The text of the Torah is divided into paragraphs, which are called Parashiyot. A Parasha is referred to as either "Petuha" or "Setuma," depending on the size of the empty space separating it from the next paragraph.
Halacha forbids the one reading the Torah from beginning an Aliya within three verses of the beginning of a Parasha, or end an Aliya within three verses of the end of a Parasha. Meaning, if he begins an Aliya in the middle of a Parasha, he must ensure that there are at least three verses in the Parasha before the place where he begins, and if he ends an Aliya in the middle of a Parasha, there must be at least three verses remaining until the end of the Parasha. The Sages enacted this prohibition in order to prevent the misconception that an Aliya may consist of fewer than three verses. If the reader begins an Aliya within three verses of the beginning of a Parasha, somebody who enters the synagogue at that point might think that the previous Aliya consisted of only two verses. He will not realize that the previous Aliya had started before the beginning of this Parasha. Likewise, if the reader ends an Aliya within three verses of the end of a Parasha, somebody who leaves at that point might think that the next Aliya will consist of only two verses, not realizing that the reader will begin the Aliya a verse or several verses earlier than the point at which this Aliya ended. In order to avoid this misconception, it was established that an Aliya cannot begin within three verses of the beginning of a Parasha, or end within three verses of the end of a Parasha.
If, however, the reader made a mistake and began an Aliya within three verses of the beginning of a Parasha, or ended within three verses of the end of the Parasha, the reading is perfectly valid.
This applies even to the Maftir reading, even though it is the last Aliya. Thus, for example, Parashat Ekeb concludes with a paragraph that consists of only four verses, and the reader should read all four verses for Maftir.
The exception to this rule is the first Aliya, the Aliya of Kohen. For example, the custom among Syrian Jewish communities is to end the first Aliya of Parashat Tesaveh with the words "Le’chabod U’l’tifaret," after just two verses after the beginning of that paragraph. (The Aliya begins with two verses in the previous paragraph.) This is permissible, since it is the first Aliya and everybody knows that the reading began at the beginning of Parashat Tesaveh, and not at the beginning of this paragraph.
Every Aliya should both begin and end on a positive note, meaning, with content that is positive and encouraging, and not with something negative. It should be noted that descriptions of enemy nations flourishing are considered "negative" in this regard, and descriptions of enemy nations’ downfall are considered "positive."
The division of Aliyot that is printed in Humashim is not binding in any way. This division was not made in the time of Moshe Rabbenu, or even during the time of Hazal, and was introduced later. As such, we are not bound by this system. It is entirely permissible to divide the Aliyot differently, and in some instances it is even preferable to do so. For this reason, it is permissible to begin an Aliya within three verses of the beginning of the Aliya printed in the Humash, and to end an Aliya within three verses of the end of the Aliya printed in the Humash. This is the ruling of Hacham David Yosef, in Halacha Berura. Although the Hafetz Haim (Rav Yisrael Meir Kagan of Radin, 1839-1933), in Sha’ar Ha’siyun (138:1), ruled that this should not be done, because people mistakenly afford Halachic significance to the conventional division of Aliyot, Hacham David notes that Sephardic authorities did not accept this ruling.
Summary: It is forbidden to begin an Aliya within three verses of the beginning of a paragraph, or to end an Aliya within three verses of the end of a paragraph. Nevertheless, if this was mistakenly done, the reading is perfectly valid. Every Aliya should begin and end with content that is positive and encouraging. The division of Aliyot that is printed in Humashim is not Halachically binding
