Current positioning systems are not primary, relativistic systems. Nevertheless, genuine, relativistic and primary positioning systems have been proposed recently by Bahder, Coll et al. and Rovelli to remedy such prior defects. These new designs all have in common an equivariant conformal geometry featuring, as the most basic ingredient, the spacetime geometry. We show how this conformal aspect can be the four-dimensional projective part of a larger five-dimensional geometry. Our aim has been then to explore all of the geometric, physical consequences of this projective geometry in such spacetime context and ask for a physical process, the implementation of which could reveal this fifth dimension. We find that the latter is physically obtained from a fifth time stamp effectively yielded from a new localization protocol that we present jointly with this projective geometry. In particular, this leads to a new surprisingly result in the complete integrability of the Einstein equations while the well-known incompleteness of their usual non-projective version is unquestioned. Also, it may validate completely or gives naturally true physical grounds to the old model due to Schouten, Haantjes and van Dantzig unifying electromagnetism and gravitation.