Announcement: Chris Langan has published a new paper, "An Introduction to Mathematical Metaphysics", in the journal Cosmos and History.
Distributed syntax means the universal laws of nature.
Consider a tennis ball. To perceive a tennis ball as a discrete entity quite a lot has to go on. One has to perceive all the qualities of the tennis ball that separate it from the surrounding space such as it's roundness, greenness etc. However, there is more than just the visible features of the tennis ball going into your eyes that are required for the act of perception. The numerous invisible laws of nature have to continue to be in operation. They are invisible because they are acting everywhere and anything that is everywhere just gets taken for granted in the background.
It can be said that these "everywhere" laws are "distributed" across all of the universe. Now, where the laws of the universe can be described as linguistic(in general terms)- this language has a syntax, or set of rules it obeys. So some aspects of the syntax are local-such as the greenness of the tennis ball and other aspects are distributed across all of reality(e.g. maybe a fundamental force,or the laws of thermodynamics or logic) . The latter is known as "distributed syntax".