There appears to be a consensus that there is no evidence for the Our Father being in the Mass anywhere in Christendom before about 350. Before that, it was a non-liturgical prayer used, perhaps several times a day, either privately or among groups of the Faithful. And the evidence is that during this period, when Christians shared the Our Father, they concluded it with a kiss of peace. The earliest evidence I know for this is in Tertullian (c160-225; see de Oratione PL 1 1176-9). A custom had grown up of people omitting the Peace after the Our Father when they had been fasting. Tertullian disapproves of it because it includes an inclination to boast publicly about fasting, contrary to Matth 6:16. He calls the kiss the signaculum orationis; the sealing (as a document might be sealed) or finishing-off of the prayer. Rhetorically, he asks: 'What prayer is complete when the holy kiss has been torn from it? Whom does the Peace impede as he is doing his duty towards the Lord? What sort of sacrifice is it, from which people go away without the Peace?' And a couple of paragraphs earlier, speaking about the ending of the prayer, he uses the phrase assignata oratione; 'when the prayer has been sealed'. Similarly, Origen (c185-254) , commenting on the Kiss of Peace referred to by S Paul in Romans 16 and elsewhere, describes it as happening 'after the prayers' (PG 14 1282). Since S Paul never specifies where the kiss is to be given, Origen's 'after the prayers' presumably reflects the usage of his own time.
It seems highly likely that what happened is this. When the Our Father was introduced into the Mass, it brought with it its concluding signaculum, the Kiss of Peace. Thus the Pax in the Liturgy is not, in itself, a reconciliatory preparation for Communion, but a 'signing off' from the Our Father and the Eucharistic Prayer. We find this situation reflected in the Letter of Pope S Innocent I to the Bishop of Gubbio in 416 (PL 56 515). Troublemakers in Gubbio had been saying that it was better to follow the custom of another Church as to the position of the Peace rather than that of Rome; the Pope responds ' the Pax has to be done after all the things which I'm not allowed to mention to show that the people have given their consent to everything which is done in the mysteries and celebrated in Church, and to demonstrate that they are finished by the signaculum of the concluding Pax'. The fact that he employs the very term signaculum which had been used by Tertullian suggests that we are dealing with conventional usage widespread enough to be common to Rome and North Africa and over a period of at least two centuries.
Thus the Roman position of the Peace appears to have a meaning and logic which go even beyond the introduction of the Our Father into the Mass, back to those early days when Christians met in little groups to say the Lord's Prayer together. That logic was the communal and corporate assent of God's People to the Lord's own Prayer. Of course, this does not exclude the notion of the Peace as a gesture of reconciliation among those who, as one Body, are just about to receive in the Eucharist the one Body and the one Cup of the Blood of the Redeemer. That theme is itself suggested by the last few clauses of the prayer, concerning mutual forgiveness.
But I wonder if there is a slightly different alternative narrative which might be valid here. Might the passage I have quoted from Tertullian relate not to the extraliturgical use of the Lord's Prayer among Christians, but to its use within the Mass? He does seem to be talking about something more corporate than merely a semiprivate prayergroup. And note the phrase 'What sort of sacrifice ...?' And there is a paragraph nearby where he criticises the habit of sitting down after the Peace; if the Peace simply concludes a little prayer meeting, why should the participants not be allowed to sit down once it was over?
Another And ... Having criticised his fellow Christians for witholding the Kiss so as publicly to flaunt the fact that they had been fasting, he goes on ' ... on the day of the Pasch, on which there is a rule of fasting which is common to all and as it were public, we rightly drop the kiss, because we don't care about hiding the thing [i.e. fasting] which we are doing with everybody else'. Those familiar with the traditional Roman Rite will recall that, to this day, we do not exchange the Sign of Peace at the Good Friday Mass of the Presanctified, nor at the Mass of the Easter Vigil (even though the celebrant has said the words). This is because we are all deemed to have been fasting.
Questions arise: if the Our Father was within the Mass as early as the time of Tertullian, what does this do to our understanding of the early history of the Liturgy? How are we to fit in the apparently second century evidence for the Peace coming at an earlier point in the Mass? Why should those fasting consider it appropriate to withold the Kiss? What is the relevance of all this to the Eucharistic Fast, first witnessed in North Africa at the end of the fourth century? And does the evidence we have considered derive support from Dom Gregory Dix's compelling theory about the Mass of the Presanctified (i.e., that the third century practice of Christians communicating themselves privately on weekdays from the Host which they had reserved at the Sunday Mass, and blessing by the recitation of the Lord's Prayer and then drinking a cup of wine as an 'antitype' of the Blood of Christ, is found as the Communion Rite of the traditional Roman Good Friday liturgy, simply transferred from the private to the communal context)?
I never cease to be surprised at what I find whenever I delve back into the history of the venerable and wonderful Roman Rite.
I preserve a comment from an earlier post on this subject.