Robust Jensen–Shannon Consensus Geometry for

Single-Valued Neutrosophic Information

Nabil Salman1,* Rozina Ali2

1 University of Dijlah, Iraq

2 Cairo University, Egypt

Emails: nabil.salman@duc.edu.iq · Rozyyy123n@gmail.com

Received: September 01, 2025 Revised: November 04, 2025 Accepted: January 03, 2026 ⋆ Corresponding author

ABSTRACT

Let x = (T, I,F) ∈ [0,1]3 denote a single-valued neutrosophic assessment and let wT +wI +wF = 1 with wc > 0. This paper

introduces the probability-completed embedding

Φw(x) =

􀀀

wT T,wT (1−T),wI I,wI(1−I),

wFF,wF (1−F)

∈ Δ5.

and the pullback distance

dw(x,y) = [J(Φw(x),Φw(y))/log2]1/2 ,

where J is Jensen–Shannon divergence. The construction yields a bounded metric, separates into three weighted Bernoulli

Jensen–Shannon terms, is invariant under the neutrosophic complement xc = (F,1−I,T) when wT = wF , and has the local

information metric

d2w

(x,x+δ) =

1

8log2 Σ

c∈{T,I,F}

wcδ2

c

xc(1−xc)

+O(∥δ∥3).

On this geometry, robust consensus is posed as the bounded M-estimation problem

bxτ = argmin

x∈(0,1)3

mΣ r=1 ar{1−e−τd2w

(x,xr)}.

A majorization–minimization iteration reduces each step to three one-dimensional weighted Jensen–Shannon barycenters and

decreases the objective monotonically. The induced expert weight is proportional to e−τd2w

, so strongly conflicting assessments

are downweighted without a hard rejection threshold. In a reproducible contamination study with 15 experts, 800 replications at

each of five contamination levels, and a fixed τ = 60, the proposed estimator has mean normalized Jensen–Shannon error 0.0149

at 40% oppositional contamination; the coordinate median, ordinary Jensen–Shannon barycenter, and arithmetic mean obtain

0.0380, 0.1448, and 0.1489, respectively. The contribution is therefore a metric and optimization framework for consensus itself,

rather than another ranking operator: neutrosophic disagreement is represented on a common information-geometric scale and

robust aggregation follows from a bounded variational principle.

Keywords: Single-valued neutrosophic set Jensen–Shannon divergence Robust consensus Information geometry

Group decision making Majorization–minimization

1. INTRODUCTION

A single-valued neutrosophic assessment retains three

coordinates—truth, indeterminacy and falsity—without forcing

them to sum to one [1]. This freedom is precisely

what makes the representation useful for incomplete or internally

conflicting judgments, but it also makes consensus

less straightforward than averaging a probability vector. Recent

work has developed distances, similarities, entropies,

score functions and aggregation rules for single-valued neutro-