[Rita's notes would be full of complicated theories, formulas, and magic symbols, so I'm not going to try to write it out ICly. Instead, here's a summary of the information she has on magic systems:]
Systems of Magic:
Alchemy
Type: Matter-based (?)
Origin: unknown
Requirements: unknown
Information Source: Leon
Notes: May not count as 'magic,' but could be relevant.
Card Magic
Type: Mana-based (spirits) (?)
Origin: ???
Requirements: Cards (?)
Information Source: Sakura
Blastia Magic
Type: Aer-based
Origin: Terca Lumireis
Requirements: Blastia.
Information Source: Rita Mordio
Notes: Non-sustainable. The effect of blastia use on the aer is a danger to the world.
Exsphere Magic
Type: Mana-based (also life-based?)
Origin: Tethe'alla
Requirements: Exspheres
Information Source: Raine, Kratos
Notes: Non-sustainable. Exspheres are made with human lives.
Fonon Magic
Type: Fonon-based
Origin: Auldrant
Requirements: Fonons of the necessary element
Information Source: Guy
Notes: Supposedly used to predict the future as well. Non-sustainable; reason unknown.
Heraldry
Type: Mana-based
Origin: Expel
Requirements: The Blessing of Mana (?); heraldic crests
Information Source: Leon, Claude
Notes: Was able to reproduce through the use of crests written on paper. May be able to create talismans for efficient spellcasting.
Spirit Magic
Type: Mana-based (spirits)
Origin: Terca Lumireis
Requirements: Spirits, mana.
Information Source: Rita Mordio
Stone Magic (?)
Type: unknown
Origin: unknown
Requirements: A particular type of stone that contains energy.
Information Source: Taura
Notes: The stone appears similar to blastia, but seems to have a formula applied directly onto it, giving it only one designated use. The benefit is that even those who lack magic knowledge can use it.
Witchcraft
Type: unknown
Origin: Earth
Requirements: Magic circles, candles (?), incense (??)
Information Source: Patient Library
Notes: Is this even a real magic system?
Magical/Unique Items:
Magic-Suppressing Device
Notes: Existence unconfirmed. Supposedly used to suppress magical abilities in the institute. No other information.
Portal Ring:
Notes: Has a silver band with a red gem. When the gem is broken, it teleports the wearer to another location. Can be reset with a drop of blood, which will allow the ring to later teleport to the location it was reset in.