  We study a surprising phenomenon in which Feynman integrals in
$D=4-2\varepsilon$ space-time dimensions as $\varepsilon \to 0$ can be fully
characterized by their behavior in the opposite limit, $\varepsilon \to
\infty$. More concretely, we consider vector bundles of Feynman integrals over
kinematic spaces, whose connections have a polynomial dependence on
$\varepsilon$ and are known to be governed by intersection numbers of twisted
forms. They give rise to differential equations that can be obtained exactly as
a truncating expansion in either $\varepsilon$ or $1/\varepsilon$. We use the
latter for explicit computations, which are performed by expanding intersection
numbers in terms of Saito's higher residue pairings (previously used in the
context of topological Landau-Ginzburg models and mirror symmetry). These
pairings localize on critical points of a certain Morse function, which
correspond to regions in the loop-momentum space that were previously thought
to govern only the large-$D$ physics. The results of this work leverage recent
understanding of an analogous situation for moduli spaces of curves, where the
$\alpha' \to 0$ and $\alpha' \to \infty$ limits of intersection numbers
coincide for scattering amplitudes of massless quantum field theories.
