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Horses move up to 2L when their score reaches 10. Those created with a score of 9 would be in Level 1 World. An exceptional show horse might start out with a score that puts it into 2L.

Like everything else in the game, each horse is an individual and so starts out with it's own initial score.

Here's a chart I made of level changing scores when I first started playing back in 2011. Up to Level 6, it will be accurate. Ammit made some adjustments there, which I didn't record quite a few years ago.

level grade score (top score for each level)
• 1 L 2
• 1 R 5
• 1 N 8
• 1 W 10

• 2 L 20
• 2 R 30
• 2 N 40
• 2 W 50

• 3 L 60
• 3 R 70
• 3 N 80
• 3 W 90

• 4 L 100
• 4 R 110
• 4 N 120
• 4 W 130

• 5 L 140
• 5 R 150
• 5 N 160
• 5 W 170

• 6 L ?
• 6 R ?
• 6 N ?
• 6 W ?

• 7 L ?
• 7 R ?
• 7 N 260
• 7 W 270

• 8 L 280
• 8 R 290
• 8 N ?
• 8W ?

I spread out the first two levels so that they make spaced columns, but you should be able to figure it out for the rest.