A lot of the game is running right and avoiding obstacles, so here is a quick guide on doing that more quickly:

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GENERAL MOVEMENT

A quick speed rating: Walking on up/down slopes VERY SLOW Walking on flat ground FAST Moving while in the air FASTEST

So just make sure to spend as little time as possible on the ground and especially avoid slopes

When jumping if you activated you double jump and regrab the jump key you will jump instantly when you hit the ground. This is obviously the optimaly way to do it but:

If you try to immediately jump off of an uphill slope while moving to the right there is a (seemingly random) chance that your horizontal velocity gets set to 0, loosing some time.

So avoid landing on slopes even if it's just for a single tick.

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WATER MOVEMENT

To move vertically in water you need to mash the up key. if you need to move vertically really fast (to get an IL record for example) the best way on PC is to mash the Arrow-Up and W at the same time, since both count as the same action.

When trying to move above the water like in 1-1 or 4-9 there is a small window where you can jump off the water with almost no loss of momentum. but if the distance is longer you are better of using your double jump to spend as little time in the water as possible