Are not the field equations proof of the holographic principle? The field equations (e.g. Schrödinger’s/Maxwell’s) describe the four-dimensional universe as the time evolution of a three-dimensional system.  This implies that the universe contains only three dimensions of information: the fourth is derivable deterministically (albeit not practically) from the other three via these formulas.
My understanding is that this is exactly what the holographic principle posits: that the universe contains only three dimensions of information (commonly presented as two spatial, one temporal).  Yet the holographic principle remains unproven.  Where is my misunderstanding?
 A: An intuitive description (or rather a caricature) of the dimensional reduction associated with the holographic principle run as follows.
We live in a 4D spacetime. Fields propagate over this 4D spacetime with the Big Bang providing a 3D 'initial condition' to these fields. One would therefore expect the amount of information required to specify the full 4D universe to be proportional to the 3D initial volume. 
According to the holographic principle this is incorrect. 

Suppose you take a snapshot of the universe with an omni-directional camera (see picture). The photons reaching the camera at the moment of the snapshot trace out a 3D light cone. What is 'happening' on this 3D light cone one can think of having propagated from the 3D initial volume to the light cone (in the picture three arbitrary world lines indicate this causal influence). So one would expect the information required to fully describe the light cone to be proportional to the 3D initial volume embraced by this light cone. 
This is incorrect. Rather, the holographic principle states that the information contained in the 3D light cone is proportional to the surface area of the 2D boundary to the light cone. 
In rough terms, you can think of this dimensional reduction to be related to a (non-local) field theory for the 3D light cone that forms a dual description to the local field equations 'living' in the 4D spacetime contained within this light cone. 
