Alice Carpenter was married twice. Her first husband was Edward Southworth. Their children are not Mayflower descendants. Therefore this line probably would not be approved as Mayflower. Her second husband was William Bradford. They had 3 children who do qualify as Mayflower. (Alice is discussed in a Wikipedia article about the Carpenters.)
The Mayflower society does want serious documentation for each birth, marriage, death and connection between the generations. For example, your birth certificate would have your parents' names. This connects you to the next older generation and documents your birth. You probably have to document as far back until the people are listed in the 5 generation. books. It gets expensive and time consuming to join.