Well, in terms of the actual ability demonstrated, I suck at physic intuition, I am just passionate with it on par with chem (and tried to learn but improvement is v.e.r.y.s.l.o.w.)

Acuriousmind also found I am bad at computation

And to top that my mind is incoherent, (Evidence: Acuriousmind often said "What?")

One of my GR friend describe me as follows: I have the mindset of a theorist but the tools of an engineer

My mindset is actually closer to a mathematician, due to the reluctance of taking shortcuts nor approximations

I mean, something along the lines of

how you can demonstrate the spin of an object by checking how its phase changes when you poncaire transform the particle (e.g. if you get different values like 1 ,1/2, 0 etc.) perhaps using some kind of inferometer set up (because we can only ever see phase difference) (assuming I have correctly understood you explanation on spin is a label of the representation of the group in question)

So I am guessing, if we can find something that play the role for that arbitrary divergenceless vector field term that you can add to any lagrangian but can be demonst…

https://en.wikipedia.org/wiki/Lagrangian_(field_theory)

ok... so it seems the lagrangian is the sum of all interactions in a system in the most general case

For classical mechanics, the interactions are the kinetic energy and the potential energy (will revise later why it is minus and not plus)

For relativistic mechanics, the interactions are the rest energy (corrected by time dilation along some path (represented by $\frac{d\tau}{dt}$ term(?)) and whatever itneractions due to its trajectory in spacetime

Actually, while I have rely less on pictures to visualise things now, (because of some recent inspiration) I put "visualisation" on a more abstract level:

If you have some set up in real life that is arranged in a way to simulate (as many as possible) the mathematical properties of some mathematical object, then you can "visualise" it and make laymen understand it with its full rigor presented

Side effect of this project is an increase in the number of questions and thinking on how to experimentally demonstrate something

(cont.)
Mathematically speaking, I suspect "can be drawn on paper" can be defined as follows:

There exists a map (or something more general) $f$ and some object $M$ (in order to avoid the trap of vagueness, let's stick with mathematical objects) such that $f: M \rightarrow S\subset \mathbb{R}^2$ such that the properties of $M$ is preserved by the map $f$

There's still a vague element: Interpretations. Suppose I draw a line on a piece of paper, it can mean many things, and different people will have different interpretations of it