Background: Cluster radioactivity is a rare decay mode that bridges $\alpha$ decay and spontaneous fission. Despite extensive study, the quantitative role of nuclear structure effects in this process remains unclear.

Purpose: This work aims to investigate the influence of nuclear deformation parameters and their signs on the calculated half-lives of cluster decay. It further explores how quadrupole and higher-order deformations, along with orientation effects, affect cluster preformation probabilities and decay dynamics within the preformed cluster-decay model (PCM). To enhance predictive accuracy and reveal structural correlations, symbolic regression using genetic programming (GPlearn) is applied.

Methods: The cluster-daughter interaction potential is derived from the Skyrme energy density functional using the two-parameter Fermi (2pF) model. PCM calculations explicitly include deformation and orientation degrees of freedom for cluster emissions leading to ${}^{208}\mathrm{Pb}$ and neighboring isotopes. The GPlearn model is trained to establish functional relationships between structural inputs and decay half-lives.

Results: The decay half-lives are primarily calculated using the PCM, which reproduces the available experimental data with remarkable consistency and accuracy, establishing its reliability for predictive purposes. The observed agreement between theory and experiment reflects the strength of the Skyrme-based microscopic framework, where deformation-induced modifications in nuclear densities and interaction terms directly influence the calculated decay half-lives. The analysis demonstrates that higher-order multipole deformations play a crucial role in improving the agreement with experimental half-lives by introducing significant structural corrections. While the quadrupole deformation provides a baseline contribution, the inclusion of higher-order terms substantially modifies the fragmentation potential and barrier penetrability, influencing the relative preformation probabilities of competing decay channels. The study further shows that cold elongated configurations yield half-lives much closer to experimental values compared with cold optimum orientations, emphasizing their role in reducing barrier heights at larger interaction radii. For comparison, the PCM results are also examined alongside various semi-empirical formulas and the GPlearn-based regression technique. Both comparative approaches exhibit a similar overall trend, supporting the robustness of the PCM predictions, especially in regions where experimental data are not yet available.

Conclusion: The PCM successfully captures the recent experimental trends in decay half-lives, particularly when higher-order deformations and orientations are incorporated. These effects significantly refine the calculated results and confirm the model's predictive strength across a wide range of nuclei. The GPlearn-based regression and semi-empirical formulas are employed primarily for comparison, validating the consistency and reliability of the PCM outcomes in both known and unexplored regions. This combined analysis establishes PCM as the principal theoretical framework for predicting cluster decay half-lives, supported by data-driven and semi-empirical approaches for cross-verification.

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• Binding energy & masses
• Heavy-particle decays
• Lifetimes & widths
• Nuclear decay
• Nuclear forces
• Nuclear structure & decays
• Nucleon distribution
• Machine learning
• Nuclear density functional theory