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