  On the basis of the Woodhouse causal axiomatics, we show that conformal
proper times and an extra variable in addition to those of space and time,
precisely and physically identified from experimental examples, together give a
physical justification for the `chronometric hypothesis' of general relativity.
Indeed, we show that, with a lack of these latter two ingredients, no clock
paradox solution exists in which the clock and message functions are solely at
the origin of the asymmetry. These proper times originate from a given
conformal structure of the spacetime when ascribing different compatible
projective structures to each Woodhouse particle, and then, each defines a
specific Weylian sheaf structure. In addition, the proper time
parameterizations, as two point functions, cannot be defined irrespective of
the processes in the relative changes of physical characteristics. These
processes are included via path-dependent conformal scale factors, which act
like sockets for any kind of physical interaction and also represent the values
of the variable associated with the extra dimension. As such, the differential
aging differs far beyond the first and second clock effects in Weyl geometries,
with the latter finally appearing to not be suitable.
