Average Membership Degree Curves for Modal Structure Detection and Resolution Selection

Selecting group number is ambiguous because clustering algorithms return partitions on unimodal data and group definitions may disagree. We formalise Average Membership Degree (AMD), which measures mean dominance of winning memberships over 1/c, and study the curve across candidate resolutions. Synthetic populations with independently verified modal truth were used to separate structure detection, modal-count estimation, and partition recovery. A supervised reader based on curve morphology and multistart stability achieved balanced accuracy 0.843 for distinguishing one from multiple modes, while adding data and partition diagnostics increased it to 0.947. For structured populations, exact modal-count accuracy rose from 0.608 to 0.716. The complete reader outperformed the evaluated unsupervised comparators for modal-count selection but not assignment recovery. A marine application illustrates how curve shape and partition transitions distinguish coarse and fine supported resolutions. AMD should be interpreted through its curve, refined by partition diagnostics and with confidence calibrated within the domain of application. Version and scope. Preprint based on the manuscript submitted on 11 October 2026. Scientific text, results and figures are preserved, with a dated scope note explaining historical reader versions, measurement-unit sensitivity, domain-conditional certainty and the meaning of empirical groups. Supplementary Information is included. The full computational archive is not yet publicly deposited. This is an author preprint, not a journal version of record; no claim of journal acceptance or completed peer review is made.