Glad it's helpful! Everything is a bit nebulous because things can be defined and implemented in different ways.
"Narrow syntax" can be FLN (Narrow Faculty of Language). Roughly what Chomsky seems to be trying to solve with Minimalism. I don't intend the term too precisely, because we're not sure what the boundaries are. (Syntax in general can include things like word order, but those seem to have been separated for now.)
Also, you mention "processing" with what sounds like wariness. Why?
Actually I really like research in psycholinguistics and attempts to connect syntactic theory to processing. And you can tell I like aspects of Cognitive Linguistics too. So I study Generative Grammar, but I'd skeptical about it and interested in other things.
But the wariness you correctly detected is me playing devil's advocate basically: Generative Grammar is so much about Competence that getting into Performance (processing, etc.) is sometimes like trying to mix oil and water. For example, you can actually exclude
any data at all as "just Performance" and therefore not relevant to the more abstract and not directly observable "Competence". Chomsky and others use native speaker intuition and reflection to get at grammaticality judgments for studying Competence, but of course Performance (some kind of faulty intuition I suppose) could also cloud those judgments. It makes falsifying Generative Grammar very challenging, plus the fact that "Generative Grammar" (or especially the MP) cannot be falsified overall because it is an approach rather than one theory. Adjust the theory slightly, and continue.
So, personally, I have mixed feelings about this. But most Generative syntacticians seem to like separating out any sort of processing-related arguments for theoretical discussions of narrow syntax. Personally I might encourage you to look in another direction than Generative Grammar at least for addressing some questions. (I have seen absolutely no convincing evidence that GG is psychologically/biologically real, for example!) But I don't want to discourage your interest in it, and it is a very popular theory.
There are a few linguists who are actively trying to bridge the gap between performance and competence from a Generative perspective, even some who are serious Minimalists who are doing experimental work related to that. But they're in the minority, and mostly Minimalists are serious theoreticians looking at abstract rather than experimental data.
Any version of Generative Grammar that starts to rely on processing-like arguments drifts away from its foundation.
As an example of this, we often think of derivations as directional, starting with the lexicon and an abstract structure, and then applying rules sequentially until the form we pronounce is generated. But actually Generative Grammar (despite the name!) is not directional (in most accounts). It simply describes structural relationships between words. To compare it to math, I suppose it's just like how "1+1=2" doesn't mean that you must first add 1 to 1 then get 2, but also that 2-1=1, and various other relationships. It is, overall, completely unknown how Generative Grammar is actually implemented in the brain, and the theory cannot rely on such arguments. Chomsky himself does often discuss arguments about acquisition, but those are slightly different: since children learn X, they must be capable of learning X from the available input, and therefore the theory of syntax needs to explain what their UG is like so they can do that from that input. It borders on being about Performance, but it's actually about Competence. It doesn't explain HOW they do it. It just points out that they must be able to, so their Competence logically permits it.
Personally I'd be more than happy to rely on something like frequency or memory as a factor in explaining how syntax works. I don't think we need to explain everything in an abstract theoretical way if we can instead rely on getting it for free as part of something else (like memory). Chomsky wouldn't entirely disagree with me: he calls those things "third factors". But we may disagree (or may both not yet know) about where exactly to draw that line. As a basic example, I wonder if recursion can be accomplished as an iterative performance process rather than as something that must be defined in terms of competence. That could also potentially explain phases: there simply are boundaries that we do not cross for whatever reason in performance (maybe just communicative need not applying, or maybe they're difficult to process, or something else), and then those become conventionalized (Constructionalized?) in the grammar. Chomsky instead plays with the idea of those being encoded in UG, which I am skeptical about. But something like his Phases is needed, one way or another.
In short, I agree with Chomsky about something like Merge existing. And I also think we need something beyond standard Generative approaches (like Constructions). Everything in the middle I'm currently agnostic about.
So, above, I was trying to address your questions from the perspective of GG/MP (where "processing" is irrelevant!). I personally like the idea of connecting processing to syntax, but at the same time I'm not sure of how to do it (or exactly what would be connecting to exactly what!).
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My knowledge of mathematics is limited, but I think you have a similar division, between "Analytical" solutions (like Competence, GG) and "Numerical" solutions (like Performance). The parallel isn't exact, but throwing in a Monte Carlo simulation in the middle of an Analytical problem would be about the same effect as referencing processing to explain narrow syntax in GG.
Similarly, more and more I'm starting to see different theories as not being "right" vs. "wrong" but as addressing different questions (like Analytical vs. Numerical mathematics). For example, the MP is great for explaining hierarchical structure, while Construction Grammar is great for explaining surface-level irregularities. So maybe it's not about figuring out which theory is best, but instead about which theory is best for which purpose. Chomsky's purposes are almost entirely about UG, FLN, Competence, etc. So in addition to the other many types of background reading for you to think about, also try to figure out what sort of questions you want to solve and whether those questions are the same as what Chomsky focuses on-- if so, the MP may be right for you. If not, there are other approaches. (Chomsky's work, for example, is probably the least likely to lead to a usable machine translation system today, for example. But that's not what it's for, either.)