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You have shown that IF the imposed charge and the inner surface charge are
to effect the required E=0 without the help of the outer surface charge,
then there is a unique solution, etc. But this begs the very question:
The raw physics only requires that the E=0 condition (in conducting
material) be wrought by the superposition of the effects of ALL charges...
it needs to be proven that the combination of the imposed charge and the
inner surface charge must ALONE effect the required E=0 result (without
taking into account the effects of the outer surface charge) .