Wrońskian algebra and Broadhurst–Roberts quadratic relations

Through algebraic manipulations on Wro\'nskian 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 framework, reinterpret de Rham intersection pairing through polynomial coefficients in Vanhove's differential operators, compute Betti via linear sum rules for on-shell off-shell Feynman diagrams at threshold momenta. From ideal generated Broadhurst--Roberts relations, derive non-linear diagrams, including an infinite family determinant identities that compatible Deligne's conjectures critical values motivic $L$-functions.