  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 the Wro\'nskian 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.
