  We present a formulation of scalar effective field theories in terms of the
geometry of Lagrange spaces. The horizontal geometry of the Lagrange space
generalizes the Riemannian geometry on the scalar field manifold, inducing a
broad class of affine connections that can be used to covariantly express and
simplify tree-level scattering amplitudes. Meanwhile, the vertical geometry of
the Lagrange space characterizes the physical validity of the effective field
theory, as a torsion component comprises strictly higher-point Wilson
coefficients. Imposing analyticity, unitarity, and symmetry on the theory then
constrains the signs and sizes of derivatives of the torsion component,
implying that physical theories correspond to a special class of vertical
geometry.
