Final answer:
The immediate next step in the development of a double-negative thymocyte that has completed a β-chain gene rearrangement is the expression of RAG-1 and RAG-2, causing the rearrangement of γ- and δ-chain genes.
Step-by-step explanation:
The immediate next step in the development of a double-negative thymocyte that has just completed a productive β-chain gene rearrangement is c. Expression levels of RAG-1 and RAG-2 are elevated. RAG-1 and RAG-2 are enzymes involved in V(D)J recombination, which is the process of rearranging gene segments to generate a diverse repertoire of T cell receptors (TCRs). After β-chain rearrangement, the expression levels of RAG-1 and RAG-2 are increased to initiate the rearrangement of γ- and δ-chain genes, which will then result in the generation of a functional γ:δ T cell receptor.