A force \(F_r\) on \(\psi\) along a radial line, derived from the force density, \(F_{\rho}\)
\(F_{\rho}=-\cfrac{\partial\,\psi}{\partial x}\)
\(F_r=\int{F_{\rho}}dx\)
act opposite to the direction of increasing \(\psi\). If \(\psi\) moves to higher value than a positive force is required in the positive radial direction to perform work on \(\psi\), for \(\psi\) to gain energy. This force \(F_{pinch}\) in the positive direction is countered by \(F_r\) above in the negative direction, as \(\cfrac{\partial\,\psi}{\partial x}\) is positive.
Work is done against \(F_r\). This work done increases the potential of \(\psi\).
\(F_r\) is subjected to the inverse square law. When \cfrac{\partial\,\psi}{\partial x}\) at \(a_{\psi}\) and beyond, is a constant or does not increase enough,
\(\cfrac{\partial\,\psi}{\partial x}|_{x\gt a_{\psi}}\lt\cfrac{2}{r^3}\)
where
\(-\left(\cfrac{1}{r^2}\right)^{'}_r=\cfrac{2}{r^3}\)
then \(F_r\) decreases with distance from the center of the particle. Any non zero force \(F_{pinch}\) that displaces \(\Delta \psi\) away from the center of the particle, pulls \(\Delta\psi\) away to infinity with greater and greater acceleration.
\(\Delta\psi\) is removed from the particle.
In this way, \(a_{\psi}\) the probable size of a particle was arbitrarily set at,
\(\theta_{\psi}=\pi=\cfrac{G}{\sqrt{2mc^2}}a_{\psi}\)
where \(tanh(\pi)\approx1\), for \(F_r|_{a_{\psi}}\) to increase at greater distance from the particle center, that the particle remain intact with application of small \(F_{pinch}\). \(\psi\) has minimum resistance from being pinched apart.
This does not mean that big particles of higher values of \(a_{\psi}\) beyond the \(\theta_{\psi}=\pi\) limit do not exist.
Such big particles will still interact at a constant (slightly higher) light speed limit.
