The question of whether a child's siyum counts as a valid siyum is disputed among the later poskim. B'tzel HaChochma (4:100) rules that a child who learns and completes a tractate generates a valid siyum, treating the Torah study taught to him as a mitzvah fulfilled through him, and that firstborns attending such a siyum are exempt from the Ta'anit of Erev Pesach.
By contrast, Maayan Omer (3:2:44) holds that a child's siyum is not recommended as a means of exempting firstborns from the Erev Pesach fast, because the child himself is not personally obligated in Torah study, though he notes that other authorities maintain a child who understands his learning can exempt his father through it.
Nitei Gavriel, Bar Mitzva (perek 22) offers a conceptual basis for validating a child's siyum, explaining that it can be viewed as completing the father's mitzvah of chinuch, for which a celebratory meal is made upon completion of a mitzvah.
Nishmat Shabbat (vol 6:3) brings sources supporting the broader principle that what is completed through someone's assistance, including an orphaned child, is considered completed by that person, which underlies the possibility of attributing a child's completion to another.
This question depends on the underlying obligation of children in Torah study: the Rambam rules that women, slaves, and minors are exempt from the mitzvah of Talmud Torah, while the father is obligated to teach his minor son, as derived from the pasuk "ולמדתם אתם את בניכם לדבר בם" (Mishneh Torah, Talmud Torah 1:1-2), a factor that bears on whether a minor's own completion of learning can constitute a halachically significant siyum.</br>
The practical motivation for the question is often the exemption from the fast of the firstborn on Erev Pesach, which Shulchan Arukh, Orach Chayim 470:1 establishes obtains for firstborns through a father or mother, and which a siyum is used to override, making the validity of a child's siyum directly relevant to that exemption.
Mishnah Berurah 551:73 addresses the general laws of making a siyum and the seudah that follows its completion, discussing circumstances under which the celebratory meal is or is not required, which forms part of the broader framework within which a child's siyum is being evaluated.
Magen Avraham 282:18 catalogues various categories of people deemed "chiyuvim" entitled to precedence in communal honors, a related discussion of who counts as having a claim through a mitzvah-event, bearing on the general question of whether a mitzvah performed by or through a minor can generate standing for others.
Modern poskim have addressed this question directly. Our research indicates that Igrot Moshe (Orach Chayim 1:157) is relevant, ruling that completing a book of Tanach studied in depth by a group constitutes a seudat mitzvah when studied according to an authentic interpretation of the Rishonim, since completing any mitzvah undertaken over time warrants celebration.
