In order to tolerate servers' Byzantine failures, a distributed storage service of self-verifying data (e.g., certificates) needs to make three security properties be Byzantine fault tolerant (BFT): data consistency, data availability, and confidentiality of the (signing service's) private key. Building such systems demands the integration of Byzantine quorum systems (BQS), which only make data consistency and availability be BFT, and threshold signature schemes (TSS), which only make confidentiality of the private key be BFT. Two families of correct or valid TSS-BQS systems (of which the server protocols carry all the design options) have been proposed in the literature. Motivated by the failures in finding a third family of valid server protocols, we study the reverse problem and formally prove that it is impossible to find any third family of valid TSS-BQS systems. To obtain this proof, we develop a validity theory on server protocols of TSS-BQS systems. It is shown that the only two families of valid server protocols, "predicted" (or deduced) by the validity theory, precisely match the existing protocols.