Stat Genes
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Stat Genes

Each horse has a set of 6 stat gene pairs. One pair corresponds to each of the horse's stats and bears the same name. For example, the Breaking stat gene correlates to the Breaking stat. Simple, right? These genes can be viewed if you have the Stablewide Hidden Genes purchased, which can be found in the Pro Shop in the Stable Upgrades tab.
Relation to Potential
Stat genes are very important! Stat genes can be very useful for a couple of purposes. First, you can look at stat genes on a young horse and estimate what you would expect their maximum stats and potential to be. A foals parents' producer quality can positively or negatively impact their stats.
In normal situations, where producer quality and/or bloodline have a neutral or slightly positive impact, you should expect stats to max roughly as follows:

Estimated Maxed Stats
Poor 1 - 2
Fair 2 - 3
Good 4 - 5
Excellent 6 - 7
Outstanding 7 - 9

A horse with entirely Excellent/Outstanding should max at 40+ potential under "normal" (neutral) circumstances. A horse with mostly Good genes will usually max somewhere around 30 potential. A horse with mostly poor/fair will typically max under 20 potential.
Importance in Breeding
Second, you can compare a horse's actual stats, once maxed, to their stat genes to get an idea of their parent's producing quality. If a horse has much lower stats than you'd expect based on genes, one or both of their parents are likely bad producers. If a horse has much higher stats than you'd expect based on genes, one or both of their parents are likely good producers.

These are arguably the most important genes your horse has because they determine the largest part of racing talent. However, not all horses will have stats that actually match up with what you'd expect from their genes. Though these genes provide a base, the final outcomes are impacted by other factors such as producer quality and bloodline.
Stat Increase Upon Sale
Horses have a chance to improve their stats (overall potential, not the genes themselves) when being sold or purchased by another player. Although the stats increase, it does not impact the stat genes. This means a horse can improve upon its birth potential alongside gelding, oracle and buddyworks. Retired horses no longer gain stat improvements upon being sold or purchased.
Stat Genes and Steeplechase/Driving
Stat genes have absolutely no impact on the stats a horse earns when they are transitioned to Steeplechase (Thoroughbreds & Goldbloods) or Driving (Standardbreds). Their stats are zeroed out and rerolled solely based on a gene known as Jump or Driving. High or low stats on a SC/DT horse are also NOT indicative of flat producer quality.