Schwarzian Residue of the Samuelson Obstruction in Tangent Lagrangian 2-Webs

The Samuelson condition, a classical area-ratio condition for planar Lagrangian $2$-webs, is equivalent in local web coordinates $(u,v)$ to ${\mathrm{\partial }}_u{\mathrm{\partial }}_v\log \mathrm{\mid }{J}_F(u,v)\mathrm{\mid }=0$, where $F$ is the inverse web-coordinate map and $J_F$ is its Jacobian. For the tangent-line family $L_t:y=tx+h(t)$, let $\mathrm{\Psi }(u,v)$ be the intersection map of $L_u$ and $L_v$, and set $S_h(u,v)={\mathrm{\partial }}_u{\mathrm{\partial }}_v\log \mathrm{\mid }{J}_{\mathrm{\Psi }}(u,v)\mathrm{\mid }$. Near each diagonal point $(t_0,t_0)$ with $h^{\mathrm{\prime }\mathrm{\prime }}(t_0)\mathrm{\ne }0$, the obstruction admits the decomposition $S_h(u,v)=1\mathrm{/}(v-u{)}^2+R_h(u,v)$, where $R_h$ extends smoothly across the diagonal. We call $R_h(t,t)$ the Schwarzian residue and compute $R_h(t,t)=h^{(4)}(t)\mathrm{/}(3h^{\mathrm{\prime }\mathrm{\prime }}(t))-(4\mathrm{/}9)(h^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(t)\mathrm{/}{h}^{\mathrm{\prime }\mathrm{\prime }}(t){)}^2=(1\mathrm{/}2)\{\sigma ,t\}=-{\kappa }_{\mathrm{a}\mathrm{f}\mathrm{f}}(\sigma )(d\sigma \mathrm{/}dt{)}^2$. Here $\sigma$ is the equi-affine arclength parameter of the envelope $\gamma (t)=(-h^{\mathrm{\prime }}(t),h(t)-t{h}^{\mathrm{\prime }}(t))$, and ${\kappa }_{\mathrm{a}\mathrm{f}\mathrm{f}}$ is its equi-affine curvature. Thus the universal pole accounts for the local failure of the Samuelson condition, while the diagonal finite part carries affine-projective information about the envelope. We also derive the transformation law of the Schwarzian residue under reparametrization of the tangent-line parameter.