Two quick observations, then its "quits" for Easter:
1. You may be right about a collection of "Jesus sayings" (not to be
confused with Q which traditionally is presented as the non-Markan
"sayings" material shared by Matthew and Luke). I must confess that
your hypothesis as written looks suspiciously like your S is based upon
the traditional "sayings" criteria for Q. I would want to hear more how
you define your S source.
2. I can think of no occasion where Lukan reliance upon Matthew is
indicated. As I stated previously, they do share a common (non-Markan)
source and on the occasions in which Matthew is free from Markan
influence, he is often superior to Luke. But that does not demonstrate
that Luke is dependent upon Matthew. Only that he has preserved the
material better than Luke. [BTW, I can neither think of occasions where
Matthew use of Luke can be demonstrated.]
More problematic, you have Matthew reliant solely upon Mark and the
Sayings Source. Yet, I think the evidence indicates Matthew had access
to the "good source" of Luke. I am not certain Mark did, but I have no
question that Matthew knew and used this primitive witness.
Using your symbols (I have suggested p-G [proto-Gospel] to remove the
suggestion that the shape of this unknown document(s) looked like either
Mark or Luke.
I would venture something akin to the following:
p-G + [p-G/2?] => L
p-G [or p-G/2?]=> K
K + p-K => M
One final comment. Instead of S I have indicated that both Luke and
Matthew had access to p-G and Mark (more probably) to p-G/2. This
second document may have been an existing revision of the p-G in which
the literal Hebraisms rendered in the Greek were removed. Evidence for
this may be in Luke's Doublets where they vary considerably in their
Semitic style (i.e. one more Hebraic and one more refined Greek). In
such a case, it may be that Mark had access to this revision rather than
the more primitive "Hebraic" p-G. I say this because on no occasion (or
at the mostvery, very, very rarely) that comes to mind is Mark more
Hebraic than Luke. I know that it is often said that Mark's bad Greek
is because of underlying Semitic influences (i.e. sources). It's not
true. It's just bad Greek. Either Mark has so totally re-written his
Hebraic p-G (a very remote possiblity) or he was in possession of a
revision of it that had already removed the stark Hebraisms. In any
event, Mark's own redaction contributions are all over his Gospel.
Take a look at the Cleansing episode [Luke 19:45/Mark 11:15-16/Matt
21:12]. Luke witnesses to the earliest stage of this story. Mark
inflates the tradition with clear intentexpanding the target of Jesus'
wrath, depicting him resorting to violence and in essence shutting down
the Temple. Matthew is influenced by Markbut does not follow him
fully. As is common Matthew demonstrates a "middle" (not historically
but in form) role between Mark and Luke.
As a suggested response from you, I would like to hear from you an
occasion which you think demonstrates that Luke knew and used Matthew.
HGERQH HO KYRIOS