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

Yes, given all the added steps it will sound different. And to some it may even sound nicer. Tastes differ.

Let's talk about this subject just from a musical standpoint.  In classical music, when a composer writes a score, he or she notates dynamics (forte, pianissimo, etc) to denote how loud or soft a passage is to be played.  These dynamic markings are as much a part of the musical score as the melody or harmony.  By definition, vinyl is a compressed format and cannot completely express loud or soft the way digital can.  This is not an issue in most rock music, because there are not the dynamic markers in rock charts.  In jazz, very few artists put dynamics into their performances.  Art Blakey was one who made his ensemble play with a large dynamic range (I was lucky enough to see his band live before he passed away).  This is not an attempt to plug or criticize either vinyl or digital, but to what audio should strive for from a purely musical perspective.  I hope anyone reading this gets a chance to sit front row to hear a great symphony orchestra, so you can really hear the dynamics I'm talking about.

fair

Mathematically, Fourier Analysis is a theory based on integral transforms with harmonic kernels.

No, Fourier Analysis is not a theory. It's a theorem; it can be shown to be perfect with math. Unless you can show a fault in the actual math, all of your hand waving and word salad is for naught.

Again, I'm an analog guy. But if we seek better digital, it can't be done without understanding how digital works. I'm all for hi-res and everything it takes to get the best out of it. But let's not pretend that somehow the underlying premise of digital audio is somehow broken. It isn't. Its problems are elsewhere.

G’day MikeyDee

My wife and I had the opportunity to attend a small Mozart chamber orchestra in the castle in Salzburg Austria a few years ago. That’s where sitting close within 6-15ft the small audience had a wonderfully intimate experience of dynamic range. No Hi-Fi experience can reproduce that intimate experience in a double thick stone castle, huge rectangular room with pillars etc.

Perhaps @fair , can enlighten with at least 2 or 3 of these research papers he claims are hard to find? A new paradigm with 3 decades of research that legitimately calls into question all current signal processing and hearing knowledge should have many available sources to reference.

Still I wait for this. A meta analysis of purely digital sources, some too old to be relevant due to hardware limitations and others with experimental flaws, does not support your hypothesis let alone suggest there is any new paradigm. I do appreciate the repartee as it demonstrates the vinyl argument.

This is just like the tube discussion.  Even though there are significant, identifiable differences between typical tube amplifiers and SS amplifiers with good design practices, differences that are highly audible, every discussion devolves into a debate between those who point out those differences and those who believe in some unseen, unmeasurable property that "must" exist.