What is a mathematical structure of conscious experience?

Internally we know we're conscious because we just do. We experience qualia, even though we don't know what "we", "experience", and "qualia" really mean.

Even so. For us, consciousness is just there.

Externally we attribute consciousness to certain behaviours, including (but not limited to) self-reference, goal-seeking, physical and social responsiveness, use of language, memory, and so on.

These two things are not the same. We assume other humans are conscious in the same way we are to the extent that they perform consciousness. We can't be sure, but for human interactions it's a workable proxy.

A toaster, a watch, a rock, and a web server don't perform consciousness in any way at all. They may have some kind of metaphyiscal awareness, or not, but if they do we can't see it, and without a comprehensive theory of consciousness it's parsimonious to assume they're not aware.

ChatGPT performs some elements of consciousness, but not others. It happens to perform the elements AI researchers consider important - specifically use of language. But it doesn't have a memory for previous conversations, it doesn't show any independent goal seeking, and so on.

It uses "I..." and a vast amount of human training to mimic the appearance of consciousness, and plenty of humans seem to want to believe that's enough.

But the essence of the hard problem is the difference between "performs the actions we associate with consciousness" and "is conscious in the same way we know we are."

You could argue that practically there's no difference, and from one point of view that may be true. But that doesn't work if you want to understand what's happening rather than just assuming in a "well obviously..." kind of a way.

If only because historically in science "well obviously..." has been consistently wrong.

The issue is, what else would we call the rock that does all the things a CPU does? If we can socket it into the motherboard and play games on it, what makes it so different?

I didn’t really mean to question the phrase hypothesis, rather that “lacking proof is not proof of lacking”, and why not lay out more form of curious extrapolation if that is after all, why we are here.

I suggest you look into Roger Scruton's interpretation of Wittgenstein's private language argument. I think it might go something like this:

You want an explanation for the sensation of the color red? Alright, I'll try, as long as you help me out: what exactly is the sensation of the color red? I need to have a good grasp on what we're talking about here.

Can you describe it? It's okay, take your time.

FIN

If you are controlling a computer, that control is not visible within the software of computer. The state of computer is accessible to you from "outside", say, by reading hardware state. And you can manipulate that state directly from outside.

This control and access, which needs a true observer, is consciousness. If a computer is taken as layer of reality, then this control is in outside layer of reality. But we don't stop there: It is observer in all layers of reality, whatever reality is made of. The observer can never be represented in any layer of reality.

We should consider that experience layer, like sound and vision are an independent layer of reality than underlying "material" reality. A materialist will say latter exists a priory, and thus experience is off the material layer. But we can as well argue that experience layer is a-priori. My taste sense, for e.g. has no equivalence in material reality. It deserves its own layer of reality. If I hear a voice of someone, it is singular experience rather than experience of collection of frequencies. Color like white is an experience of multiple frequencies of light. Our perceptions, happening with help of neurons, are not based on light at all. Projection is a-priory present, from which we create even light and other formulations...

A mathematical reality is in its own layer of reality. A conceptual reality of say, chair, is also in a "mental" reality.

So we should assume that all are simply abstract layers of reality. But our mental conditioning has created "things" model, just because we can touch and feel, or perceive a boundary in what we see.

As observer, which creates memory of observation, and accesses it, all in the present. An observation which is not in time and space, but creates sense of time and space out of observation.

And so on... A totally different model of reality.