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Who is the player with the biggest and fastest progression?

I mean real progression. It's too hard to believe someone rising 500 points in one month.
Sergey Karjakin became a GM aged twelve - and he did it the 'proper' way by getting a 2500+ FIDE rating and GM norms in tournaments. (As opposed to being awarded the GM title for winning a junior tournament.)

Magnus Carlsen became a GM aged 13 (again by rating and norms) and the youngest ever World Champion at 22.

Those two are certainly candidates. Although note that Carlsen took 8 years to go from total beginner to grandmaster, but nine more to get from GM to World Champion. That last 300-ish rating points seems to be the hardest.
"Magnus Carlsen became a GM aged 13 (again by rating and norms) and the youngest ever World Champion at 22."

Kasparov is still the youngest ever world champion. They were both 22, but Carlsen was a few months older when he won his championship.

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