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I'm still learning but I believe it comes down to training them every week. That ups their level and allows them to be entered into higher shows.

Many foundations gain about 3 points each week, depending on their Performance Testing score. After horses reach level 2 Local, each level/grade contains about 10 points. Here is a chart that indicates the scores when horses generally level up. Some horses level up a point higher than the chart indicates. Once horses get to level 6, the grades contain more points. I haven't ever tried to figure out the exact scores there. With the showing bonus system, it has become less important to pay attention to. This chart was made for HaJ 1. Since the game years are only half as long here, the horses train only half as long and I don't think any have yet progressed beyond level 7 and the few who reach that level are the ones with PT scores of 13 and higher. When I sort all of the horses in this game by PT score, there are only two who are that high.

level grade score
• 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