The core teaching, drawn from the Vilna Gaon's circle, is that a soul which has caused spiritual damage is not simply punished and discarded but is sent back into a body to repair what it damaged: after suffering in Gehinnom and confessing its sins, it is brought to Gan Eden and from there sent into gilgul specifically 'to repair its wrongs,' with the number and nature of subsequent gilgulim signaled by physical signs on the returning soul (Even Sheleimah 10:32).
This mechanism of repair is tied to the idea of 'visiting the sin of the fathers on the sons': when a person's own flawed conduct echoes the deeds of earlier generations because those ancestral deeds remain rooted in him, he is sent into gilgul to correct that same root, as developed through the cases of Hevel, Kayin, and Shet (Even Sheleimah 4:12, Even Sheleimah 4:14).
The Vilna Gaon on Yonah teaches that while the body, once broken, has no further repair, the soul is different — even after bearing its punishment it is never lost forever, because it can still be rectified through gilgul, citing the repeated return 'twice, even three times, with a man' as the mechanism by which the soul is restored (Beur HaGra on Yonah 1:4).
Even Sheleimah extends this further, describing how a soul that refuses to repent even after three cycles of gilgul is instead cast down into the chambers of Gehinnom and forgotten there for generations, showing that gilgul is itself offered as the normal path for a soul to fix its damage before any such finality is reached ({{the repeated 'three gilgulim' theme}} Even Sheleimah 10:25).
The broader kabbalistic sources on the sheet reinforce that every soul's incarnations, actions, and particulars from the time of Adam onward recur in each generation as part of this ongoing process of rectification, consistent with the Gaon's framework of gilgul as corrective (Beur HaGra on Sifra DeTzniuta 5:5, Even Sheleimah 1:12).
Sha'ar HaGilgulim likewise explains that one who did not complete his rectification in a first gilgul must return a second time to complete it, even if what remained lacking was minor, and that the merit earned in the second body services the completion of the soul's original mission — the same principle of return-to-fix found in the Gra sources (Sha'ar HaGilgulim 4).
Rabbeinu Bahya, writing earlier than the Gra but reflecting the same kabbalistic tradition he draws on, describes how after souls receive their reward or punishment in Gan Eden or Gehinnom they return after a time to another body to receive what is fitting for them measure for measure, a cycle he identifies with what Chazal call techiyat hameitim (Rabbeinu Bahya, Devarim 33:6).
The Gra's comments on Yonah 4:3 further distinguish between souls who merited not needing a second gilgul at all and those who 'come back' a second time, framing the debate in terms of whether it is better for a soul to have been created and risk needing further rectification through gilgul (Beur HaGra on Yonah 4:3).
In an unrelated note on Mishlei, the Gra discusses three angels who assess a person at the time of death — one of his deeds, one of his sins, and one representing the Torah he studied before birth — which bears on judgment generally though not directly on the fix-through-gilgul theme (Beur HaGra on Mishlei 6).
A Chabad teaching recorded in the Igros Kodesh states explicitly that the overwhelming majority of cases of a soul's gilgul occur specifically in order to rectify what was not yet completed in its first descent into a body, and that this bears on which bodily aspects rise at techiyat hameitim (Igros Kodesh letter 209).
Nishmat Chayyim situates the entire discussion of gilgul hanshamot as a major topic addressed from Scripture and from the cases of Adam, Shet, David, and Reuven, and frames it as part of the rational grounds for the soul's survival after death, together with the many questions the topic raises (Nishmat Chayyim, Fourth Treatise 6:1, Nishmat Chayyim, Introduction 8).
Scans of the Gra's comments on Mishlei further tie the return of souls to the body to specific scriptural moments — such as Avraham's standing at a place and Yitzchak's going out at night — and to the principle that divine judgments operate measure for measure, so that a person ultimately repays for his deeds through this process (Gra on Mishlei 3, Gra on Mishlei 11).
A passage touching on fear of sin and the retention of learning appears among the Even Sheleimah selections but does not itself address the gilgul mechanism (Even Sheleimah 8:8).
