  We investigate the structure of the three-qubit magic Veldkamp line (MVL).
This mathematical notion has recently shown up as a tool for understanding the
structures of the set of Mermin pentagrams, objects that are used to rule out
certain classes of hidden variable theories. Here we show that this object also
provides a unifying finite geometric underpinning for understanding the
structure of functionals used in form theories of gravity and black hole
entropy. We clarify the representation theoretic, finite geometric and physical
meaning of the different parts of our MVL. The upshot of our considerations is
that the basic finite geometric objects enabling such a diversity of physical
applications of the MVL are the unique generalized quadrangles with lines of
size three, their one point extensions as well as their other extensions
isomorphic to affine polar spaces of rank three and order two. In a previous
work we have already connected generalized quadrangles to the structure of
cubic Jordan algebras related to entropy fomulas of black holes and strings in
five dimensions. In some respect the present paper can be regarded as a
generalization of that analysis for also providing a finite geometric
understanding of four-dimensional black hole entropy formulas. However, we find
many more structures whose physical meaning is yet to be explored. As a
familiar special case our work provides a finite geometric representation of
the algebraic extension from cubic Jordan algebras to Freudenthal systems based
on such algebras.
