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This study presents a holistic picture of the preformation of nuclearclusters with credence to the kinematics of their emissions. Besides thefitting of the preformation formula to reproduce the experimental half-lives,we have investigated the interrelationship between the parameters involved inthe cluster decay process for medium, heavy and superheavy nuclei. Based on theestablished conceptual findings, we propose a new cluster preformationprobability ($P_0$) formula that incorporates all influential parameters of thecluster radioactivity and thus has an edge over the existing formulae in theliterature. Further, we hypothesize that a fraction of the decay energy isneeded for cluster formation within the parent nucleus. The proposed formulaopens a new paradigm to separately estimate the energy contributed during thecluster formation from its emission and thus shows that the contribution of theQ-value splits into three major parts accounting for the energy contributedduring the cluster preformation, its emission and recoil of the daughternucleus. Moreover, the expression $P_0$ is adept at accommodating the theorizedconcept of heavy particle radioactivity (HPR). The result reveals that, like$\alpha$-decay, a proper estimation of the $P_0$ and $Q$-value in the clusterstudies are enriched with qualitative information about the nuclear structure.However, from the analysis, the Geiger-Nuttall law is not the best compromisein 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 theinclusion of the proposed formula, the half-life predictions from bothmicroscopic R3Y and phenomenological M3Y NN potentials closely agree with theavailable experimental data and that the slight variation can be traced totheir peculiar barrier characteristics.

Quest for a Universal Cluster Preformation Formula: A new paradigm for estimating the cluster formation energy