What does it mean for a brain to "have more information" or be "more complex," than a volume of gas? My big question is: are there set definitions for the concepts of "information," or "complexity?"
Before the background, here are my basic sub-questions:

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*Can a brain or any other complex system be said to contain "more information," than another higher mass/energy system that is considered "less complex?" If so, why?


*If the answer to Question 1 is yes, how do more complex systems contain more information with less mass or energy? Is this because information is related to discerniblity and would this imply that emergence generates new information because it allows for new types of discerniblity? Or does this imply information is a somewhat subjective (but not arbitrary) measure like entropy?


*Is there a firm, formal definition of complexity? All the definitions I've seen seem nebulous.
Background:
In his book "Out Mathematical Universe," Max discussed a hypothetical consciousness detector. This detector would look at two things to determine if a system was conscious:

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*How much information it stores about the world.

*How complex it is.

First, how can a brain, or any complex system, have "more information," than another system with a higher mass/energy? Doesn't energy = information, with black holes representing a maximum information density? I've seen papers that try to calculate the total information content of the visible universe, and they assign discrete information values to elementary particles. This being the case, wouldn't a more massive and higher energy system necessarily have more information than a system with less mass and energy? This entails that 30 kilograms of hydrogen gas in a large vat would have more information than a brain.
This makes sense to me in that it would require more information to encode the position of every particle in the gas over a given length of time. This is because the gas has more particles and because the gas can be arranged in vastly more configurations. The brain can only be in a narrow range of configurations that represent a functioning brain, most configurations are just a random soup of molecules.
The only way I can see this not holding is if we say the brain has more types of things that constitute its building blocks, which gives it more possible states it can be in (e.g. a soup of just the letter H, for hydrogen gas, versus a smaller soup of all letters- the second can have fewer letters but a higher entropy). However, this implies that information is created by emergence and is destroyed when emergent structures disappear. But since the atoms in both structures are reducible to the same subatomic particles, where does the new information come from?

Second, books I've read say information isn't created or destroyed. However, in books on information theoretic approaches to the sciences, authors frequently talk of information being "lost," "disappearing," or "not being recorded." I'm guessing this is really just shorthand for "the information is not trackable for us," or is the idea that emergence actually creates new information, and so the destruction of emergent phenomena means information is actually lost?
I've seen information defined as "the difference that makes a difference," or "asymmetry." This seems to imply that information depends on indistinguishability, and which system is interacting with which other systems, rather than being a static value. But this makes information a somewhat subjective measure then, since definitions for systems are arbitrary, right?
Example: If you mix two non-reactive gasses, the particles are not very discernable. If you instead mixed a highly reactive gas into your original system, that gas is more discernable to that system.

Final question: complexity seems to have a fairly nebulous definition in the sciences. Complex systems are robust and adaptive, they do not dissipate due to small changes in the enviornment. Complex systems are not too ordered (too much order means everything can be explained with a simple algorithm) and not too chaotic.
Boyle's Law is an emergent phenomena that comes from lots of interactions (networking) and gasses adapt to readily to the environment. So what is really meant by complexity and why is a brain more complex than a large volume of gas? How can a brain have more memory if the gas has more information? Obviously the information in the gas is still coming from, and so is "about," the enviornment. It's just not organized in a way that is easy for us to extract. Or do lower entropy systems like brains somehow record more about the enviornment? In any event, the emergence of Boyle's Law seems more adaptive than a brain, involves more information, and is more networked, so how is it less complex? To be sure, the gas is way easier to model, but only because we don't care about where specific particles are. So, is something more complex because more states equate to "meaningful," differences?

Any help is appreciated. I find these topics very interesting, but sources seem to keep disagreeing with each other (and themselves) on these points.
 A: Since we define information (technically, Shannon entropy) as $-\sum_ip_i\log p_i$ for a discrete distribution with probability masses $p_i$, the information present in a given example depends on (i) which variable's distribution is considered and (ii) which information characterizes a distribution for that variable (consider e.g. the distinction between a Bayesian prior and a Bayesian posterior). Bearing all this in mind, let's go through your post:

are there set definitions

Only with the aforementioned disclaimers.

Can a brain or any other complex system be said to contain "more information," than another higher mass/energy system that is considered "less complex?" If so, why?

Because of what we "care about". If there are $n$ possible states each of probability $\frac1n$, the information is $n\times-\frac1n\log\frac1n=\log n$. If there are $k$ particles with such states independently, we replace $n$ with $n^k$, i.e. multiply the information by $k$. So I understand why you expect larger systems to have more information. But the above $k$-particle calculation counts microstates, many of which may be indistinguishable. Equivalence classes of these are called macrostates. Tracking the macrostate instead means considering a different variable, with much less Shannon entropy. With a brain, we care about much more than a macrostate quantifying how common different energy levels are per the largely identical particles of a gas; we care about the knowledge stored on each neutron, which is a microstate concern.
(With a gas, we care about some continuous quantities, such as the energy per particle. This complicates the definition of Shannon entropy, but not in any manner that need concern us here.)

Is this because information is related to discerniblity

As noted above, you're right, although as a matter of jargon we usually refer to indistinguishability (or maybe fungibility) rather than discernibility.

and would this imply that emergence generates new information because it allows for new types of discerniblity?

That's one way to think about it. To take a better understood system than a brain, a hard drive stores data at specific stable locations so it can be read later, so those locations become distinguishable by memory addresses even if they basically have the same physical composition. In particular, we care about bit-string microstates, not simply a "how many $1$s on the hard drive" macrostate, which is similar to what we have in mind for a gas. I won't attempt to summarize how far neuroscience shows that's a good analogy for what a brain does.

Or does this imply information is a somewhat subjective (but not arbitrary) measure like entropy?

I like that description, especially for its second part: subjective (but not arbitrary). Describing gases with macrostates but data storage mechanisms (artificial or evolved) with microstates is useful and convenient, and therefore not arbitrary. The word subjective in this context means it's relative to whether we choose a macrostate-labelling or microstate-labelling variable.

Is there a firm, formal definition of complexity? All the definitions I've seen seem nebulous.

Defining complexity is indeed a bit trickier than defining information, mainly because a number of concepts are sometimes convenient to give the former label. In this context, I'd recommend the Kolmogorov complexity as one of the choices closest to what you have in mind. It too is relative to certain choices we made; you can probably guess which, when I say its definition is "the length of the shortest possible description". The shortest description of a hard drive's state has to characterize its (possibly compressed) microstate. By contrast, if you want a mole of nitrogen you don't care about constantly fluctuating particle positions we can't control or measure well anyway.

Doesn't energy = information, with black holes representing a maximum information density?

That question clearly contradicts itself: if only the total mass-energy mattered, why would black holes be better than alternatives? BHs do in fact maximize the microstates for a given mass-energy, albeit not in a controllable way one could exploit for data storage; it's more analogous to a gas than a hard drive.

this implies that information is created by emergence and is destroyed when emergent structures disappear. But since the atoms in both structures are reducible to the same subatomic particles, where does the new information come from?

As with so many concepts in information theory and thermodynamics, it comes down to how useful the result is. You can use the information on a hard drive in several ways, and can't do the same with a gas. Formatting the drive "destroys" its information in the sense that, no matter what particle physics has happened, there is no longer useful information there.
