Has anyone been able to define well or measure differences between vinyl and digital?

fair

... A mathematical theorem is like a part of a legal contract: its words have precise meaning, often unexpectedly different from their everyday meaning; and it has small print ...

A math theorem is a principle that can be proven with math. The Fourier Transform is a theorem. It has been proven by both math and in actual implementation - it's the basis of both digital and analog audio. Your alternative paradigms aren't consistent with Fourier. You're probably having fun with your fanciful imaginings but that's all they are. If you want to make actual progress you'll have to accept the math. Anything else is futile and more than a bit silly.

@johnread57 , I was out a few messages ago.

I'm telling you that the errors are signal dependent, you are quoting this, and then are asking me to tell you exactly how big the errors are? 

You know, serious people run long simulations to answer this question for specific digitization schemes and sets of representative signals.

To quote @cleeds these are fanciful imaginings, not serious discussion points. Serious discussion points would come with serious analysis or serious links showing that analysis. 

@cleeds

A math theorem is a principle that can be proven with math. The Fourier Transform is a theorem. It has been proven by both math and in actual implementation - it's the basis of both digital and analog audio. Your alternative paradigms aren't consistent with Fourier. You're probably having fun with your fanciful imaginings but that's all they are. If you want to make actual progress you'll have to accept the math. Anything else is futile and more than a bit silly.

Let's make it more interesting. Shall we?

You likely know who Bob Stuart is:

Despite his lifelong achievements and indisputable expert status, audiophile community met MQA format he co-created with significant controversy, e.g. 

I propose reading a peer-reviewed paper he co-authored:

And then telling us what Tidal did wrong to further stir up the MQA controversy. And perhaps what Tidal and others can do in the future to improve the MQA situation. You may think it can't be improved, and this would be interesting to hear too. 

I hope the ensuing discussion would be very relevant to topic at hand, because Tidal and other "Ultra HD" music streaming services hasn't yet made a dent in LP sales, but I believe they may do so in the near future.

@thespeakerdude : Very curious from your username: what are your own speakers? Thanks you

@johnread57 , I came across this NPR interview again, has some good comments on this topic. I am going to copy some and a link to the whole article.

https://www.npr.org/2012/02/10/146697658/why-vinyl-sounds-better-than-cd-or-not

METCALFE: Well, I think it has a lot to do with the fact that I'm primarily a recording engineer, as far as working with music. And it's - the closer thing to what I'm sending into the recorder is very much what I'm getting back out. With analog formats, although the sound can be very pleasing in certain styles, it's definitely imparting its own sound on it. And I think, to an extent, it's that sound that some people are really drawn to. But it's nice as an engineer to have the confidence of knowing that what I'm putting into - in most cases these days, the computer - is pretty close to what I'm going to get out.

OLIVE: Well, I mean, I grew up listening to records up until about '85, when the CD was already out. And I was involved in testing loudspeakers up at the National Research Council in Canada. And we were testing cartridges at that time, and it was quite apparent that the amount of distortion coming out of these devices was very high compared to CD. So what we found was that vinyl was a limiting factor in our ability to do accurate and reliable listening tests on loudspeakers, and we had to find a more reliable and more accurate medium.