Why chess ratings across websites and Fide,USCF cant be mathematically compared.

 intrigued by the number of people wanting conversions between rating. i tried to find a mathematical base point to even compare the both ratings. see, the glicko system, which lichess and chess.com uses has a factor called R.D (relative difference) and Time decay.While elo doesnt have these factors. Rating difference is the range of rating change keeping in mind the uncertainity of result/ Which means its a list of total possibilities of rating change if you won, lost or drew, with subject to the opponents, rating and R.D, which is in turn affected by his time decay. Time decay refers to the increase of Rating difference if you have not played for a long time, or decrease of R.D when u play constantly. R.D is a big factor with Time decay as a sub factor, making the formula more complex than complex,
Thus we conclude we cant compare player skill between lichess and Fide/Uscf.

We have to keep in mind that time decay, which affects the R.D of two players comparing their ratings on different sites, is different and has infinite permutations and combinations, making comparisions kind of pointless.

When we talk about chess.com, we are  taking into account that chess.com follows the glicko system too, BUT, the rating starts from 1200. Now in general math, the difference in starting points is 300, BUT, in such a complex math formula with sets, rating uncertainities, and time decay affecting the R.D , the difference of 300 points does not mean 300 at all! this being a complex formula, if the difference in 2 starting ratings would be 200 (supposedly), the R.D would change in an indefinite manner. Thus even if we try to compare the lichess ratings to chess.com ratings, the fact that chess.com ratings start 300 points before lichess rating and a factor of time decay affecting the R.D in turn. makes them truly, apples and oranges.
The whole mess of rating systems being non-comparable is because of the time decay which affects the R.D

Now if we talk about surveys, they are just information which change from time to time with change in rating trends, and cannot give a satisfying idea. Rating trends change in a day to day manner, and its reall difficult to keep track of conversions for the whole community, because each person plays in different gaps of time.

I will keep editing this as i get more information

I would like to make it clear that the interests of this post are purely mathematical

In regards to your point on starting ratings, for chess com they can be between 400 and 2000

https://support.chess.com/article/671-how-do-i-create-an-account

Well @Martin_Stahl, thank you for bringing that point up. This will in-turn increase the deviation of rating comparisions mathematically. That however does not affect the mathematical point I have proved here 

That information is actually EXTREMELY important. It stands to reason, that a self-chosen initial rating rarely will be accurate.

Lots of humble experts (2000) start accounts as intermediates (1200), and they will kick down lots of innocent REAL intermediates to the 900's.

"Lots of humble experts (2000) start accounts as intermediates (1200), and they will kick down lots of innocent REAL intermediates to the 900's."

Ahhh seal clubbing brings back such fond memories for me when I first started playing on here.

Chess ratings across websites and FIDE, USCF can be mathematically compared.
It is enough to take the established ratings of players who are rated in two or more different pools and with the different rating systems and then fit a conversion formula.

The reason why the ratings differ is not so much the system elo or glicko or the start rating 1200 or 1500, or the RD, or the time decay, but rather the different pools.

Say there are two schools and the pupils of each schools play tournaments among themselves and they calculate ratings. They both use the same rating system with the same formulas and the same start rating etc. Once the ratings are established, they start to play between the two schools. It will soon become clear, that a 1700 rated player from school A is much stronger than a 1700 rated player from school B, so all those of school A will start to gain rating and all those from school B will start to lose rating the more they play between schools.