Derivative Sums of Balanced Gamma Quotients and Multiple Zeta Values: Five Conjectures of Zhi-Wei Sun

Abstract

We introduce a uniform reduction for derivative sums of balanced gamma quotients. For exponent data $(a_i,e_i)$ satisfying $\sum_i e_i a_i=0$, the translation-dependent gamma prefactor is governed by the characteristic power sums $\chi_m=\sum_i e_i a_i^m$ through $\log C(u)=\sum_{m\ge2}(-1)^m\chi_m\zeta(m)u^m/m$. This separates the universal gamma contribution from a hypergeometric coefficient-extraction problem and organizes three exponential families. For $\chi_m=2-2^m$, diagonal and symmetric specializations of a four-parameter Wilf--Zeilberger identity prove Conjectures 4.2 and 4.3 of Zhi-Wei Sun. For $\chi_m=3-3^m$, exact span certificates in the coefficient spaces of Au's Example IV prove corrected forms of Conjectures 4.4 and 4.5. For $\chi_m=4^m-10\cdot2^m+16$, a half-integer specialization of Au's first $1/\pi^4$ construction, combined with the diagonal transformation in his Example VI, proves Conjecture 4.6 through weight eleven. The transformed sides reduce to ordinary multiple zeta values, and every computer-assisted acceptance test is exact: explicit rational WZ certificates and separately implemented MZV certificate checkers use no numerical recognition, PSLQ, or conjectural MZV dimensions. We also identify four errors in the printed statements of Conjectures 4.2--4.5.