Foundations for Gravitational Potential Energy Evaluation

We investigate the evaluation of the effective gravitational binding energy in heterogeneous stellar systems by comparing a permutation-invariant estimator with heuristic formulations based on sequential aggregation. While the exact method provides a physically consistent reference independent of population ordering, the heuristic estimator introduces a path dependence that strongly influences the reconstructed dark-to-baryonic mass ratios. A systematic exploration of permutations quantifies this sensitivity and shows that an optimized ordering substantially reduces the systematic bias, producing an error distribution that is approximately Gaussian and centered near zero. When applied to galactic rotation data, the heuristic approach achieves significantly higher resolution rates than the exact estimator. These results highlight a fundamental trade-off between physical invariance and algorithmic sensitivity, and suggest that the trajectory through population space is an intrinsic component of sequential gravitational energy models.