ESP in this RP is close to magic in general meaning.
ESP emits/possess their own [Genetic Pure Energy] (Lets call it [GPE] for now) which count as different form of usual [Pure Energy]. To put it in perspective, [Pure Energy] is neutral-type attack, meanwhile GPE is a specified-type, like elemental attack you can say. This what makes Esper can only possess one power, like either telekinesis or pyrokinesis.
Esper use their [GPE] as their power-source and the convert it as form of their power. I had mentioned why Esper can't be a Mage or use a Madou before, that also because [GPE] are not a "blank" energy and cannot be converted as any other form than by the possessor (Esper's brain) itself.
In case of Espers activating their power, their body is their power-source and their brain is the converter. Esper's brain will convert/change the [GPE] inside their body and change it to the "manifested" form which is their power.
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Magic meanwhile require a Madou. Madou itself can be in various form and shapes. Either its a bracelet, headband, or even earring, your choice. Madou is mechanism, not the machine itself. Madou is like a magic wand. You hold it, chant a spell, and it will cast magic. Madou works similarly. For example, let's say you want to cast Magic A and B in order with a gun-type Madou:
Magic A: Hold the gun, chant "By my will, may ye be enlighten the holy ring!", pull the trigger, and it will shoot light magic.
Magic B: Hold the gun, chant "By my will, may ye be perished by the wrath!", pull the trigger, and it will shoot fire magic.
As for what the Madou is and what Magic, is up to you. My character use bracelet Madou and its triggered when he swings his hand in certain motion. You must have different trigger for each magic, be it the chant or how you pull the trigger (slow, fast, rapidly, etc).
Magic operated by Madou is based on the user's brainwave or any other medium that Madou use to detect the user's will (like pressure, force, breathing, movement, etc). What Madou do is to detect "what magic this person want to use and can use" and will execute it accordingly. If a person attempting to cast a magic that is too hard for him it will either go rampage or won't even executed at all.