The Groups Change When the Rules Change: What Clustering Actually Finds

Clustering is often presented as the discovery of groups already present in unlabeled data. In practice, a partition is produced only after a representation, algorithm family, resolution, objective, and validation rule have been fixed. Agreement on one of those choices does not guarantee agreement on the others. This paper evaluates 84 clustering configurations on the 307 development vehicles from the complete-case Auto MPG data. Four representations inherited from Paper 21 are combined with K-means, Ward agglomerative clustering, and full-covariance Gaussian mixtures across K values from 2 through 8. Silhouette, Calinski–Harabasz, Davies–Bouldin, and perturbation stability remain internal selection rules. Origin is withheld until external audit, and 85 vehicles from 1980–1982 remain a diagnostic temporal population. Silhouette selects K=2 in all 12 representation–algorithm groups, but the four validation rules disagree in 6 of the 12 groups. Among the 12 silhouette-selected partitions, mean pairwise adjusted Rand agreement is 0.734 and the minimum is 0.212. Agreement on the number of clusters therefore does not imply agreement on cluster membership. Within the silhouette-selected set, the largest development origin NMI is 0.309 for raw Gaussian mixture clustering with K=2, whose silhouette score is 0.535. Across all 84 candidates, the post hoc external maximum is 0.313 for the raw Gaussian mixture with K=7. However, that partition has a silhouette score of 0.001, was not selected internally, and does not replace the silhouette-selected solution. Mean perturbation stability across the selected partitions is 0.962, with a minimum of 0.775. Mean origin NMI falls from 0.235 in development to 0.032 in the temporal population. The findings show that internal compactness, perturbation stability, external agreement, and temporal transport are distinct properties. None alone establishes natural groups or a uniquely correct partition. The deposit includes the complete paper, source code, data, machine-readable results for all 84 candidates, validation-selection records, pairwise partition-agreement outputs, five generated figures, 20 deterministic perturbation audits per candidate, temporal diagnostics, a SHA-256 integrity manifest, and an automated reproduction verifier.