Not All Edge Errors Are Equal: Incidence-Preserving Optimal Transport for Neural TSP Heatmaps

Neural TSP solvers commonly train edge heatmaps with pointwise losses that ignore the geometry of wrong neighbour choices. GeoEdge-OT instead compares, per city, incident-edge distributions over neighbour coordinates. For valid tours, we prove $|L(T)-L(T')|\le D_E(T,T')\le D_{\mathrm{Geo}}(T,T')\le\eta(T,T')$. On $240{,}000$ 2-opt perturbations of LKH reference tours, $D_{\mathrm{Geo}}$ has Spearman $\rho=0.985$ with reference-relative cost increase and $52\%$ lower median bound slack than $\eta$. Its Sinkhorn relaxation lowers raw TSP-50/100 gaps versus continued cross-entropy under checkpoint-conditional paired-instance intervals; the only resolved weighted-$L_1$ comparison favors weighted $L_1$ on clustered-50.