I may be missing something, but it's not obvious to me from a physics perspective that running up hills is less impact-on-the-joints per effort, though from experience I agree it's true. Maybe I don't understand how impact is measured.

As long as your mass is rising at a constant rate, ascending vs. descending doesn't change the downward force your feet must exert. Yeah, there will always be some vertical acceleration, like when cresting a hill, but unless one is running staircases with landings every 10 steps this wouldn't be representative of the workout.

Inexperienced musing about other possible explanations:

Is it simply that effort-per-step goes up, so that fewer steps are required to achieve your goal? OTOH I've also been told that one should take faster, shorter steps while ascending, which would somewhat counter this effect. OTOOH maybe a short step is better on the joints than a long step. Then opting for steps that are both short and effortful could reduce a workout's total joint-impact.

Oh, and obviously you (and your legs?) don't move as fast. Maybe that is another contributing factor.

Did I get that right, or am I missing something?

I'm super excited for this! First impressions of site:

• The colorful graphs are eyecatching and fun. Some graphs have too much going on, with all the markets and different resolution criteria. Some visual simplification desired.
• It seems that platforms get different colors in different graphs? I desire each platform to have a consistent style.
• Icons rather than platform names all over the place would feel good
• I love the "China position paper" annotation. I want to see lots of that.
• On the US debt graph, one of the vertical line annotations is probably not displayed correctly: 'In 'In pri...
• Some underlined text is an actual link to another site, but other underlined text is an expandable section, which causes me confusion.

This is a something I frequently get hung up on: If the AGI is highly intelligent and socially manipulative, but lacks good motor skills/advanced robotics, doesn't that imply that it also lacks an important spatial sense necessary to understand, manipulate, or design physical objects? Even if it could manipulate humans to take arbitrarily precise physical actions, it would need pretty good spatial reasoning to know what the expected outcome of those actions is.

I guess the AGI could just solve the problem of human alignment, so our superior motor and engineering skills don't carelessly bring it to harm. 

Thanks for being patient with my questions. I'm definitely not solid enough on these concepts yet to point out an exploit or misalignment. It would be super helpful if you fill out your LVPM playground page in the near future with a functional AMM to let people probe at the system.

Flipping is the only symmetry that exists for unidimensional cases.

If you have multiple dimensions, you have a whole continuum of symmetries, because you can continuously rotate the dimensions into each other.

What do you mean by unidimensional cases? So like if the binary LVPM $Y$ is made up of binary markets $X_1$ to $X_N$ , I would have called that an N-dimensional case, since the PDF is N-dimensional. How many symmetries and "possible convergences" does this $Y$ have?

So this is in fact true: LVPMs will not be totally unique, but instead have symmetries where some options do equally well. The exact type of symmetry depends on the kind of market, but they mainly look like what you describe here: swapping around the results such that yes becomes no and no becomes yes.

I'm not a savvy trader by any means, but this sets off warning bells for me that more savvy traders will find clever exploits in LVPMs. You can't force anyone to abide by the spirit of the market's question, they will seek the profit incentive wherever it lies.

(I feel somewhat satisfied that the "question reversal" symmetry is ruled out by the market maker restricting the possible PDF space (in half along one specified dimension, IIUC). I'm curious how that would be practically implemented in payouts.)

 I worry about these other possible symmetries or rotations, which I can't yet wrap my head around. I would love an illustrating example showing how they work and why we should or shouldn't worry about them.

So if I'm understanding correctly, you're saying a LVPM market Y could be displayed without a title, and traders would still converge toward finding the joint probability distribution function? So instead of making a LVPM titled "Will Ukraine do well in defense against Russia?", I could make a LVPM that says "Hey, here are a bunch of existing objective questions that may or may not be correlated, have at it", and it would be just as functional?

If so that's pretty neat. But what stops Trader1 from interpreting Y as "Will Ukraine do well in defense against Russia?" while Trader2 treats it as "Will Russia do well in the invasion of Ukraine?" IIUC, these two traders would make nearly opposite trades on the conditional probabilities between each X${}_i$ and Y.