  This study presents a holistic picture of the preformation of nuclear
clusters with credence to the kinematics of their emissions. Besides the
fitting of the preformation formula to reproduce the experimental half-lives,
we have investigated the interrelationship between the parameters involved in
the cluster decay process for medium, heavy and superheavy nuclei. Based on the
established conceptual findings, we propose a new cluster preformation
probability ($P_0$) formula that incorporates all influential parameters of the
cluster radioactivity and thus has an edge over the existing formulae in the
literature. Further, we hypothesize that a fraction of the decay energy is
needed for cluster formation within the parent nucleus. The proposed formula
opens a new paradigm to separately estimate the energy contributed during the
cluster formation from its emission and thus shows that the contribution of the
Q-value splits into three major parts accounting for the energy contributed
during the cluster preformation, its emission and recoil of the daughter
nucleus. Moreover, the expression $P_0$ is adept at accommodating the theorized
concept of heavy particle radioactivity (HPR). The result reveals that, like
$\alpha$-decay, a proper estimation of the $P_0$ and $Q$-value in the cluster
studies are enriched with qualitative information about the nuclear structure.
However, from the analysis, the Geiger-Nuttall law is not the best compromise
in the clustering due to the non-linearity between $\log_{10}T_{1/2}$ and
$\sqrt{Q}$, unlike in $\alpha$-decay. We have demonstrated that with the
inclusion of the proposed formula, the half-life predictions from both
microscopic R3Y and phenomenological M3Y NN potentials closely agree with the
available experimental data and that the slight variation can be traced to
their peculiar barrier characteristics.
