I've already offered multiple time to reject the concepts of determinism and randomness. That leaves us with no concepts whatsoever, since no one in this conversation (or elsewhere) has offered a coherent concept to replace or augment those two.
I'd like to finally share with you the "third possibility" which you have also asked for many times, and I'll do it without rejecting your own definitions (of the supposed "dichotomy" of determinism vs randomness).
You have explicated or implied these definitions in multiple places, so this quote of yours is simply a representative one, with all emphases mine:
The only sort of causality I know of is deterministic causality, where a set of events cause a new event in exactly one way. The laws of physics describe how the causes work. If an event occurs without any causes, then it is an arbitrary event.
OK, so, let's look at the causal models which we can make, given your definitions, to explain any given event, E2:
Firstly, there is your deterministic causal model: E2 is in some sense a
necessary consequence of a prior event, E1. i.e. it is in some sense
impossible for E1 to effect anything
other than E2 (this follows from the first emboldened part of your above quote).
We might symbolise this as (where "nec" stands for "necessarily"):
E1 ->(nec) E2.
Secondly, there is the "arbitrary" causal model, where "arbitrary" is for you synonymous with "random". In this case, E2 has no prior cause, E1 (per the second emboldened part of your above quote). It simply occurs. We might symbolise this as:
-> E2.
Now, you ask for a third causal model which leaves room for free will. Fine - here it is, symbolically:
E1 ->(n.n.) E2.
Here, "n.n." stands for "not necessarily". In other words, E2 is caused by E1, but not necessarily, i.e. not deterministically - it is a "freely caused" event, which leaves room for it to be a "freely willed" one.
If you can see a way around this, especially without changing your definitions, then please share it, but, given your definitions, you seem to have boxed yourself in.