  We give a geometric characterization of the quantitative non-integrability,
introduced by Katz, of strong stable and unstable bundles of partially
hyperbolic measures and sets in dimension 3. This is done via the use of higher
order templates for the invariant bundles. Using the recent work of Katz, we
derive some consequences, including the measure rigidity of $uu$-states and the
existence of physical measures.
