Why adding medium guns to a carrier is costly

Should Queen Elizabeth-class carriers receive 3-inch or 5-inch defensive guns?

A 3-inch gun is the more plausible option, but its support burden makes additional air-defense escorts a better investment.

A 3-inch weapon would make more sense than a 5-inch gun for local defense, but either requires a new training and support chain if the navy uses it nowhere else. The carrier must surrender internal volume to ammunition, spares, maintainers, and the mounting itself. More air-defense frigates would strengthen the escort screen without removing other carrier functions to add a gun late.