Why capital ships received 5.25-inch guns

Why did the Royal Navy develop the 5.25-inch gun for capital ships while retaining smaller dual-purpose weapons elsewhere?

Cruiser and capital-ship status pushed designers toward a gun larger than the destroyer's weapon, even though a five-inch system had originally been preferred.

The 5.25-inch choice does not show that every sub-five-inch weapon lacked surface power. The failed five-inch programme had been the more natural technical path. Once destroyers were expected to carry weapons around five inches or below, however, a cruiser or capital-ship secondary was expected to look and feel more powerful. The resulting 5.25-inch system carried a classification and prestige logic alongside its intended dual-purpose role.