What is a casket called when the rim, crown, and pies are formed as one unit?

The term used for a casket where the rim, crown, and pies are formed as one unit is known as a "perfection full couch." This type of casket is characterized by its design, where the lid encompasses the entire surface rather than being divided into sections.

In this design, the seamless construction allows for a more aesthetically pleasing appearance, giving a sense of unity and completeness to the casket. It's often employed for aesthetic reasons, as well as for practical considerations during the process of viewing and memorialization.

The concept of "full couch" indicates that the casket lid can be opened fully, which is commonly preferred for display purposes during funerals. While other types of caskets exist, such as the "half couch," which has a lid that is divided into two sections, it is the "perfection full couch" that specifically refers to the unified and seamless construction mentioned in the question.