Volume 11 • Issue 1 • PP: 25–31 • 2026
A Neutrosophic Layer for Fuzzy c-Means Clustering: Score Function Theory, Metric Properties, and Diagnostic-Ambiguity Quantification on Breast Cancer Data
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© 2026 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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Fuzzy c-means clustering assigns every data point a degree of membership to each cluster but offers no separate account of how ambiguous that assignment is. This paper builds a single-valued neutrosophic layer on top of the classical fuzzy c-means membership distribution, representing each point by a truth-membership Ti (its strongest cluster membership), an indeterminacy Ii (the normalized Shannon entropy of its full membership vector), and a falsity-membership Fi = 1−Ti. Four results are proved: the fuzzy c-means update equations are re-derived from the Lagrangian stationarity conditions of the underlying constrained optimization; the resulting (Ti, Ii,Fi) triplet is shown to be bounded and to attain its extremes exactly at crisp and maximally ambiguous membership distributions; a score function combining the three components is shown to be strictly monotone in each; the natural root-mean-square distance between two neutrosophic triplets is shown to satisfy the metric axioms; and, for the two-cluster case specifically, indeterminacy is proved to be an exact deterministic function of truth-membership, so that a third, genuinely independent source of information requires three or more clusters. Every result is checked numerically, including a direct verification of the two-cluster degeneracy result to floating-point precision. Applied to the Breast CancerFuzzy c-means clustering assigns every data point a degree of membership to each cluster but offers no separate account of how ambiguous that assignment is. This paper builds a single-valued neutrosophic layer on top of the classical fuzzy c-means membership distribution, representing each point by a truth-membership Ti (its strongest cluster membership), an indeterminacy Ii (the normalized Shannon entropy of its full membership vector), and a falsity-membership Fi = 1−Ti. Four results are proved: the fuzzy c-means update equations are re-derived from the Lagrangian stationarity conditions of the underlying constrained optimization; the resulting (Ti, Ii,Fi) triplet is shown to be bounded and to attain its extremes exactly at crisp and maximally ambiguous membership distributions; a score function combining the three components is shown to be strictly monotone in each; the natural root-mean-square distance between two neutrosophic triplets is shown to satisfy the metric axioms; and, for the two-cluster case specifically, indeterminacy is proved to be an exact deterministic function of truth-membership, so that a third, genuinely independent source of information requires three or more clusters. Every result is checked numerically, including a direct verification of the two-cluster degeneracy result to floating-point precision. Applied to the Breast CancerWisconsin Diagnostic dataset (569 cases, 30 measured features), the clustering recovers the malignant/benign partition with 91.4% accuracy and an adjusted Rand index of 0.683, matching a hard k-means baseline on point accuracy; the neutrosophic layer nonetheless adds diagnostic information the hard baseline cannot provide, since indeterminacy is significantly higher for misclassified cases than for correctly classified ones (Mann–Whitney U-test, p < 10−18), correctly flagging the cases nearest the decision boundary as the ones most likely to be wrong.Wisconsin Diagnostic dataset (569 cases, 30 measured features), the clustering recovers the malignant/benign partition with 91.4% accuracy and an adjusted Rand index of 0.683, matching a hard k-means baseline on point accuracy; the neutrosophic layer nonetheless adds diagnostic information the hard baseline cannot provide, since indeterminacy is significantly higher for misclassified cases than for correctly classified ones (Mann–Whitney U-test, p < 10−18), correctly flagging the cases nearest the decision boundary as the ones most likely to be wrong.
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