The office, position, or authority of a comptroller; the period during which someone serves as comptroller.
From 'comptroller' plus '-ship' (state, position, or office), following the standard English pattern for creating nouns denoting positions of authority or status.
The '-ship' suffix is incredibly productive in English for positions of power and prestige—comptrollership sounds grander than 'comptroller job,' which is why official documents always use it.
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