Fifth-Order Derivative-Sum Identities for Two Balanced Gamma Quotients: Bilateral Summation and Level-2 WZ Reduction

We prove two derivative-sum conjectures of Zhi-Wei Sun for balanced gamma quotients. For the first family, a generator of Guillera and Campbell is completed to a bilateral very-well-poised series. Dougall's bilateral summation evaluates the completed part, while two transformations of Chu and Zhang reduce the negative tail to a value and a first derivative of a rapidly convergent well-poised series. Dougall's nonterminating summation and an Euler sum then determine the complete fifth-order translated jet. For the second family, Au's diagonal WZ identity converts the translated sum to a hypergeometric deformation. Its fourth and fifth coefficients are expressed as level-2 colored multiple zeta values and reduced by exact double-shuffle and distribution certificates. These arguments prove all four claimed derivative identities for each gamma-quotient family; the second proof also gives the companion zeroth- and first-derivative values. All termwise differentiations and bilateral rearrangements are justified by normal or absolute convergence, and the finite certificates are supplied in independently checkable rational form.