Why was sovereignty over Ducal Prussia important for the Hohenzollern dynasty?

Sovereignty over Ducal Prussia was a critical milestone for the Hohenzollern dynasty for several key reasons:

• Independence from Suzerainty: By securing full sovereignty through the Treaty of Wehlau (1657) and the Treaty of Oliva (1660), the Hohenzollerns ended Polish suzerainty over the duchy. This was uniquely important because, unlike their other territories that remained part of the Holy Roman Empire under the theoretical authority of the emperor, Ducal Prussia existed outside the empire's jurisdiction.123
• Foundation for Royal Status: Because Ducal Prussia lay outside the Holy Roman Empire, it provided the necessary legal basis for the Hohenzollern electors to elevate their status. In 1701, Frederick I was able to crown himself "king in Prussia" at Königsberg, a title that would have been impossible to claim for his German lands within the imperial structure.14
• Rise to Great Power: The union of the duchy with Brandenburg was fundamental to the dynasty's rise as a great power in Europe, allowing for the centralization of administration and the development of a powerful, independent military machine.13