  We relate generalized Lebesgue decompositions of measures in terms of curve
fragments (Alberti representations) and Weaver derivations. This correspondence
leads to a geometric characterization of the local norm on the Weaver cotangent
bundle of a metric measure space $(X,\mu)$: the local norm of a form $df$ sees
how fast $f$ grows on curve fragments seen by $\mu$. This implies a new
characterization of differentiability spaces in terms of the
$\mu$-a.e.~equality of the local norm of $df$ and the local Lipschitz constant
of $f$. As a consequence, the Lip-lip inequality of Keith must be an equality.
We also provide dimensional bounds for the module of derivations in terms of
the Assouad dimension of $X$.
