# Does the universe obey the holographic principle due to Stokes' theorem?

Does the universe obey the holographic principle due to Stokes' theorem?

\begin{equation} \int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega. \end{equation}

Can this theorem be enough proof of our Universe being a hologram – the choice of $\omega$ and $\Omega$ is completely arbitrary!

No, it cannot be enough. Stokes' theorem says that the volume ($\Omega$) integral of $d\omega$, a form that is the exterior derivative of another one (of $\omega$), may be written as a surface integral. But it doesn't allow us to rewrite the volume integral of a general integrand (which isn't the exterior derivative of anything) such as the Lagrangian density ${\mathcal L}$ as a surface integral. So the Stokes' theorem is useless for dealing e.g. with the action $S$ that defines the dynamics of a general theory in the volume.
One should mention that when the action is topologically invariant, ${\mathcal L}$ may indeed be locally written as a "total derivative", and in that case, the theory has indeed a provable relationship with lower-dimensional theories (a major example is Chern-Simons theory in 3 dimensions and the related WZNW theories in 2D). But the general theories we know – the Standard Model coupled to gravity – aren't of this special type, at least not manifestly so. What's happening in the volume is general – we surely do care about values of some fields such as the electric field in particular places of the volume – and there apparently isn't any "counterpart degree of freedom" on the surface that we could associate it with.