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Time-space cubes are a player growth feature that increases your overall combat power. Boosting them is the key to being able to challenge higher floors and levels of Cross Field.
They can be accessed from the Main Menu > Growth > Time-Space Cube, once you have unlocked the Cross Field dungeon.
Each time-space cube has its own base stats which feed into your combat power. Each face of the cube can equip a time-space fragment, which have their own base stats that also feed into your combat power. The cube has 6 faces, 2 of which are unlocked when you acquire it. The Aquarius cube unlocks additional faces at levels 30, 60, 115, and 165, while the Guardian Ruins cube unlocks additional faces at levels 50, 100, 150, and 200.
On the status screen, you can see the base stats of your cube (the blue values under the cube represent the important time-space pierce/resist values), as well as all of the time-space fragments you have in your inventory. You can also equip fragments on any unlocked faces.
Increasing the base level of the cube requires a certain amount of frozen time-space energy, dropped by monsters from its particular Cross Field dungeon. The amount of energy needed per level increases as the cube's level increases.
Leveling up the cube will boost all of its base stats by a small amount per level.
Fragments are disk-shaped objects which are rare drops from Cross Field monsters. They range in base rank from I to XII. Monsters within a floor all drop fragments of the same ranks; in higher levels, monsters can also drop boxes which roll lower rank fragments. Boss monsters can drop rare rank fragments.
Each rank has a different base pierce/resist value, which can be boosted by awakening the fragment. However, this will only get you so far. The most efficient grind is to boost low level fragments just enough so that you can challenge the highest floors in a level, then farm those levels to maximize high rank fragment drops. Fighting bosses will also give you a good chance at rare fragments, which have better base stats than the equivalent normal fragment. You will need to equip and awaken these fragments significantly if you want to get enough pierce/resist value to enter the next level.
Fragments have up to 3 stat boosts, which are rolled randomly between stat type (such as HP, attack, or evasion) and stat value. The value range for the boost is based on the rank of that fragment; higher fragments have both a higher minimum and maximum value for stat boosts. If you roll a max value, it will be noted in the fragment's info screen.
Fragments start with 1 boost, and gain a second boost at awakening level 3 and a third boost at awakening level 7, making these key levels to awaken to.
Awakening a fragment will increase its base stats as well as its time-space pierce/resist value, up to level 10. Each level requires an increasing number of the same rank fragment as fodder. After the first level, each attempt to awaken has a chance to fail, which increases as the levels get higher.
For the Aquarius cube, after level 3, there is also a chance to break the fragment. You will need to use time-space scrolls to repair the fragment so that you can continue to awaken it (similar to using anvils on equipment.) The type of scroll depends on the rank of the fragment. Scrolls can be purchased in the Cross Fields Aquarius token shop.
The Guardian Ruins cube fragments do not have a chance to break when awakening them, but also require more copies per level.
You can reroll a fragment's stat boosts by feeding it a fodder fragment of the same rank. Go to the fragment stat change screen, select a fragment, and then which boost slot to reroll. The game will roll a new boost, and you will then have the option to accept the new one, or keep the original one. The fodder fragment is consumed no matter what you choose.
If you have extra fragments of lower ranks you don't want to keep, you can disassemble them from the main status screen. Each one will give you a small amount of frozen time-space energy.