How should a score be interpreted w.r.t. proportionality?

There are many proportional representation schemes using ranked or approval ballots. Also, there are many PR rules which try to 'spread out' a single voter's voting power over the candidates on their ballot. For example, you might think of SPAV as 'spreading out' a vote over all previous winners & potential new candidates equally, so if a voter has 2 winners already, they can only contribute 1/3 to another winner. Similarly, STV just gives the full voting power of a voter to their top choice until it gets eliminated (or fractional surplus handling happens).
At a very nonrigorous handwavy level, I think of the ranked or approval ballots as a voter saying how they're comfortable having their vote spread:
with an approval ballot, I think the interpretation is quite clear as "give my voting power to any of these candidates as such to elect as many as possible." you can axiomatically characterize proportionality via measures like PJR
with a ranked ballot, you might interpret it as "give my vote 100% to as a high a choice of mine as will benefit me" and can axiomatically characterize proportionality via measures like PSC
But how should score ballots be interpreted in the context of proportional representation? what are the natural extensions of the axioms for proportional approval? I know there has been a little bit of discussion around questions like this in the context of Sequentially Spent Score whether to use scaling or capping, but I have not yet seen a satisfying conclusion.
In general, there is a ton of existing literature on proportional elections with approval ballots and I'd really like to know what results carry over or what to expect from proportional scored elections.

But how should score ballots be interpreted in the context of proportional representation?
There is a whole electowiki page about this topic. https://electowiki.org/wiki/Cardinal_PR
what are the natural extensions of the axioms for proportional approval?
I spent some time on this a while ago. The claim was that RRV was the natural extension of SPAV which maintained the Thiele ethos. I don't think that is true so I invented a new system to show what should be done.
https://electowiki.org/wiki/Single_distributed_voteI know there has been a little bit of discussion around questions like this in the context of Sequentially Spent Score whether to use scaling or capping, but I have not yet seen a satisfying conclusion.
I agree and I invented SSS. I feel like capping is more true to the philosophy but scaling gets better results in some specific examples. The Philosophy of vote unitarity is incomplete. It was a counter point to Parkers Sequential Monroe Voting which tried to do what STV does but with score. The "give my vote 100% to as a high a choice of mine as will benefit me" philosophy seems like it would polarize so I did not like it. SSS is intended to be unpolarized.
I am trying to get the authors of this paper to opensouce their code so we can test more systems for bias
In general, there is a ton of existing literature on proportional elections with approval ballots and I'd really like to know what results carry over or what to expect from proportional scored elections.
This is the absolute cutting edge of research. Equal vote is planning on forming another committee to tackle this topic. Would you have time and ability to contribute? I have still not found the time to publish the findings of the first committee because I am too busy with Canadian PR Campaigns.
Another question to answer is how to compare the amount of PR accross systems. Galligher index is bad but what is the nonpartisan alternative. My preference would be to base it in Winner Set Stability but I do not know how to do that. Also, what is the level of PR for single winner systems. Does STAR give higher PR than FPTP? look at this post.
https://www.votingtheory.org/forum/topic/101/whatlevelofprdodifferentsystemsget?_=1643416428211 
Another question to answer is how to compare the amount of PR accross systems. Galligher index is bad but what is the nonpartisan alternative
Agreed, it's not super obvious how to compute proportionality. Things like utility variance or fraction of electorate with/without a winner among their top choices work to a point, but do not seem particularly theoretically motivated. The state of affairs here is much simpler with approval ballots!
This is the absolute cutting edge of research. Equal vote is planning on forming another committee to tackle this topic. Would you have time and ability to contribute?
Glad to hear I am not the only one with these questions. I could be interested in contributing. Would that look like more simulations or trying to axiomatically characterize these methods or both? Simulations can be great tools but one thing I've found is that they frequently have so many tunable parameters that just by changing a few numbers you can benefit one voting rule or another...

Things like utility variance or fraction of electorate with/without a winner among their top choices work to a point, but do not seem particularly theoretically motivated.
If you look at some of the metrics I used in this simulation from the last Equal Vote Committee you will see that I used such metrics. This may be a nonsolvable problem for 3 reasons:
 First if there was a metric to maximize we could just make a system that tried all permutations and take the one which maximised the metric. Warren Smith attempted to find such a method here but gave up.
 Secondly, PR has historically been about representing parties fairly not representing people fairly. Even if you came up with a good metric people would fight you on it.
 Lastly, there are theories for the party list case what show that you cant have it all. For example the Balinski–Young theorem. Winner set stability is likely the best definition of PR but it does not always have a solution. There may literally be no answer.
. I could be interested in contributing. Would that look like more simulations or trying to axiomatically characterize these methods or both?
Honestly, since it would be all volunteer it up to you. The mandate last time was to come up with a PR system for Equal Vote to endorse. We ended up with Allocated Score but some of the reasons for that are not 100% proven. Any evidence towards the goal of justifying what the best system is would be great. If you are not a coder or a mathematician there is still lots to do. I would love somebody to write up some of the results for publication.
Simulations can be great tools but one thing I've found is that they frequently have so many tuneable parameters that just by changing a few numbers you can benefit one voting rule or another...
Agreed. There are ways around that but in the end simulations never represent reality. Actual mathematical proofs are better but tend to lack applicability.
Anyway, if you are interested write to Sara (sara@equal.vote) and I(keith@equal.vote).

@keithedmonds I emailed you, let me know if you didn't receive it.

@brozai Received!