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There are 8 Gym skills you can improve on. Leveling these up will increase flat stats for your pokemons, and every 10 levels you unlock a 'core'. They will add percentual stats according to the level of inlaid core. You can gain additional stats from 'refine'.

Gym Stats

Gym 1 - Santalune Gym: HP

Gym 2 - Cyllage Gym: P-Atk

Gym 3 - Shalour Gym: Sp-Atk

Gym 4 - Courmarine Gym: P-Def

Gym 5 - Lumiose: Sp-Def

Gym 6 - Laverre: SPD

Gym 7 - Anistar: Crit

Gym 8 - Snowbelle: Res

Core Slots

Level Santalune Cyllage Shalour Courmarine Lumiose Laverre Anistar Snowbelle
10 HP P-Atk Sp-Atk P-Def Sp-Def SPD SPD HP
20 P-Def SPD SPD HP HP P-Atk P-Atk P-Def
30 Sp-Def P-Atk Sp-Atk P-Def Sp-Def Sp-Atk Sp-Atk Sp-Def
40 HP P-Def Sp-Def Sp-Def P-Def SPD HP SPD
50 SPD Sp-Def P-Def P-Def SP-Def HP P-Def P-Atk
60 HP Sp-Atk HP HP SPD Sp-Atk
70 P-Def
80
90
100

Cores can range from white (lowest) - green - blue - purple - to orange (highest).

Refine

You can 'refine' the stats in each Gym by rolling the stats of different Attributes. This will also consume Gym Score Points. As you reach the higher end of each attribute range (90?) you will unlock higher ranges. This will be shown by the attribute's color as well, changing from blue (lowest) - purple - to orange (highest). The according slot to the hightened attributed will take on this color as well.