AI 中文总结
该研究针对四个经典族计算Witten ζ函数的第二极壁周期，推导射影积分的简化形式，得到A、B、C型的正弦权重及D型的有限分圆乘积，且无法直接推广到例外型。
AI 中文摘要
设Φ为秩r的不可约简约晶根系，N为正余根的数量。已有定理（arXiv:2608.16363，定理3.3）将2/h以下的第一极确定为q₂=(r-1)/(N-1)，并将其留数表示为附属于优势腔单壁的周期之和。我们针对全部四个经典族计算该壁周期和。恒等式q₂(N-1)=r-1消去了每个壁积分中的径向变量。所得射影积分在A_r型中简化为混合Dotsenko-Fateev腔，在B_r和C_r型中简化为这些腔连同一个Selberg端点，在D_r型中简化为单个腔族。腔递推在A、B、C型中给出显式正弦权重；在D型中，单位根处的终止基本超几何和简化为有限分圆乘积。我们还证明，对应的例外壁限制并非反射排列，因此该特定简化无法直接推广到例外型。
英文摘要
Let Phi be an irreducible reduced crystallographic root system of rank r, and let N be the number of positive coroots. A previous theorem (arXiv:2608.16363, Theorem 3.3) identifies the first pole below 2/h as q_2 = (r-1)/(N-1) and expresses its residue as a sum of periods attached to the simple walls of the dominant chamber. We evaluate that wall-period sum for all four classical families. The identity q_2 (N-1) = r-1 removes the radial variable from every wall integral. The resulting projective integrals reduce to mixed Dotsenko-Fateev chambers in type A_r, to those chambers together with a Selberg endpoint in types B_r and C_r, and to a single chamber family in type D_r. The chamber recurrences give explicit sine weights in types A, B, and C. In type D, a terminating basic-hypergeometric sum at a root of unity reduces to a finite cyclotomic product. We also show that the corresponding exceptional wall restrictions are not reflection arrangements, so this particular reduction does not extend directly to the exceptional types.
