The price of regularizing squared distance in hyperbolic optimal transport

Changing a transport cost to obtain regularity also changes the task being solved. We quantify this price for squared distance on hyperbolic space, whose negative curvature violates the Ma–Trudinger–Wang (MTW) condition. Among normalized radial costs in dimension $d\ge 3$, critical log-cosh minimizes uniform deviation from squared distance among full-A3w costs, uniquely on each bounded distance interval. A prescribed transverse MTW floor admits an explicit sharp envelope with interval rigidity. Its weakest full margin and mixed-Hessian condition number obey $m_D^{\star }({\mathsf{K}}_D^{\star }{)}^2=\Lambda$, exposing the sensitivity cost of this optimal repair. Positive full margins require additional distortion: finite directional candidates and a sixth-order obstruction sharpen analytic bounds, while calibrated log-cosh supplies a feasible family-wise frontier. The critical repair's distortion and conditioning penalties vanish in the flat limit. In a fixed three-target OT problem, new-seed experiments show that repair changes the optimal transport regions at a measurable original-objective cost. Separate inversion experiments recover the predicted local sensitivity. Together, the theory and experiments distinguish the geometric benefit of regularization from its task and numerical prices.