Beam Me Up Scotty!

If there is no constrain on $x_z$ other than,

$m_{\rho}c^2-\int^{x_a}_{0}{\psi}dx\ge0$

since,

$x_a=2x_z$

$m_{\rho}c^2-\int^{2x_z}_{0}{\psi}dx\ge0$

from the post "We Still Have A Problem", then $x_z$ can take on all values from

$0\lt x_z\lt x_{z\,max}$

where,

$m_{\rho}c^2=\int^{2x_{z\,max}}_{0}{\psi}dx$

A particle under no force from its surroundings will tend towards maximum $\psi$, being mass-less.  Under the action of a force (or to provide a centripetal force) the particle will change its $\psi$ such that,

$F=-\cfrac{\partial\,\psi}{\partial\,x}=F_{external}$

the force $F$ as a result of $\psi$ equals the external force as required.

The particle's mass changes as the forces around it.  In the case of a photon oscillating between one time dimension and one space dimension, it experiences less constrain by way of the space dimension than a charge or gravity particle that oscillate between two space dimensions.  So, a photon fully manifest itself as $\psi$.  A charge or gravity particle experiences more constrain in two dimensions cannot manifest fully as $\psi$ and so has mass.

If this is true, we can distill a photon to have mass by applying a force in the one space dimension that it is oscillating.  And since a photon can display negative $\psi$ effects, an electron (charge) or gravity particle can be made mass-less in a stream of photons, where the particle fully manifest itself as $\psi$.

Under the right conditions we can all turn into pure energy.