Geometry from Information: Paper 18 — Quantum Optimal Transport and the Gravitational Metric: W_2^Q-Hessian = BKM at de Sitter (Draft)

We establish that the Bogoliubov–Kubo–Mori (BKM) metric of the Geometry from Information programme is the Hessian of relative entropy in the quantum Wasserstein-2 geometry of Carlen–Maas. The main result is: Hess_{W2Q}[S_rel]_{rho_dS}(h,h) = G_BKMg_dS + O(l_P^2/l^2) for all transverse-traceless h. Two corollaries follow: (i) Open Problem OP-rg of the Master Paper is closed — the locality of G_BKM[g_dS] is a consequence of the W2Q geodesic structure at de Sitter, not of RG flow; (ii) the island formula extremisation is equivalent to a W2Q-optimal partition of the Cauchy slice. Two open problems remain: exactness of the identity (OP-18.1) and extension of W2Q to the Type II∞ setting (OP-18.2).Keywords:quantum optimal transport, Wasserstein metric, BKM metric, relative entropy, de Sitter, von Neumann algebras, island formula, Carlen-Maas, Type II infinity, gravitational metricRelated works:Is part of: 10.5281/zenodo.19034071 (Master Paper)