Do you believe in free will?

[Perhaps]

So it's random then. Because numbers and time have mechanism - they're quite deterministic (in that they exhibit regularities which we can write down, known as "mathematics" and "physics"). The specific ontology is irrelevant, however interesting it may be to some.

Further, whilst time and numbers would appear to be independent of matter, brains are not. Again: "mind" is an article of rhetoric, not a scientifically or even philosophically valid term. Since minds are never discussed without brains to host them (or equivalently: this has never been observed, however easily it can be hypothesised), they are contingent upon matter in a way that numbers and time are not (or if they were, would void your argument).

To be without a mechanism doesn't necessitate that it is random. If something still interacts with causation then it can maintain its ability to be non-random (choices), while being without a specific mechanism (cause and effect created through a deterministic material world). As for the supposed mechanisms of numbers and time, they are ascribed by us to make sense of a non-material abstraction. The only reason 2 comes after 1 is because that is how we make sense of numbers; the same idea is true for time (although, as was said earlier this is a much deeper topic). These non-material subjects are used in a material world only after we have confined them within material constraints so that we can make sense of them.

"Non-physical" = "Magic Pixies".

There's no evidence for a non-physical agent, no need to invoke one, and when you do you just end up with absurdities - like brains being irrelevant to minds.

Once again, how do you provide physical evidence for a non-physical subject? We observe the effects of free will daily while being contained within a deterministic world; we also abstract the possibilities of things which can't be physically observed all the time (dimensional analysis and the topology of other dimensional objects). I've already addressed the idea that the non-material can be dependent on a physical object, while maintaining its individual identity.

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Which means that the conscious mind must have an identity separate from the material mechanism. While its existence must be dependent on the material mechanism, its identity must be independent of it. A parallel here would be "cause and effect". While the existence of the effect is dependent upon the cause, its identity is independent of it. Here's where you have to provide the justification of how an entity can be conscious if the mechanism for consciousness is not a part of its identity.

Is it not true that cause exists as an identity separate from effect, yet they are dependent on one another? An entity cannot be conscious if the mechanism for consciousness is not physically present, that doesn't mean that the identity of the conscience is dependent on the existence of a physical mechanism. I can still talk about the thoughts and choices (conscience) made by Socrates without his physical mechanism (brain) being present. The conscience retains its identity while being causally linked to the physical brain.

Exactly how does its non-material nature allow it to interact with causation while remaining independent of it - that is the question here.

I can abstract about methods through which it may be possible that a non-material nature allows for interaction with physical causation, but they would simply be abstractions. The question of remaining independent of causation has been half answered already. The conscience is dependent on the physical brain, just as effect is dependent on cause, but past this dependency I don't believe that it is tethered to the line of causation which I think you are leaning towards. To illustrate this differently, if I die so does my conscience, but if an apple falls down beside me it doesn't effect my conscious decision in a deterministic manner - that is my belief. My justification for it not providing a deterministic choice is simply that I believe I have the ability to pick it up, or leave it, or get up and walk away, or... etc..

Then your belief has no place in the discussion, does it? As you say here - you cannot demonstrate how the non-material might exist independently of the material. You cannot demonstrate how it'd work. You cannot demonstrate how it'd interact with causation, if it actually existed. Your position is unjustified by your own admission. All you can demonstrate is that you believe in free-will, which is accomplished by saying "I believe in free-will". Without justification, that is meaningless.

1. The non-material can exist as an identity separate from the material, but its utilized existence is dependent on the material (I can't consciously act if I don't have a brain).
2. The conscious would be outside the deterministic causation of the material world (aside from being dependent on the brain), while still providing interaction with the physical world's causation the same way numbers and time are enacted upon the physical world.
3. The existence of free will isn't easily demonstrated if the observers already believe it to be determined materially. Just as I can't illustrate to you the fourth dimension, but I can draw you the shadow of a hypercube; I can't demonstrate to you the existence of the non-material conscience, but I can show you its effects (free will).