A Valkyrie core is a white sphere whose size ranges from a golf to a tennis ball. When synched it is absorbed into the person’s body.
A Valkyrie Core essentially comes with an ever expanding hammerspace that can store any nonliving objects into it as long as it has space. Any object can be repaired or improved while in storage space and any technology can be integrated allowing it to be used when either stored or expressed externally. Cores tend to develop personalities overtime but other than Durga’s occasional shows of sapience only Kandakara has demonstrated an actual human like personality during its failed attempts to communicate with Koujirou.
Impeller fields are capable of all sorts of bullshit. They essentially act as exotic force fields at their most basic level. Your Impeller also allows you for the manipulation of object within its radius including the human body. It also allows you to perform various forms of spatial manipulation, such as teleportation, at the cost of current Impeller strength. Your Impeller constantly recharges, though the amount it recharges and your Impeller’s total strength is determined by a combination of the total number of hours synched to your Valkyrie Core, compatibility rating, total equipment integrated, and the complexity of that equipment.