*The last two bullet points. Meta-consciousness and self-consciousness

I meant if you had any suggested rewords, because there don't seem to be any perfect definitions of these concepts.

"Easy problems of consciousness" is an established term that is a bit better-defined than consciousness. By transcending, I just meant beyond what can be explained by solving the easy problems of consciousness

This was actually what I meant by a version of panpsychism that seemed to be the natural conclusion of humans having subjective experiences, but a conclusion I want to see if I can avoid.

I tried some different definitions of consciousness while writing this point, until settling on "able have subjective experiences that transcend the 'easy problems of consciousness'"

Do you have any suggestions for making this more precise?

I'd like to explore these in more depth, but for now I'll just reduce all the angles you provided to the helpful summaries/applications you provided. I'll call the perspective of going from adult human to zygote the "physical history" and the perspective of going up the ancestral tree as the "information history" (for simplicity, maybe we stop as soon as we hit a single-celled organism).

• Sentience: This feels like a continuous thing that gets less and less sophisticated as we go up the information history. In each generation, the code gets a little better at using the laws of physics and chemistry to preserve itself. Of course if one has a threshold for what counts as sentience, it will cross it at some point, but this still strikes me as continuous.
• Wakefulness: This would strike me as a quantized thing from both the information and physical history perspective. At some point in both histories, the organism/cell would pick up some cyclic behavior.
• Intentionality: I'd need to look more at this, because my interpretation of your first sentence doesn't make sense with the second.
• Phenomonal, Self-Consciousness, Meta-Consciousness: Definitely quantized in both perspectives

When I was thinking of subjective experience, I think the only concepts here that are either weaker or stronger than what I had in mind are the last two. For the rest, I think I can both imagine a robot that satisfies the conditions and imagine a conscious being that does not satisfy the condition.

But the last two still feel too strong. I will think more about it.

That's a bit of a long read, and both your endorsement and the title seem too strong to be believable. If a few more people endorse that it's worth reading, I'll give it a go!

Very nice! Notice that if you write $r=j-k,$ $I$ as $D^{-1}$, and play around with binomial coefficients a bit, we can rewrite this as:
 

(By the way, how do you spoiler tag?)

This is true, but I'm looking for an explicit, non-recursive formula that needs to handle the general case of the kth anti-derivative (instead of just the first).

The solution involves doing something funny with formal power series, like in this post.

Here's a puzzle I came up with in undergrad, based on this idea:

Let $f\left(x\right)$ be a function with nice derivatives and anti-derivatives (like exponentials, sine, or cosine) and $p\left(x\right)$ be a polynomial. Express the $k$th anti-derivative of $p\left(x\right)f\left(x\right)$ in terms of derivatives and anti-derivatives of $p\left(x\right)$ and $f\left(x\right)$.

Can provide link to a post on r/mathriddles with the answer in the comments upon request

Suppose we don't have any prior information about the dataset, only our observations. Is any metric more accurate than assuming our dataset is the exact distribution and calculating mutual information? Kind of like bootstrapping.