  An immense class of physical counterexamples to the four dimensional strong
cosmic censor conjecture---in its usual broad formulation---is exhibited. More
precisely, out of any closed and simply connected 4-manifold an open Ricci-flat
Lorentzian 4-manifold is constructed which is not globally hyperbolic and no
perturbation of it, in any sense, can be globally hyperbolic. This very stable
non-global-hyperbolicity is the consequence of our open spaces having a
"creased end" i.e., an end diffeomorphic to an exotic ${\mathbb R}^4$. Open
manifolds having an end like this is a typical phenomenon in four dimensions.
  The construction is based on a collection of results of Gompf and Taubes on
exotic and self-dual spaces, respectively, as well as applying Penrose'
non-linear graviton construction (i.e., twistor theory) to solve the Riemannian
Einstein's equation. These solutions then are converted into stably
non-globally-hyperbolic Lorentzian vacuum solutions. It follows that the
plethora of vacuum solutions we found cannot be obtained via the initial value
formulation of the Einstein's equation because they are "too long" in a certain
sense (explained in the text). This different (i.e., not based on the initial
value formulation but twistorial) technical background might partially explain
why the existence of vacuum solutions of this kind have not been realized so
far in spite of the fact that, apparently, their superabundance compared to the
well-known globally hyperbolic vacuum solutions is overwhelming.
