Dear Marc,
I appreciate the walk through your mindscape! It's a pleasant and instructive journey through another line of thought.
Now, to answer you.
"You mentioned that you had trouble with the idea that a Maxiverse without any limitation on the kind of admissible mathematical structures (in particular, no conservation laws) could be a viable hypothesis". No, not at all. While I do believe (admittedly, I also like to believe) that since the universe looks explainable then it may actually be explainable, my question was aimed at understanding the structure that you describe, and wasn't at all an implied criticism. For what I know, nonexistence cannot be proved, which means that even if we find a unique TOE it will not mean that it's impossible for the Maxiverse to be out there.
You mentioned another essay which I just read at your suggestion. While I do rather fall on their side of the fence from the perspective of hoping our universe is not just made from meaningless coincidences, I can't agree with the formulation of the argument. They say (pg 5, column 2 and a little in column 1) that the Tegmarkverse fails because we can measure our constants with high precision, a better precision than anthropic reasoning asks for. Aside from the fact that this mixes two separated ideas in a way that none of them was made for, that is wrong and wrong. The weak anthropic principle asks for regular worlds up to a certain threshold where they can maintain life, but without necessarily needing high measurement precision, but that doesn't forbid universes that manifest regularity up to 10100 decimal places or whatever they damn well please. The Tegmarkverse is democratic in the sense in which allows observers to exist in worlds that don't have that regularity, but also doesn't forbid the nice smooth ones. Both a mathematical universe and the WAP allow for the existence of awesome universes like ours. Because a true A implies a true B (anthropic -> not necessarily too smooth), it doesn't mean that a false B implies a false A (very smooth -> not anthropic); it's not an iff relation. Then they carry on with false dichotomy when they say (pg 5, column 6) that "since the laws... are not picked randomly, they can only be purposefully chosen". Well. Why aren't they random? In an infinite number of universes, the chance for either single one of them to exist it's exactly zero. Weird values are not predominant; they are equal to the number of fine-tuned values, because, just like you said, the probabilities can't be calculated. And even if they aren't random, that doesn't exclude a third possibility, that there is just one recipe for making a universe, so there is no choice. So no, it's not between random picking and purposeful picking, there are third options and maybe even more options that we didn't consider so far. It's not important if there are other options or not, since I already gave an example thus proving the dichotomy false. A lesser objection (because it's a lesser part of their argument) is the cosmic observer hypothesis (pg 4, col 1) where they say that there's a difference between cosmic observers and minimal observers, which is that the cosmic observers have theoretizable worlds. Well we don't have an explanation for a lot of constants, do we? That rather places us in the untheoretizable pile. So this argument rather works against their line of thought than for it. We can't argue that our universe has a special logical structure because some day we might have an explanation for it, because that's selling the hide of the bear that's still free in the forest. We will only be able to say that we are "cosmic observers" in the day when we will have a single TOE which proves the theoretizabiilty hypothesis, and would prove the random/purposeful dichotomy false. It's a catch 22! I'm sorry to say these things, I respect the work of the authors and I respect them and their opinions and I reserve the right to change my opinion upon future extra input, but I can't agree to these specific arguments because of the way they are built. Your Maxiverse is safe, and if it isn't, it's not because of the Pythagorean Universe. Actually in the reply I left for you on my page, I played a bit with weird conceivable parts of a multiverse, which in a certain sense says that those universes are theoretizable without necessarily being appropriate for life.
Warm regards,
Alma