As someone who has 2000 points in puzzles my Elo is still an embarrassing 500 and going down. The reason being probably that there is a difference between the fluidity of an actual game and that of a puzzle where you know there will be a good move, when in games as a noob, you don’t really know if you are in a winning position, and if you are, you also don’t know how to turn it into a checkmate in 3. I am speaking for myself of course.
Estimated Elo of a player vs puzzle rating?

I don't know how puzzle ratings are calculated but they seem completely out of sync with everything else.
In puzzle battles, you get given puzzles with a rating well over your puzzle battle rating.
That kind of shows even chess.com knows their puzzle ratings are a bit off.

puzzle battles are just head-to-head 3-minute rushes: they start with about a 100-rated puzzle & then climb roughly 50 points in difficulty with every new puzzle (whether you got the last one correct or not), with a puzzle rating maximum of 4000 that should be reached by an estimated score of 80 on survival mode (5-min if you're ray robson). 😆
your puzzle BATTLE rating is an indicator of how well you perform with solving in this format relative to other players' abilities, while the PUZZLE rating is a reflection of how difficult the puzzle is to solve & believe me that they get HARD in the 3-4 k range... a decent score of 40 in puzzle battle would only require that you solve puzzles with barely HALF the max difficulty (2140 according to my current survival).
think of YOUR puzzle ratings as being relative to THE puzzle ratings: anybody with a legitimate score of 4000 or above is about as good as you can get. 😁

I'm 475 rapid and 1409 puzzles (rapid fluctuates but not often and puzzles fluctuates pretty often).
Highest rapid ever: somewhere in the high 500s
Highest puzzles ever: Low 1600s

i wanna ask: i am very low rated at puzzle battle, i wanna play against my friend who is higher rated, he says that he can't play, is it because we don't have the same elo in puzzles?

i wanna ask: i am very low rated at puzzle battle, i wanna play against my friend who is higher rated, he says that he can't play, is it because we don't have the same elo in puzzles?
No

you can challenge a friend at puzzle battle, but i believe the algorithm will attempt to match you with a similarly-rated player in randoms.

puzzle battles are just head-to-head 3-minute rushes: they start with about a 100-rated puzzle & then climb roughly 50 points in difficulty with every new puzzle (whether you got the last one correct or not), with a puzzle rating maximum of 4000 that should be reached by an estimated score of 80 on survival mode (5-min if you're ray robson). 😆
your puzzle BATTLE rating is an indicator of how well you perform with solving in this format relative to other players' abilities, while the PUZZLE rating is a reflection of how difficult the puzzle is to solve & believe me that they get HARD in the 3-4 k range... a decent score of 40 in puzzle battle would only require that you solve puzzles with barely HALF the max difficulty (2140 according to my current survival).
think of YOUR puzzle ratings as being relative to THE puzzle ratings: anybody with a legitimate score of 4000 or above is about as good as you can get. 😁
Well it might work for high rating, but at lower ratings like me it seems kind of skewed.
I'm at 250-300 elo in puzzle battle, according to that system I should expect to be scoring 4-5 most games (if that), but in actuality I'm scoring 12-15, and so do other people of my rating.
I'm 1400ish in "puzzles" BTW.
Puzzle battle ratings are actually based on people going head-to-head with each other so seem more objective than the "puzzle" rating from who knows where; so it seems to me the difficulty of low-elo puzzles is highly overstated.

fear gives wings to the feet. ~aesop
most of the easier puzzles (say under 1000) are based on simple pattern recognition, demand very little calculation & thus are solved near-instantly. 🙂

fear gives wings to the feet. ~aesop
most of the easier puzzles (say under 1000) are based on simple pattern recognition, demand very little calculation & thus are solved near-instantly. 🙂
Indeed, many of the easier puzzles are pattern recognition based, but that requires being both good at pattern recognition and knowing patterns to look for.
If feel like doing some of chess.com's checkmating lessons has helped me learn what patterns to look for.
But in connection with elo raring, I understand that normally elo rating can be used to cslcuate the chances of a player beating another player?
Puzzle battles seem an objective way of working that out, but your "puzzle rating" obviously can't be worked out the same way.
It seems to me it would make more sense if the average puzzle rating is close to their puzzle battle rating; even though one is more time pressured than the other.
Also, at minimum the puzzles you can reach in your puzzle battles should be a lower or comparable rating to your puzzle battle elo.
I'd say if I havea typical puzzle battle-to regular puzzles ratio, really ~1400 puzzles should be rated as ~300.
If my regular puzzles are particularly better than my puzzle battles, I still think at minimum ~700 puzzles should be rated as ~300, as you can reach this puzzle battle elo by routinely beating puzzles of this level within the puzzle battles.
Obviously this may require also changing puzzle battles themselves because it might make them too difficult for low-rated puzzle solvers, perhaps they could increase in puzzle rating more slowly on the first few puzzles?
Looking at higher-rated games maybe it might make sense to continue increasing the rate as the difficulty goes up, as the top-rated players don't seem to have too much difficulty with their highest-rated puzzles in their puzzle battles.
2000 blitz 3000 in puzzles
3000 blitz 4000 in puzzles