The relationship between continuous causal diagrams and the modern laws of physics that you described was fascinating. What's the mainstream status of that?

space and time look like a simple generalization of discrete causal graphs to continuous metrics of relatedness and determination

Mind officially blown once again. I feel something analogous to how I imagine someone who had been a heroin addict in the OB-bookblogging time period and in methadone treatment during the subsequent non-EY-non-Yvain-LW time period would feel upon shooting up today. Hey Mr. Tambourine Man, play a song for me / In the jingle-jangle morning I'll come following you.

finitely Turing-compute a discrete universe with self-consistent Time-Turners in it

In computational physics, the notion of self-consistent solutions is ubiquitous. For example, the behaviour of charged particles depends on the electromagnetic fields, and the electromagnetic fields depend on the behaviour of charged particles, and there is no "preferred direction" in this interaction. Not surprisingly, much research has been done on methods of obtaining (approximations of) such self-consistent solutions, notably in plasma physics and quantum chemistry. just some examples.

It is true that these examples do not involve time travel, but I expect the mathematics to be quite similar, with the exception that these physics-based examples tend to have (should have) uniquely defined solutions.

There are precedents and parallels in Causal Sets and Causal Dynamical Triangulation

CDT is particularly interesting for its ability to predict the correct macroscopic dimensionality of spacetime:

" At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time"

Scott Aaronson showed that time loop logic collapses PSPACE to polynomial time.

It replaces the exponential time requirement with an exactly analogous exponential MTBF reliability requirement. I'm surprised by how infrequently this is pointed out in such discussions, since it seems to me rather important.

It is rarely appreciated that the Novikov self-consistency principle is a trivial consequence of the uniqueness of the metric tensor (up to diffeomorphisms) in GR.

Indeed, given that (a neighborhood of) each spacetime point, even in a spacetime with CTCs, has a unique metric, it also has unique stress-energy tensor derived from this metric (you neighborhoods to do derivatives). So there is a unique matter content at each spacetime point. In other words, your grandfather cannot be alternately alive (first time through the loop) or dead (when you kill him the second time through the loop) at a given moment in space and time.

The unfortunate fact that we can even imagine the grandfather paradox to begin with is due to our intuitive thinking that that the spacetime is only a background for "real events", a picture as incompatible with GR as perfectly rigid bodies are with SR.

I've seen academic physicists use postselection to simulate closed timelike curves; see for instance this arXiv paper, which compares a postselection procedure to a mathematical formalism for CTCs.

I tend to believe that most fictional characters are living in malicious computer simulations, to satisfy my own pathological desire for consistency. I now believe that Harry is living in an extremely expensive computer simulation.

I know that the idea of "different systems of local consistency constraints on full spacetimes might or might not happen to yield forward-sampleable causality or things close to it" shows up in Wolfram's "A New Kind of Science", for all that he usually refuses to admit the possible relevance of probability or nondeterminism whenever he can avoid doing so; the idea might also be in earlier literature.

that there is in fact a way to finitely Turing-compute a discrete universe with self-consistent Time-Turners in it.

I'd thought about that a long time previously (not about Time-Turners; this was before I'd heard of Harry Potter). I remember noting that it only really works if multiple transitions are allowed from some states, because otherwise there's a much higher chance that the consistency constraints would not leave any histories permitted. ("Histories", because I didn't know model theory at the time. I was using cellular automata as the example system, though.) (I later concluded that Markov graphical models with weights other than 1 and 0 were a less brittle way to formulate that sort of intuition (although, once you start thinking about configuration weights, you notice that you have problems about how to update if different weight schemes would lead to different partition function values).)

I think there might have been an LW comment somewhere that put me on that track

I know we argued briefly at one point about whether Harry could take the existence of his subjective experience as valid anthropic evidence about whether or not he was in a simulation. I think I was trying to make the argument specifically about whether or not Harry could be sure he wasn't in a simulation of a trial timeline that was going to be ruled inconsistent. (Or, implicitly, a timeline that he might be able to control whether or not it would be ruled inconsistent. Or maybe it was about whether or not he could be sure that there hadn't been such simulations.) But I don't remember you agreeing that my position was plausible, and it's possible that that means I didn't convey the information about which scenario I was trying to argue about. In that case, you wouldn't have heard of the idea from me. Or I might have only had enough time to figure out how to halfway defensibly express a lesser idea: that of "trial simulated timelines being iterated until a fixed point".

You can do some sort of lazy evaluation. I took the example you gave with the 4x4 grid (by the way you have a typo: "we shall take a 3x3 Life grid"), and ran it forwards, and it converges to all empty squares in 4 steps. See this doc for calculations.

Even if it doesn't converge, you can add another symbol to the system and continue playing the game with it. You can think of the symbol as a function. In my document x = compute_cell(x=2,y=2,t=2)

I haven't yet seen anyone else point out that space and time look like a simple generalization of discrete causal graphs to continuous metrics of relatedness and determination, with c being the generalization of locality.

Yeah, this is one of the most profound things I've ever read. This is a RIDICULOUSLY good post.

I think there might have been an LW comment somewhere that put me on that track or maybe even outright suggested it

I certainly made a remark on LW, very early in HPMoR, along the following lines: If magic, or anything else that seems to operate fundamentally at the level of human-like concepts, turns out to be real, then we should see that as substantial evidence for some kind of simulation/creation hypothesis. So if you find yourself in the role of Harry Potter, you should expect that you're in a simulation, or in a universe created by gods, or in someone's dream ... or the subject of a book :-).

I don't think you made any comment on that, so I've no idea whether you read it. I expect other people made similar points.

I don't totally understand it, but Zuse 1969 seems to talk about spacetime as a sort of discrete causal graph with c as the generalization of locality ("In any case, a relation between the speed of light and the speed of transmission between the individual cells of the cellular automaton must result from such a model."). Fredkin and Wolfram probably also have similar discussions.

I'm not sure how to respond to this; the ability to compute it in a finite fashion for discrete universes seemed trivially obvious to me when I first pondered the problem. It would never have occurred to me to actually write it down as an insight because it seemed like something you'd figure out within five minutes regardless.

"Well, we know there are things that can't happen because there are paradoxes, so just compute all the ones that can and pick one. It might even be possible to jig things such that the outcome is always well determined, but I'd have to think harder about that."

That said, this may just be a difference in background. When I was young, I did a lot of thinking about Conway's Life and in particular "garden of eve" states which have no precursor. Once you consider the possibility of garden of eve states and realize that some Life universes have a strict 'start time', you automatically start thinking about what other kinds of universes would be restricted. Adding a rule with time travel is just one step farther.

On the other hand, the space/time causal graph generalization is definitely something I didn't think about and isn't even something I'd heard vaguely mentioned. That one I'll have to put some thought into.