Bava Kamma Chapter 9, Mishnah 6. The Mishnah continues to address the law of a person who swore falsely and subsequently wishes to return the stolen property and achieve atonement. The principle that emerges from it is simple: a person cannot bring his asham offering, an asham gezeilos, after having sworn falsely, unless he has returned the stolen item to the person from whom he stole it. However, this law applies only when there remains an amount worth a perutah to be returned, and this amount worth a perutah must be from the principal - the original value of what was stolen - and not from the chomesh, that additional fifth. The chomesh itself does not prevent atonement: one who has not paid it has indeed not done what is required of him properly, but he can still bring the asham gezeilos and progress toward atonement, at least partial atonement.
The cases where "eino tzarich leilach acharav" - he does not need to go after him:
"Nasan lo es hakeren velo nasan lo es hachomesh" - The robber returned what he stole or its value, but did not give the victim the additional chomesh (twenty-five percent).
"Machal lo al hakeren velo machal lo al hachomesh" - The victim forgave the robber for the robbery itself and told him: "There is no need to return what you stole from me, but I do request the additional chomesh." Since the obligation for the principal is nullified, there is no longer a duty to return it.
"Machal lo al zeh veal zeh chutz mipachos mishaveh perutah bekeren" - The victim forgave everything, both the original amount and the chomesh, and all that remained was a debt of less than a perutah's worth of the principal.
In all these cases, the Mishnah rules: "Eino tzarich leilach acharav" - the robber is not required to go after the victim and make up the shortfall before bringing the asham.
To illustrate: a robber stole a dollar from a victim and now must return it. The two assume the exchange rate stands at three shekels and sixty agorot, and the robber pays based on this assumption. When he arrives home and checks, he finds that the exchange rate is three shekels and sixty-two agorot, and it turns out he was short a few agorot. He does not need to go after the victim and make up these agorot before he brings his asham.
The cases where "tzarich leilach acharav" - he must go after him:
"Nasan lo es hachomesh velo nasan lo es hakeren" - The robber paid the additional chomesh, but not the principal amount.
"Machal lo al hachomesh velo machal lo al hakeren" - The victim told him: Leave the additional chomesh, but give me the principal amount.
"Machal lo al zeh veal zeh chutz mishaveh perutah bekeren" - The victim forgave both the robbery and the addition at once, except for a perutah's worth of the principal, which he requests back.
In all these, "tzarich leilach acharav" - the robber is obligated to transfer to the victim that remaining perutah's worth of the principal, before he can bring his asham gezeilos.
In summary: The Mishnah teaches that returning the principal is a prerequisite for bringing the asham gezeilos, whereas the chomesh is not. As long as a perutah's worth remains of the principal - whether it was not paid or not forgiven - the robber must go after the victim and return it before bringing his sacrifice; but if the principal was paid or forgiven, or if less than a perutah's worth remains of it, he does not need to go after him, and he may bring his asham and achieve atonement.