one extending wall] remains valid even if he did not yet fulfill his plan to extend the second wall to that area.[9]
[1] See Admur 631:8-9; Michaber 631:6-7; Sukkah 19a
[2] Admur 631:8; Michaber 631:6; Sukkah 19a
[3] The intent here of emphasizing the middle wall is seemingly in regards to a three walled sukkah in which case it is only possible to have two extending walls if the Sechach protrudes past the middle wall. However, by a four walled sukkah every wall can potentially be considered the middle wall.
[4] Admur ibid adds after making the above statement, “Such as if behind the middle wall of the sukkah there are canes of the Sechach sticking out.” It is unclear as to what this statement is coming to accomplish or clarify which was already known in the previous statement. However, seemingly the intent here is to emphasize the middle wall when in reference to a three walled Sukkah as explained in the previous footnote.
[5] The reason: As in such a case that area contains all the criteria of a Kosher Sukkah, as it has three walls and is 7Ă—7, and has majority shade. Now although the middle wall was not made for this Sukkah but rather for the main Sukkah, nevertheless it is valid to serve as a third wall also for the smaller Sukkah. [Admur ibid; Michaber ibid]
[6] Admur 631:9; Michaber 631:7; Raba and Rav Yosef in Sukkah 19a; See Piskeiy Teshuvos 631:4
[7] Must the extended area of wall and Sechach contain Hechsher Sukkah [i.e. 7×7]? Some Poskim rule that the area that extends with the third wall is valid even if it is less than 7×7. [Chazon Ish 144:1, brought in Piskeiy Teshuvos 631:3, and so is evident from Aruch Hashulchan 631] However, see Admur ibid who writes “And one wall continues from the Sukkah with these canes of Sechach which have Hechsher Sukkah.” Now, it is unclear if this wording of “Hechsher Sukkah” is going on the extended area of Sechach, which means that it is 7×7 Tefachim [and if it is less, then this extended area is Pasul!] or if it is going on the original Sechach itself, and Admur is saying that since the original Sechach contains Hechsher Sukkah, therefore the extended Sechach and wall are also Kosher, even if they are less than 7×7 [as learns Chazon Ish]. [See Sukkah ibid that the entire Chidush is that even though it does not contain Hechsher Sukkah it is valid, and see Rashi ibid who explains this. However, it is not clear as to Rashi’s intent, and as to if he refers to the Sechach not being 7×7, or to the fact the 7×7 Sechach which extends does not contain walls. Indeed, the Maharsha ibid explains it to refer to the walls, while the Aruch Laner on Sukkah 4a explains it to refer to the Sechach. This would then create a Nafka Mina regarding if the extended space is valid even if it is less than 7×7. Vetzaruch Iyun! See Yagdil Torah Israel 1 p. 84; See also Piskeiy Teshuvos ibid footnote 28 who concludes that it must as the very least contain 6×6 Tefachim!
[8] For example, if after erecting the long wall that stretches throughout the length of the entire Sechach he then went ahead and made a wall parallel to it which is longer than seven Tefachim, then one can claim that since he could’ve sufficed with making that third parallel wall simply seven Tefachim long, therefore by him making it longer than this amount that could’ve sufficed he has revealed his intent to extend this wall to the full length of the Sechach in order not to sit in an area under the Sechach with an open side which does not have two walls that extend with it. If one accepts this claim than one can further argue that this single wall that extends together with the canes protruding past [the area of] Sechach [which has two parallel walls] is not considered at all part of the sukkah, and is rather its own domain and it is thus found that these extending canes only contain one wall, which is the wall that extends together with them [and thereby this area of the sukkah should be invalid]. [Admur ibid; See Michaber ibid; M”A 631:4; Taz 631:5; Rosh 1:35]
[9] The reason: We do not claim [that since he could’ve sufficed with making that third parallel wall simply seven Tefachim long, therefore by him making it longer than this amount that could’ve sufficed he has revealed his intent to not to sit in an area under the Sechach with an open side which does not have two walls], as we do not follow his expression of opinion on this matter at all (as the making of a sukkah does not need any prior intent to be valid as explained in chapter 635:1), and it is thus considered as if he did not express any opinion at all regarding that all the extending Sechach should be validated with this short third parallel wall even though it does not extend throughout the entire length of the Sechach as does its parallel wall, as explained in chapter 630:2 [that it suffices to have two walls and third wall of a single Tefach]. [Admur ibid]