A review of the evidence so far suggests that Luke got his sayings material from an early form of Matthew (Matthew I), in which the list of Beatitudes was minimal and individual sayings and parables had not yet been "sermonized."
We know that an intermediate form of Matthew (Matthew II), in which the "sermonizing" had already taken place (leaving, however, telltale introductory formulas indicating that the sayings and parables had once been separate), and containing a somewhat fuller but still far from complete list of the Beatitudes, once existed. We know this because such a text underlies the Hebrew Matthew given by Shem-Tob ben-Isaac ben-Shaprut in his fourteenth-century polemical work Even Bohan.
In the latest version of Matthew (canonical Matthew, which I call Matthew III), The "sermonizing" has been done and the telltale introductory formulas edited out. The list of the Beatitudes is complete,
Our evidence for Matthew I is the Gospel of Luke.
Our evidence for Matthew II is the Shem-Tob Hebrew Matthew, which survives in at least twenty-eight manuscripts.
The existence of Matthew III is self-evident.
There are, thus, no hypothetical sources required in this solution of the "Synoptic Problem."
The above evidence does, however, require that we make a slight change to our diagrammatic representation of this hypothesis. Here is the new diagram:
(to be continued)