Ahem.

Everyone and their dog has their own hypothesis of how magic works, and not
all of them are compatable (indeed, almost none are completely compatable).
The problem with rigorous definition of what is by nature a plastic thing,
is that the definitions almost always lead to contradictions.

As a physics guy, I can usually see such contradictions in many magic
hypothesis, such that it's clear that the inventor did not spend much
effort exploring the consequences of his definition. The problem with the
"virtual energy field" hypothesis of summoning is that it would lead to the
conclusion that summoning spells should be absolutely safe; any attempt of
the summoned creature to leave the energy field is thwarted by the fact
that the rules of manaphysics does not allow such a thing to happen. This
means that all stories of chaos caused by demons breaking out of their
summoning circles are bullshit. All such summonings are essentially
temporary bridges between worlds surrounded by a ward that prevents escape.

The notion of "sacrificing part of one's soul" sounds a bit unnerving to
me. What if you can't get to the apple to recover your little bit of soul?
What happens when this occurs once too often? Also, what about creatures
that supposedly have no soul, yet can work magic anyway, such as vampires?
Do these guys actually have a soul, can't work magic, or don't exist in the
first place? A hypothesis of magic that allows soulless creatures to work
magic cannot use soul-power to fuel it.

Anyway, rigorously defined magic systems lead to contradictions unless they
are very carefully thought out. Also, such systems would inevidably lead to
conclusions that disallow some things from existing or happening. The way I
see it, if you're going to rigorously define magic, you'd better be
prepared to accept the consequences of your definition. If not, then you
might as well not have wasted the effort.

My $0.02.

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## # ##   ###    #####   ## # ##  Wielder of ANVIL and SPAM!
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"What kind of stupid alien escapes in a car?!"
                                     -- Kizuki Mitsuha, Silbuster ch.1