The definition of arbitrary is very screwed up.
Consider half of a sliced bagel. Also consider a bread crust of the same bread, of roughly the same thickness. I assume the purposes of either is to be consumed for enjoyment and nourishment; bagels and bread should have nutrition and a pleasant flavour. (And their, say, aerodynamics, are irrelevant for all intents and purposes.)
There is no real difference between the two; there is no purpose-significant action which is served better by one or the other. They have the same nutrients because they're the same bread, they can both hold spreads at roughly the same ratio, and they can both be properly toasted in a toaster with a 'bagel' setting, which only toasts one side. No matter what you intend for that chunk of dough, your purpose will be served. The only possible difference is if you just have a raw preference for having a hole in your bread, or simply prefer 'b' noises.
(While Anglophone society considers these subjective purposes to be non-practical, the union of subjective and objective makes clear that a subjective, non-rational preference is an intent or purpose. Nevertheless, it is true that this is the only purpose against which the bagel and bread crust differ.)
In sum, the distinction between a bread crust and a bagel is an arbitrary distinction. There is no functional difference marked by this distinction. No matter how you define the difference, it makes no difference; the properties of the bread crust and the bagel remain identical. (If we were to try to systematize the difference, to strictly define it, we would find we needed a different name for every chunk of dough ever, because none of them are exactly the same shape, and even still this is an 'irrelevant' and 'irrational' difference, with regard to the purposes of food, assuming we drop blatantly repellent forms like swastika-bread.)
And now I can break the concept for you. Consider instead half a positive parabola. You can, if you want, define an upper portion where the slope is very large and a lower portion where the slope is smaller. Again, however, the choice is arbitrary; as you slide the definition up and down, there is no quality that suddenly changes for it to catch onto. However, this kind of arbitrary is completely the opposite of the previous kind of arbitrary. Instead of the difference making no difference, every tiny change makes a difference. Every time you move the difference between steep and gentle, the functions of both change.
In fact, I can say that the first example is arbitrary because it makes no difference where you put the distinction, and the second is arbitrary because it's never arbitrary where you put the distinction. It makes me giggle so I'm typing it again; it's arbitrary because it's never arbitrary.
Consciousness vs. Math
And thus part of my problem trying to explain why if one equation is conscious, every equation is conscious. Yes, it makes a difference on where you put it; but it is never arbitrary in the first sense, which means it is arbitrary in the second sense. No matter where on the parabola you define 'steep,' to properly define it, every part of the parabola is steep, to a degree, even at the base where that degree is zero at one and exactly one point.
So I guess if one equation is conscious, the situation is even worse than every equation is conscious. Rather, almost every equation is conscious except a few which completely change depending on which equation you define as conscious first. Even if you found something in nature described by a zero-consciousness equation, it would be more like sleeping than dead, and you'd be able to awaken it with a single poke. It would be very unstable. On top of all this, consciousness would still be either acausal or an epiphenomenon. It would have to either change the behavior of the equations that were highly conscious, that is, away from their purely mathematical behavior, or it would be unnecessary for calculating dynamics.
This is because adding consciousness in as a fifth force in nature forms an infinite regression. The existing equations of motion would all be assigned a consciousness number, which would determine their interactions to the consciousness field. But this field would also be described by an equation, which would have such a number, but the interaction of the field with itself would change the equation, changing the number. Which would cause an interaction of the field with itself, changing the number. And so on. Attempting to make consciousness mathematical either annihilates consciousness into epiphenomena, or leads to the equation tying itself into knots until it disappears.
A further problem is where to divvy up nature into equations, so that we can assign them numbers.
So my inner critic is bugging me, but it a good point. I need to make this explicit. Let us assume that we have a fully mathematical formula for consciousness. First, determinism is strictly true. The equation cannot deviate from itself; your 'choices' in each moment are a function of the state the moment before. Choices do not truly exist, just a sensation we call 'choice' erroneously.
Second, 'happy' = '3.' Perhaps '90=theory of relativity.' (Of course single numbers would be much more fundamental units of thought, but this isn't material to my argument.)
Okay. Why? Why is 'happy' 3 and not 4? Or 70? These things are first, arbitrary in the first sense, and second, you do not need these labels to fully describe the physics.
This is important, so let's do a second example. What about operators? You have the Schrodinger wave function of consciousness, and you use the 'happiness' operator on it to find the degree of happiness. This number then feeds into interactions with other parts of consciousness. But, again, this label is arbitrary. It may empirically be that 'happiness' corresponds to this operator, but there is no reason to use this label. We don't need to know what sensation it corresponds to; we just need the operator and the mathematics to describe its interactions. Consciousness becomes merely epiphenomenon.
Consider the opposite situation, the momentum operator, p=ħ/i d/dx. We also don't need to know that it is momentum for it to work, but this is fine because it is a number that only references other numbers. Its essence of momentum-ness isn't critical to its definition, the way the essence of happiness is critical to its definition. The defintion is, relatively, completely backwards; momentum is defined as the thing (ħ/i)(d/dx) picks out, while what picks out happiness would have to be empirically tested. If we've made a mistake with (ħ/i)(d/dx) and find that what it picks out doesn't have the properties of momentum we want, we don't need to change it at all; we just pick a new name, use it for exactly the same things as we did before, and find the operator that we do want to call momentum. If we find that the happiness operator doesn't pick out the properties of happiness, we immediately realize all our data indicate that it does and we're just screwed. And, as expected from epiphenomenal consciousness, it doesn't matter; oh, this happiness indicator that corresponds to resported happiness isn't happiness, but all our predictions come out true anyway. This property truly is a frivolous extra property.
Consciousness cannot be mathematical. Therefore, it cannot be physical.