AI 中文总结
该研究引入平衡伽玛商导数和的统一归约方法，证明孙智伟四个猜想的修正形式，指出其印刷版错误，该方法适用于猜想4.6，所有计算机辅助步骤精确。
AI 中文摘要
我们为平衡伽玛商的导数和引入了一种统一归约方法。对于满足∑ᵢ eᵢaᵢ=0的指数数据(aᵢ,eᵢ)，整个依赖平移的伽玛前因子由特征幂和χₘ=∑ᵢ eᵢaᵢᵐ通过恒等式log C(u)=∑ₘ≥2 (-1)ᵐχₘζ(m)uᵐ/m决定。这将解析伽玛贡献与超几何系数提取问题分离，并在单一框架下解释孙智伟四个猜想中出现的逆4族和逆27族。对于分支χₘ=2-2ᵐ，四参数Wilf–Zeilberger恒等式的对角和对称特化证明了猜想4.2和4.3；对于分支χₘ=3-3ᵐ，Au的示例IV系数空间中的精确跨度证书证明了猜想4.4和4.5的修正形式。变换后的边约化为权重至多9的普通多重zeta值。所有计算机辅助步骤均为精确的：两个分支使用独立的有理证书工具链，未使用数值识别或PSLQ。我们还指出了印刷版猜想中的四个错误，并表明相同的平衡商归约适用于猜想4.6，分离出缺失的WZ输入。
英文摘要
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 $χ_m=\sum_i e_i a_i^m$ through $\log C(u)=\sum_{m\ge2}(-1)^mχ_mζ(m)u^m/m$. This separates the universal gamma contribution from a hypergeometric coefficient-extraction problem and organizes three exponential families. For $χ_m=2-2^m$, diagonal and symmetric specializations of a four-parameter Wilf--Zeilberger identity prove Conjectures 4.2 and 4.3 of Zhi-Wei Sun. For $χ_m=3-3^m$, exact span certificates in the coefficient spaces of Au's Example IV prove corrected forms of Conjectures 4.4 and 4.5. For $χ_m=4^m-10\cdot2^m+16$, a half-integer specialization of Au's first $1/π^4$ 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.
Comments36 pages, 7 tables. v2 adds the proof of Conjecture 4.6, restores its exact ancillary certificate branch, updates the title and abstract from four to five conjectures, and corrects the companion-paper citation
