Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions

We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. subsequently use this geometric characterization to answer several questions in analysis. Notably, it follows the Lipschitz-free space $\mathcal{F}(M)$ over $M$ is dual if and only 1-unrectifiable. Furthermore, we establish determinacy principle for Radon-Nikod\'ym property (RNP) deduce that, any complete $M$, pure 1-unrectifiability actually equivalent some well-known Banach properties such RNP Schur property. A direct consequence complete, 1-unrectifiable isometrically embeds into with RNP. Finally, provide possible solution Whitney by finding rectifiability-based description 1-critical spaces, prove following: bounded turning tree fails be each its subarcs has $\sigma$-finite Hausdorff 1-measure.