  The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic
notion for a metric-measure space to have Ricci-curvature bounded from below
and dimension bounded from above. We prove that it is enough to verify this
condition locally: an essentially non-branching metric-measure space
$(X,{\mathsf d},{\mathfrak m})$ (so that $(\text{supp} \; {\mathfrak
m},{\mathsf d})$ is a length-space and ${\mathfrak m}(X) < \infty$) verifying
the local Curvature-Dimension condition $\mathsf{CD}_{loc}(K,N)$ with
parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$, also verifies the global
Curvature-Dimension condition $\mathsf{CD}(K,N)$. In other words, the
Curvature-Dimension condition enjoys the globalization (or local-to-global)
property, answering a question which had remained open since the beginning of
the theory.
  For the proof, we establish an equivalence between $L^1$ and $L^2$
optimal-transport-based interpolation. The challenge is not merely a technical
one, and several new conceptual ingredients which are of independent interest
are developed: an explicit change-of-variables formula for densities of
Wasserstein geodesics depending on a second-order temporal derivative of
associated Kantorovich potentials; a surprising third-order theory for the
latter Kantorovich potentials, which holds in complete generality on any proper
geodesic space; and a certain rigidity property of the change-of-variables
formula, allowing us to bootstrap the a-priori available regularity. As a
consequence, numerous variants of the Curvature-Dimension condition proposed by
various authors throughout the years are shown to, in fact, all be equivalent
in the above setting, thereby unifying the theory.
