Through algebraic manipulations onWrońskian matrices whose entries are reducible to Bessel moments, we present a new analytic proof of the quadratic relations conjectured by Broadhurst and Roberts, along with some generalizations. In the Wrońskian framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove’s differential operators, and compute the Betti intersection pairing via linear sum rules for on-shell and off-shell Feynman diagrams at threshold momenta. From the ideal generated by Broadhurst–Roberts quadratic relations, we derive new non-linear sum rules for on-shell Feynman diagrams, including an infinite family of determinant identities that are compatible with Deligne’s conjectures for critical values of motivic $L$‑functions.

中文翻译：

---

Wrońskian 代数和 Broadhurst-Roberts 二次关系

通过对 Wrońskian 矩阵的代数操作，其条目可简化为贝塞尔矩，我们提出了 Broadhurst 和 Roberts 推测的二次关系的新解析证明，以及一些概括。在 Wrońskian 框架中，我们通过 Vanhove 微分算子中的多项式系数重新解释 de Rham 交集对，并通过阈值动量下壳上和壳外费曼图的线性求和规则计算 Betti 交集对。从 Broadhurst-Roberts 二次关系生成的理想中，我们推导出了壳上费曼图的新非线性求和规则，包括与 Deligne 对动机 $L$ 函数的临界值的猜想兼容的无限行列式恒等式。