It takes seriously the idea of obtaining non-classical logic from General Relativity. That is, propositions determining states do not necessarily obey the distributive law. Typical 2-valued propositions ask whether or not a certain region contains a particle. This is an argument for the requirement of higher dimensional toposes in gravity, because the logic of a 1-topos is always distributive, whether or not it is Boolean. Recall that distributivity in general is naturally expressed by a map $ST \Rightarrow TS$ for two monads $S$ and $T$.
The basic idea bears a little resemblance to Louis Crane's geometrization proposal for four dimensional spin foam models, which allows for a relaxation of the manifold condition at a point. Hadley only discusses manifolds, for ordinary GR, as if quantization of gravity were unnecessary, but no analysis of solutions to Einstein's equations is actually given. His conclusion is that gravitons do not exist, because as 3-geons they would lack the topological structure needed to localise them.
Note that Hadley's later papers have even more grandiose titles, without much accompanying mathematical analysis.