It's really kind of stunning how many people have just lost their donuts over the question asked the OP--or the inability to answer the question. Apparently the question threatens people's religion.
"Lost their donuts"? I'm not familiar with that expression. But I don't think I have lost anything. My 'religion' (if that's what my current opinion about how the world might really work is) being based on what might be termed "panexperiential physicalism" would certainly benefit if it could be established that 'consciousness' (whatever that is) is a 'fundamental phenomenon' (whatever that means). That is to say that if everything physical somehow existed in a fundamental universal "field of consciousness" it would be far easier to see how everything physical somehow experiences the world. So my 'religion' is certainly not threatened by this idea.
Unfortunately, I cannot see how something fundamentally immaterial and aphysical could possibly have causal efficacy in the real physical world. (Perhaps that is just metaphysical myopia on my part, but I don't think so). If (fundamental) 'consciousness' is, as you suggested in the OP "like energy" then "like energy" it should be a physical property of the physical universe, and like energy, it should be associated in a coherent way with the matter that it has causal influence over.
Clearly, consciousness does have causal influence in the physical world - I doubt that there could be any serious question about that. Bach presumably did not compose his Cantatas as a result of the deterministic processes of particle physics going on in his body. But then are we to accept that they were merely "downloaded" (somehow) to his brain from an ethereal and immaterial 'consciousness' with no input from the physical reality of Bach's brain/body? Clearly there has to be a third option here. A 'middle way' in which consciousness is both a cause and an effect of physical reality - don't you think?
In the OP you wrote:
...there is just no amount or complexity of neuronal electrical activity that logically produces mental phenomena.
I am taking this (in part) to imply a rejection of the idea of 'radical emergence' - i.e. you don't believe that 'consciousness' can simply emerge from an entirely non-conscious 'substratum' of physical reality - regardless of how complex the 'substratum' might appear to be? I agree with that. I flatly reject this kind of radical emergence.
So what are we left with? A fundamental reality that is both 'physical' (perhaps energy-like) and 'mental' (perhaps mind-like). This is the bipolar 'philosophy of organism' that Whitehead proposed and my guess is that quantum mechanics is an approach to understanding this scientifically (complicated by the fact that the manifest world also displays matter-energy duality). But we are not there yet (and there may yet be some startling surprises in store). What this view really says is that each real entity (which is not really an "entity" as such but an "occasion" or "event" or, in succession a "process") is always simultaneously
both what it IS (physically) and WHAT it is (mentally) - it's physical reality
and its mental description.
On this way of looking at it, 'consciousness' - in the sense you used it in the OP is not fundamental
per se but is an aggregation of the 'mental poles' of the 'actual occasions of experience' that make up this or that 'organism'. It is, I suppose, at one and the same time, fundamental and emergent - fundamental in the sense that it, at the most basic level, pervades the entire reality of the physical world and 'inhabits' even the most fundamental units of that reality, emergent in the sense that in combination, these 'bits' of reality can and do produce phenomenal expressions of 'mental' experientiality that may appear novel or radically emergent but that are really just an extension of the experiential-relational reality that exists at the most fundamental levels.
From an evolutionary point of view, it stands to reason that nature would have found a way to exploit the 'mental poles' of physical reality and the human brain is, perhaps (but the jury is definitely still out IMO), its greatest success to date.
I get that you were asking for a valid deductive syllogism and I have not provided one - I can't. But neither can one be constructed to establish an origin of consciousness that does not have a physical basis. As I suggested earlier, I suspect that the latter task would remain beyond us even if the conclusion were true. What I have provided, I believe, is a valid abductive argument (reasoning to the most likely account) for taking human consciousness (which is what you were really asking about) to be a highly-evolved and well-developed expression of the fundamentally experiential nature of physical reality.
Of course you do not have to agree - but if you want to dismiss this abduction as invalid, you now have the task of having to provide either a simpler, more likely or more elegant account (preferably one that does not include donuts) of how human consciousness arises from a non-physical reality (if that is what you are really suggesting).