Flatness of the single‑u connection in differential‑equation representation

Explain why the connection one-form B governing the differential system dI_U = B · I_U for the single‑u contribution to the reduced triangle integral is not flat on the full kinematic manifold X, in contrast to the flat nilpotent connection for the factorised term, and identify conditions under which flatness fails or could be restored.

Background

To evaluate the reduced triangle integral, the authors recast the measure via a contour integral and decompose the integrand using partial fractions, turning the u,v integrations into iterated integrals expressible in terms of logarithms and dilogarithms. They then organise these contributions using first-order differential equations with exact connection one-forms.

For the factorised term, the associated connection is strictly upper triangular (nilpotent) and flat, confirming its interpretation as an iterated integral. For the single‑u term, however, the derived block-triangular connection B is not flat on the full kinematic manifold X, and the authors state they do not understand this difference, suggesting a deeper geometric or analytic reason tied to the dependence on ratios like V/U and W/U and their pull-back to the complex parameter w.