"If your life choices led you to a place where you had to figure out anthropics before you could decide what to do next, are you really living your life correctly?"

– Eliezer Yudkowsky

To revisit our premises: Why should we think the end result is achievable at all? Why should it be possible to usefully represent the universe as an easily interpretable symbolic structure?

First, I very much agree with the sentiment quoted above, so we aren't quite doing that here. Most of the actual reason is just: it sure looks like that's the case, empirically. As I'd argued before, human world-models seem autosymbolic, and the entirety of our (quite successful) scientific edifice relies on something-like-this being true. I think the basic case is convincing enough not to require much further justification.

But can we come up with theoretical justifications regarding why the universe might have ended up with this structure?

Here's one one approach: it's because there's an anthropics-based selection pressure on universes to be structured this way. Three lines of reasoning:

1. Because well-abstracting universes are simpler: their histories have a shorter description length. (In the initial pitch, I'd argued that aiming for simplicity gets us interpretability. Same logic holds: if anthropics reasoning prefers simple universes, it prefers well-abstracting universes.)
2. Because "coarse" agents – agents implemented on abstract high-level variables, which can only exist in well-abstracting universes – pool the realityfluid from all "low-level" agents corresponding to them, and are therefore the more likely viewpoints (see this post).
3. Because embedded agents could only survive in well-abstracting universes to begin with.

I go full anthropics-weirdness in this section, taking the premises of Tegmark IV as granted. I primarily view that type of reasoning as a useful tool for organizing one's thinking and informing one's intuitions in internally consistent ways. The conclusions don't rely on anthropics' metaphysical premises being literally true. (Indeed, I expect this can all be translated into more mundane reasoning about Solomonoff-induction approximation / generalized Occam's razor: reasoning about what theories about the universe's structure we should consider more likely a priori. But the language would grow cumbersome.)

4.1. Well-Abstracting Histories Are Simpler

Consider a universe whose initial conditions and physics unroll into a history that has certain robustly reoccurring high-level regularities/emergent dynamics. In other words, it has "abstraction levels", which follow different high-level "laws" and across which "high-level histories" play out. The tower of abstractions can have many levels, of course (fundamental particles to molecules to cells to organisms to populations).

Suppose that the high-level history is generated by simple laws as well. (Chemistry, evolution, microeconomics.) Then, we could exploit it to compress the description of the full-fidelity/lowest-level history. Instead of having to use an encoding that specifies the full state at any given moment, you could point to the high-level history, then "clarify" the low-level states using "correction terms". This would work inasmuch as the high-level history distills the information that is repeated in several low-level variables.

(If we have two random variables $X$, $Y$ with nonzero mutual information $I(X;Y)$, we have $H\left(XY\right)=H\left(X\right)+H\left(Y\right)-I(X;Y)>H\left(X\right)+H\left(Y\right)$, because sum of individual descriptions double-counts the shared information. Describing the low-level state directly is equivalent to using the sum of individual descriptions.)

Alternate reasoning: Any description of a low-level history necessarily describes the high-level history, so a longer-description high-level history necessarily means a longer-description low-level history. Now consider a history without robust abstraction levels. This is isomorphic to it having a high-level history which takes a random walk through some state-space, which blows up its description length.

(Note that high-level histories don't have to be absolutely robust: abstractions can be leaky or even sometimes destroyed. Governments fall apart, the orbital dynamics of star systems change. In this frame, it only means you have to spend bits on "manually encoding" discontinuous jumps through the high-level state-space ("abstraction leaks", high-level state momentarily becoming dependent on low-level details), or on switching the function generating the high-level history ("abstraction break", the high-level emergent dynamics fundamentally changing). As long as each high-level dynamic stays around for a nontrivial interval and largely explains (an aspect of) the low-level dynamics within this interval, it still serves to compress the low-level history.)

All this naturally works better if we have many layers of abstraction: intermediate levels compress lower levels and are in turn compressed by higher levels.

So: if we assume some process in Tegmark 4 that picks which histories to implement, instead of implementing all histories blindly and uniformly, the histories with "tall" abstraction towers would be picked more often / have more realityfluid / are anthropically selected-for. (In our case, this process is maybe the interference patterns between Everett branches.)

I. e.: this is the same logic I initially used to argue that aiming for a low-description-length representation of our universe would recover a well-structured representation. If reality "prefers" universes that are simple, it prefers universes that can be compressed well, which means it prefers well-abstracting universes.

Sidenote: This potentially sheds light on the induction problem. If large intervals of history have to approximately agree with the output of a simple program, then looking at the past data and inferring its generators would yield you some generators which are still "active" in your time. Hence, compressing the past lets you predict the future.

4.2. Coarse Agents

If you sampled an agent from the anthropic prior, what sort of agent would you expect to see?

Consider "coarse" agents: agents that are implemented on some high-level abstract variables, "live within" high-level histories, and frequently discard low-level details of their observations.

The observation-history of a coarse agent would correspond to a set of lower-level observation-histories; to a set of lower-level agents which are approximately similar to each other. Mechanistically, the observation-streams would "diverge" the moment two agents perceive some detail that differs between their universes, but would then "merge again" once they forget this detail. Coarse agents would then necessarily hoard more realityfluid than any "precise" agent. (See a more gradual introduction of the ideas here. Also, note that this doesn't give you any weird anthropic superpowers. Like, this idea feels weird, but as far as I can, it still adds up to normality.)

Note, however, that such agents could only live in well-abstracting universes. After all, as a premise, we're assuming that they're able to discard detailed information about their observations and still navigate their universe well. This naturally implies that there are some high-level laws under which the agents live, instead of being exposed to the raw low-level dynamics.

This also means we can expand the arguments above. We can count all approximately similar universal histories as "the same" history, for the purposes of adding up the realityfluid of the embedded agents. That biases the anthropic prior towards well-abstracting histories even further: they're not only simpler by themselves, but they also embed a simpler type of agent. (The relevant notion of "approximately similar" here is provided by the particulars of the agent's vantage point/at what abstraction level it's embedded.)

4.3. Only Well-Abstracting Worlds Permit Stable Embedded Agents

Consider:

• In a universe lacking high-level emergent patterns, with incompressible low-level dynamics, you have to pay attention to the entire low-level state in order to predict the future.
• Reliably navigating/controlling an environment requires having access to a structure isomorphic to its model.
• Embedded agents are distinguished by being smaller than the universes they inhabit.

Therefore, if a universe's dynamics aren't (approximately) compressible, embedded agents cannot (even approximately) model them, cannot navigate that universe, and cannot survive/thrive in it.

I'm not entirely confident in the specifics of that theoretical reasoning; my anthropics-related arguments may contain errors. Still, I think the overall intuitive story is convincing: histories with higher levels of organization are meaningfully simpler than histories without them, and inasmuch as we put any stock in arguments for simplicity, we should expect to find ourselves in a universe with a well-abstracting history.