Caratheodory's formulation of second law of thermodynamics, also referred to as Caratheodory's principle states
In any neighbourhood of any thermodynamic state $P$ there exist states which are adiabatically inaccessible from $P$,
where adiabatic accessibility means that states can be connected by paths satisfying $DQ = 0$. Caratheodory's theorem states then that this is a sufficient condition for an integrating factor to exist, so that
$$DQ = \tau \,\text{d}\sigma$$
or in other words that $DQ=0$ describes a foliation (planar arrangement) of states' space.
I am looking for an intuitive proof/argument for this implication (Caratheodory's principle $\Rightarrow$ foliation). In the literature, I was only able to find rather technical proofs, not helping with intuition, or arguments that seem wrong (but it may very well be that I am missing something). For example, Chandrasekhar in his Introduction to the Study of Stellar Structure writes that in a given neighbourhood of $P$ points, adiabatically accessible from $P$ must form a hyperplane, since they cannot form anything of smaller dimension as $DQ$ is already an infinitesimal hyperplane element, nor can they form anything of higher dimension as it would contradict Cartheodory's principle. How do we however exclude the possibility of those points forming two surfaces tangential at $P$?