A Neutrosophic Layer for Fuzzy c-Means Clustering:

Score Function Theory, Metric Properties, and

Diagnostic-Ambiguity Quantification on Breast Cancer Data

Takaaki Fujita1,*

1 Independent Researcher, Tokyo, Japan

Received: August 28, 2025 Revised: November 01, 2025 Accepted: January 02, 2026 ⋆ Corresponding author

ABSTRACT

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

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.

Keywords: Neutrosophic sets Fuzzy c-means Score function Indeterminacy Distance metric Breast cancer

diagnosis

1. INTRODUCTION

Fuzzy c-means clustering generalizes hard clustering by allowing

each data point to hold a graded membership in every

cluster rather than belonging exclusively to one; the resulting

membership vector is informative about cluster structure but

is not, by itself, a statement about how certain the assignment

is. Two points can have the same strongest membership value

and yet differ sharply in how the remaining membership mass

is distributed across the other clusters – one concentrated in a

single competitor, the other spread almost uniformly – and a

fuzzy c-means output alone does not distinguish them. The

single-valued neutrosophic set, which represents an object

by independent degrees of truth, indeterminacy, and falsity