EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7120 ISSN 1307-5543 – ejpam.com Published by New York Business Global Neutrosophic Statistical Manifolds: A Unified Framework for Information Geometry with Uncertainty Quantification Abed Al-Rahman M. Malkawi1,∗, Ayat M. Rabaiah1 1 Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman 11953, Jordan Abstract. This paper introduces a novel framework integrating neutrosophic logic with infor- mation geometry, establishing the foundation of neutrosophic statistical manifolds. We define a neutrosophic MR-metric structure on statistical manifolds, incorporating truth, indeterminacy, and falsity membership functions to quantify distributional similarity, epistemic uncertainty, and dissimilarity. The proposed structure generalizes the Fisher–Rao metric through a symmetric triple-based formulation using Jensen–Shannon divergence. We prove that the triplet (T , I,F) satisfies all axioms of a neutrosophic MR-metric space and derive explicit relations between the contraction constant R and the curvature of the underlying statistical manifold. Several ap- plications are explored, including Gaussian and categorical models, hypothesis testing, model selection, geometric machine learning, and quantum information geometry. This work bridges fixed-point theory in generalized metric spaces with statistical inference under uncertainty, of- fering a robust tool for uncertainty-aware data analysis. 2020 Mathematics Subject Classifications: 53B12, 62B11, 46S50, 03E72, 68T37 Key Words and Phrases: Neutrosophic logic, information geometry, MR-metric spaces, Jensen–Shannon divergence, statistical manifolds, uncertainty quantification, fixed point theory 1. Introduction The study of generalized metric spaces has been a fertile area of research in pure and applied mathematics, with significant contributions to fixed-point theory and its applications. The concept of b-metric spaces was introduced by Bakhtin [1] and later formalized by Czerwik [2], providing a framework for handling non-linear contraction mappings. Subsequent extensions, such as Gb-metric spaces and Ω-distance mappings, have further enriched the theory [3–11]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7120 Email addresses: a.malkawi@aau.edu.jo, math.malkawi@gmail.com (A. Malkawi), a.rabaieha@aau.edu.jo (A. Rabaiah) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 2 of 16 In parallel, the notion of MR-metric spaces was introduced by Malkawi et al. [12] as a generalization of standard metric spaces, enabling the analysis of triple-based geomet- ric structures. This has led to numerous fixed-point results under various contraction conditions [13–23]. Recent work has also explored the intersection of metric spaces with fuzzy and neutrosophic logic. For instance, Hazaymeh and Bataihah [24] and Bataihah and Hazaymeh [25] introduced neutrosophic fuzzy metric spaces, while Malkawi [26, 27] extended fixed-point theory to neutrosophic MR-metric settings. On the other hand, information geometry—the study of statistical manifolds endowed with the Fisher–Rao metric—has provided deep insights into the geometric structure of probability distributions. Divergence measures such as the Kullback–Leibler divergence and Jensen–Shannon divergence play a central role in this field. Recent work by Malkawi and Rabaiah [28, 29] has begun to explore the connections between MR-metric spaces and information-theoretic divergences. This paper unifies these two streams of research by introducing neutrosophic sta- tistical manifolds—a structure that combines the triple-based geometry of MR-metric spaces with the uncertainty-handling capabilities of neutrosophic logic. Our work is also influenced by applications of generalized metric spaces to fractional differential equations [30–39] and cyclic mappings [5, 11, 40]. The main contributions of this paper are: • The definition of a neutrosophic MR-metric structure on statistical manifolds. • A proof that the triplet (T , I,F) satisfies neutrosophic metric axioms. • Explicit links between the contraction constant R and curvature. • Detailed examples and applications in statistics, machine learning, and quantum information. Definition 1. [12] Consider a non-empty set X 6= ∅ and a real number R > 1. A function M : X× X× X → [0,∞) is termed an MR-metric if it satisfies the following conditions for all v, ξ, s, `1 ∈ X: • M(v, ξ, s) ≥ 0. • M(v, ξ, s) = 0 if and only if v = ξ = s. • M(v, ξ, s) remains invariant under any permutation p(v, ξ, s), i.e., M(v, ξ, s) = M(p(v, ξ, s)). • The following inequality holds: M(v, ξ, s) ≤ R [M(v, ξ, `1) +M(v, `1, s) +M(`1, ξ, s)] . A structure (X,M) that adheres to these properties is defined as an MR-metric space. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 3 of 16 Definition 2. [27][Neutrosophic MR-Metric Space (NMR-MS)] A 9-tuple (Z,M, T ,F , I, •, �, R, ?) is called a Neutrosophic MR-Metric Space if: (i) Z is a non-empty set. (ii) M : Z × Z ×Z → [0,∞) is an MR-metric satisfying: (M1) M(υ, ξ,=) ≥ 0, (M2) M(υ, ξ,=) = 0 ⇐⇒ υ = ξ = =, (M3) Symmetry under permutations, (M4) M(υ, ξ,=) ≤ R [M(υ, ξ, `) ? M(υ, `,=) ? M(`, ξ,=)], R > 1. (iii) T ,F , I : Z × Z × (0,∞) → [0, 1] are neutrosophic functions satisfying: (N1) T (υ, ξ, γ) = 1 ⇐⇒ υ = ξ (Truth-Identity), (N2) T (υ, ξ, γ) = T (ξ, υ, γ) (Symmetry), (N3) T (υ, ξ, γ) • T (ξ,=, ρ) ≤ T (υ,=, γ + ρ) (Triangle Inequality), (N4) limγ→∞ T (υ, ξ, γ) = 1 (Asymptotic Behavior). (iv) • (t-norm) and � (t-conorm) are continuous operators generalizing fuzzy logic. (v) ? is a binary operation generalizing addition (e.g., weighted sum). 2. Main Results This section presents the core theoretical contributions of this work. We begin by introducing the neutrosophic statistical structure defined on smooth manifolds of prob- ability distributions. This framework integrates an MR-metric constructed from the Jensen–Shannon divergence with neutrosophic membership functions designed to quan- tify truth, indeterminacy, and falsity. We subsequently establish the mathematical con- sistency of this structure and investigate its geometric characteristics, particularly its connections to the Fisher–Rao metric and the α-connections in information geometry. Theorems and lemmas are formulated to rigorously characterize these relationships, ac- companied by proofs that validate the fulfillment of all neutrosophic metric axioms. Definition 3 (Statistical Manifold with Neutrosophic Structure). Let P be a smooth manifold of probability distributions p(·; θ) parameterized by θ = (θ1, . . . , θn) ∈ Θ ⊆ Rn. A neutrosophic statistical structure on P is given by: (i) Neutrosophic MR-Metric: For p, q, r ∈ P, define: M(p, q, r) = DJS(p‖q) +DJS(q‖r) +DJS(r‖p) where DJS is the Jensen-Shannon divergence: DJS(p‖q) = 1 2 DKL ( p ∥∥∥∥p+ q 2 ) + 1 2 DKL ( q ∥∥∥∥p+ q 2 ) A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 4 of 16 (ii) Neutrosophic Membership Functions: T (p, q, γ) = exp (−γ · JSD(p‖q)) I(p, q, γ) = 1− ∣∣∣∣H(p)−H(q) maxr∈P H(r) ∣∣∣∣ · exp (−γ · |DKL(p‖u)−DKL(q‖u)|) F(p, q, γ) = 1− T (p, q, γ)− I(p, q, γ) where H(p) is the Shannon entropy, u is the uniform distribution, and JSD is the normalized Jensen-Shannon divergence. (iii) Operations: The t-norm • is the product t-norm a • b = ab, and ? is the standard addition. Theorem 1 (Well-Defined Neutrosophic Structure). The triplet (T , I,F) defines a valid neutrosophic structure on the statistical manifold P, satisfying all axioms of a Neutro- sophic MR-Metric Space. Proof. We prove each axiom systematically: Part 1: Neutrosophic Axioms Verification (i) Truth-Identity: T (p, q, γ) = 1 ⇐⇒ p = q T (p, q, γ) = 1 ⇐⇒ exp (−γ · JSD(p‖q)) = 1 ⇐⇒ JSD(p‖q) = 0 ⇐⇒ p = q (since JSD is a metric) (ii) Symmetry: T (p, q, γ) = T (q, p, γ) T (p, q, γ) = exp (−γ · JSD(p‖q)) = exp (−γ · JSD(q‖p)) = T (q, p, γ) since JSD is symmetric. (iii) Triangle Inequality: T (p, q, γ) • T (q, r, ρ) ≤ T (p, r, γ + ρ) Using the product t-norm a • b = ab: T (p, q, γ) • T (q, r, ρ) = exp (−γ · JSD(p‖q)) · exp (−ρ · JSD(q‖r)) = exp (−γ · JSD(p‖q)− ρ · JSD(q‖r)) Since JSD is a metric, it satisfies the triangle inequality: JSD(p‖r) ≤ JSD(p‖q) + JSD(q‖r) Therefore: T (p, r, γ + ρ) = exp (−(γ + ρ) · JSD(p‖r)) ≥ exp (−(γ + ρ) · [JSD(p‖q) + JSD(q‖r)]) A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 5 of 16 = exp (−γ · JSD(p‖q)− ρ · JSD(q‖r)) · exp (−ρ · JSD(p‖q)− γ · JSD(q‖r)) Since exp (−ρ · JSD(p‖q)− γ · JSD(q‖r)) ≤ 1 for all γ, ρ > 0 and p, q, r ∈ P, we have: T (p, r, γ + ρ) ≥ exp (−γ · JSD(p‖q)− ρ · JSD(q‖r)) = T (p, q, γ) • T (q, r, ρ) (iv) Asymptotic Behavior: lim γ→∞ T (p, q, γ) = { 1 if p = q 0 if p 6= q This follows because: lim γ→∞ exp (−γ · JSD(p‖q)) = { exp(0) = 1 if JSD(p‖q) = 0 (i.e., p = q) 0 if JSD(p‖q) > 0 (i.e., p 6= q) Part 2: Indeterminacy and Falsity Properties (i) Indeterminacy Bounds: 0 ≤ I(p, q, γ) ≤ 1 I(p, q, γ) = 1− ∣∣∣∣H(p)−H(q) maxr∈P H(r) ∣∣∣∣ · exp (−γ · |DKL(p‖u)−DKL(q‖u)|) Since 0 ≤ ∣∣∣H(p)−H(q) maxH ∣∣∣ ≤ 1 and 0 ≤ exp(·) ≤ 1, we have 0 ≤ I(p, q, γ) ≤ 1. (ii) Consistency: T (p, q, γ) + I(p, q, γ) + F(p, q, γ) = 1 by construction. This completes the proof that (T , I,F) forms a valid neutrosophic structure on the statistical manifold P. Theorem 2 (MR-Metric as Symmetric Fisher-Rao Analog). The MR-metric M is a symmetric generalization of the Fisher-Rao metric, with the following relations: (i) Local Expansion: For infinitesimally close distributions p(θ), p(θ + dθ), and p(θ + dφ): M(p(θ), p(θ + dθ), p(θ + dφ)) = 1 2 [ gij(θ) dθ idθj + gij(θ) dφ idφj + gij(θ) (dθ i − dφi)(dθj − dφj) ] +O(‖dθ‖3), where gij is the Fisher–Rao metric tensor. (ii) Curvature Relation: The contraction constant R is bounded by: R ≥ 1 + 1 4 max p,q,r∈P R(p, q, r)√ g(p, p)g(q, q)g(r, r) where R is the sectional curvature tensor of the statistical manifold. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 6 of 16 Proof. Part 1: Local Expansion and Fisher-Rao Connection Consider the Taylor expansion of DJS around θ: For p = p(θ), q = p(θ + dθ), r = p(θ + dφ), we have: DJS(p‖q) = 1 8 gij(θ)dθ idθj +O(‖dθ‖3) DJS(q‖r) = 1 8 gij(θ)(dφ i − dθi)(dφj − dθj) +O(‖dφ− dθ‖3) DJS(r‖p) = 1 8 gij(θ)dφ idφj +O(‖dφ‖3) Therefore: M(p, q, r) = DJS(p‖q) +DJS(q‖r) +DJS(r‖p) = 1 8 [ gijdθ idθj + gij(dφ i − dθi)(dφj − dθj) + gijdφ idφj ] +O(‖d‖3) = 1 4 [ gijdθ idθj + gijdφ idφj + 1 2 gij(dθ i − dφi)(dθj − dφj) ] +O(‖d‖3) This shows that M captures the Fisher-Rao geometry in a symmetric, triple-based formulation. Part 2: Curvature and Contraction Constant Using the generalized triangle inequality for M : M(p, q, r) ≤ R [M(p, q, s) +M(p, s, r) +M(s, q, r)] For infinitesimal triangles, the worst-case ratio occurs when the manifold has maxi- mum sectional curvature. By the generalized law of cosines on Riemannian manifolds: For a geodesic triangle with vertices p, q, r and a point s on the geodesic between p and r, we have: d2(p, r) = d2(p, q) + d2(q, r)− 2d(p, q)d(q, r) cos(∠pqr) + 1 3 RijklX iY jXkY l +O(d5) where X,Y are tangent vectors, and Rijkl is the Riemann curvature tensor. Translating this to our MR-metric context: M(p, q, r) = 3 2 [ d2(p, q) + d2(q, r) + d2(r, p) ] +O(d4) M(p, q, s) +M(p, s, r) +M(s, q, r) = 3 2 [ d2(p, q) + d2(q, s) + d2(s, p) + · · · ] The curvature correction appears at fourth order. Maximizing over all configurations gives: A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 7 of 16 R ≥ 1 + 1 4 max R(X,Y,X, Y ) ‖X‖2‖Y ‖2 − 〈X,Y 〉2 where the maximum is taken over all linearly independent tangent vectors X,Y . Lemma 1 (Differential Geometric Structure). The neutrosophic statistical manifold (P,M, T , I,F) inherits a rich differential geometric structure: (i) Connection: The α-connection ∇(α) is related to the neutrosophic structure via: Γ (α) ij,k = Ep [∂i∂j`p · ∂k`p] + 1− α 2 Ep [∂i`p · ∂j`p · ∂k`p] where `p = log p. (ii) Divergence: The Jensen-Shannon divergence is a symmetric Bregman divergence: DJS(p‖q) = 1 2 [DKL(p‖m) +DKL(q‖m)] , m = p+ q 2 Proof. Connection Structure: The Fisher-Rao metric is: gij(p) = Ep [∂i`p · ∂j`p] The α-connection coefficients are: Γ (α) ij,k(p) = Ep [ (∂i∂j`p + 1− α 2 ∂i`p∂j`p)∂k`p ] For our neutrosophic structure, the MR-metric M induces a connection that inter- polates between the α = −1 (mixture) and α = 1 (exponential) connections, with the indeterminacy I quantifying the uncertainty in this interpolation. Divergence Properties: The Jensen-Shannon divergence has the key properties: • Symmetry: DJS(p‖q) = DJS(q‖p) • Positivity: DJS(p‖q) ≥ 0 with equality iff p = q • Convexity: Jointly convex in p and q • Boundedness: 0 ≤ DJS(p‖q) ≤ log 2 These properties ensure the well-definedness of our neutrosophic structure. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 8 of 16 Theorem 3 (Detailed Curvature-Contraction Relation). The contraction constant R in the NMR-MS structure is explicitly related to the curvature of the statistical manifold: R = 1 + 1 2 max p∈P X,Y ∈TpP R(X,Y,X, Y ) +R(X,Y, Y,X) ‖X‖2‖Y ‖2 − 〈X,Y 〉2 + ε(I) where ε(I) is a correction term depending on the indeterminacy: ε(I) = 1 4 Ep,q,r[I(p, q, γ) + I(q, r, γ) + I(r, p, γ)] Proof. We analyze the curvature effects through several steps: Step 1: Riemannian Geometry Framework Consider the statistical manifold as a Riemannian manifold (P, g) with Fisher-Rao metric. The sectional curvature for a 2-plane spanned by orthonormal vectors X,Y is: K(X,Y ) = R(X,Y,X, Y ) Step 2: MR-Metric Expansion For small geodesic triangles, expand M using the metric and curvature: M(p, q, r) = 3 2 [ d2(p, q) + d2(q, r) + d2(r, p) ] − 1 8 [K(X,Y ) +K(Y, Z) +K(Z,X)] ·Area2 +O(d6) where X,Y, Z are tangent vectors along the triangle edges. Step 3: Worst-Case Contraction The contraction inequality becomes tightest for triangles maximizing the curvature terms. The worst-case ratio is: M(p, q, r) M(p, q, s) +M(p, s, r) +M(s, q, r) ≤ 1 + 1 2 maxK +O(d2) Taking the supremum over all configurations gives the stated bound. Step 4: Indeterminacy Correction The neutrosophic indeterminacy I introduces additional uncertainty in the metric relations. This can be modeled as a stochastic correction to the curvature: R̃ = R+ δR, ‖δR‖ ∝ E[I] This leads to the ε(I) correction term, which quantifies how epistemic uncertainty affects the geometric structure. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 9 of 16 3. Applications and Examples Having established the theoretical foundations of neutrosophic statistical manifolds, we now turn to their practical implications. The following section provides detailed examples and applications across a range of domains. We explore Gaussian and categor- ical statistical manifolds, demonstrate how the neutrosophic structure enhances model selection and hypothesis testing, and illustrate its utility in geometric machine learning and quantum information geometry. Each example includes explicit computations and visualizations to aid intuition and demonstrate applicability. 3.1. Gaussian Statistical Manifold Example 1 (Univariate Gaussian Distributions). Consider the family of univariate Gaussian distributions parameterized by θ = (µ, σ): p(x;µ, σ) = 1√ 2πσ exp ( −(x− µ)2 2σ2 ) . • Fisher–Rao Metric: The metric tensor in coordinates (µ, σ) is: ds2 = gµµdµ 2 + 2gµσdµdσ + gσσdσ 2 = 1 σ2 dµ2 + 2 σ2 dσ2. This induces a hyperbolic geometry on the half-plane (µ, σ) ∈ R× (0,∞). • Neutrosophic MR-Metric: For three Gaussians p, q, r, we compute: M(p, q, r) = DJS(p‖q) +DJS(q‖r) +DJS(r‖p). For infinitesimally close distributions p(µ, σ), q(µ+dµ, σ+dσ), r(µ+dµ′, σ+dσ′), a second-order expansion yields: M(p, q, r) ≈ 1 4 [ gijdθ idθj + gijdφ idφj + 1 2 gij(dθ i − dφi)(dθj − dφj) ] , confirming the local dominance of the Fisher–Rao geometry. • Neutrosophic Membership Functions: – Truth-Membership: Measures similarity via JSD. T (p, q, γ) = exp (−γ · JSD(p‖q)) . For example, if p = N (0, 1), q = N (0.1, 1.1), then JSD(p‖q) ≈ 0.0023, so T (p, q, 10) ≈ exp(−0.023) ≈ 0.977. – Indeterminacy-Membership: Captures entropy and divergence differences. I(p, q, γ) = 1− ∣∣∣∣H(p)−H(q) maxr∈P H(r) ∣∣∣∣ · exp (−γ · |DKL(p‖u)−DKL(q‖u)|) , where H(p) = 1 2 ln(2πeσ 2) and u is the uniform distribution over a sufficiently large interval. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 10 of 16 – Falsity-Membership: Defined as F = 1− T − I. • Curvature and Contraction Constant: The Gaussian manifold has constant negative sectional curvature K = −1 2 . Applying Theorem 3: R ≥ 1 + 1 4 max |R| √ gppgqqgrr ≈ 1 + 1 4 · 1/2 1 = 1.125. This indicates a mild contraction requirement due to the hyperbolic geometry. µ σ p q r K = −1 2 3.2. Categorical Distributions (Simplex Geometry) Example 2 (Finite Discrete Distributions). Let P = {p = (p1, . . . , pn) : pi > 0, ∑ pi = 1} be the (n− 1)-dimensional probability simplex. • Fisher–Rao Metric: This is the spherical metric induced by the embedding pi = x2i with ∑ x2i = 1. The metric is: ds2 = 4 n∑ i=1 dx2i = n∑ i=1 dp2i pi . The manifold is a portion of a sphere with radius 2, hence has constant positive curvature. • Neutrosophic Structure: – MR-Metric: For categorical distributions, DJS has a closed form. For example, for n = 3, let p = (0.5, 0.3, 0.2), q = (0.4, 0.4, 0.2), r = (0.6, 0.2, 0.2). Then: M(p, q, r) = DJS(p‖q)+DJS(q‖r)+DJS(r‖p) ≈ 0.024+0.018+0.022 = 0.064. – Truth-Membership: T (p, q, γ) = exp(−γ·JSD(p‖q)). For γ = 10, T (p, q, 10) ≈ exp(−10 · 0.008) ≈ 0.923. – Indeterminacy-Membership: Reflects entropy differences. For p and q above, H(p) ≈ 1.029,H(q) ≈ 1.055, so the entropy difference is small, leading to high indeterminacy if the distributions are otherwise distinct. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 11 of 16 • Contraction Constant: For the positive-curvature simplex, the contraction con- stant R is larger. The curvature is K = 1 4 , so: R ≥ 1 + 1 4 · 1/4 1 = 1.0625. However, the presence of boundaries (some pi → 0) increases the effective R in practice. 3.3. Exponential Family and α-Connections Example 3 (Exponential Family). Consider an exponential family: p(x; θ) = exp (θ · T (x)−A(θ) + lnh(x)) . • Fisher–Rao Metric: gij(θ) = ∂i∂jA(θ). • Neutrosophic α-Connection: The α-connection coefficients are: Γ (α) ij,k(θ) = 1− α 2 ∂i∂j∂kA(θ). Our neutrosophic structure naturally incorporates this via the indeterminacy func- tion I, which can be linked to the deviation from the Levi-Civita connection (α = 0). For instance, define a weighted indeterminacy: I(α)(p, q, γ) = |α| · (1− T (p, q, γ)) + (1− |α|) · I(p, q, γ). This blends the ”geometric uncertainty” (α-deviation) with the ”information un- certainty” (entropy differences). 3.4. Application to Model Selection and Hypothesis Testing Corollary 1 (Neutrosophic Bayesian Information Criterion (NBIC)). In model selection, the standard BIC is BIC = −2 lnL+ k lnn. We propose a neutrosophic adjustment: NBIC = −2 lnL+ k lnn+ λ · (1− E[T ]− E[I]), where E[T ] and E[I] are average truth and indeterminacy memberships over the model’s parameter space, and λ is a tuning parameter. This penalizes models with high epistemic uncertainty or low truth membership (poor fit). Example 4 (Hypothesis Testing). Consider testing H0 : θ = θ0 vs. H1 : θ 6= θ0. The classical p-value can be enriched with neutrosophic memberships: • Truth-Membership (T ): Likelihood of data under H0. • Indeterminacy-Membership (I): Function of the Fisher information at θ0; high indeterminacy suggests the test is less informative. • Falsity-Membership (F): Evidence against H0. A decision rule could be: Reject H0 if F > τF and I < τI , where τF , τI are thresholds. A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 12 of 16 3.5. Geometric Machine Learning [Uncertainty-Aware Deep Learning] In variational autoencoders (VAEs), the latent space is often a Gaussian manifold. Our neutrosophic structure can quantify uncertainty in the latent representations: • Let z1, z2 be latent codes for two inputs. • Define T (z1, z2, γ) based on their JSD in the data space (via decoder). • Define I(z1, z2, γ) based on entropy of the latent distributions. • The triplet (T , I,F) provides a nuanced similarity measure for clustering or anomaly detection. Example 5 (Neutrosophic t-SNE). Modify the t-SNE algorithm to use the neutrosophic MR-metric M instead of Euclidean distance. The joint probabilities become: Pij = exp ( −M(pi, pj , pref)/2σ 2 )∑ k 6=l exp (−M(pk, pl, pref)/2σ2) , where pref is a reference distribution. This incorporates three-way relationships and uncertainty into the visualization. 3.6. Physical and Quantum Applications Remark 1 (Quantum Information Geometry). In quantum mechanics, states are density matrices ρ. The Jensen–Shannon divergence can be extended to quantum JSD [M. B. et al.]. Our neutrosophic framework then applies to the manifold of quantum states: • MR-Metric: M(ρ, σ, τ) = DJS(ρ‖σ) +DJS(σ‖τ) +DJS(τ‖ρ). • Truth-Membership: T (ρ, σ, γ) = exp(−γ ·DJS(ρ‖σ)). • Indeterminacy-Membership: Can be linked to quantum entropy S(ρ) = − tr(ρ ln ρ) and coherence measures. • Contraction Constant R: Related to the curvature of the Bures metric, which is the quantum analog of Fisher–Rao. This provides a novel tool for analyzing quantum phase transitions and decoherence. 4. Conclusions This paper has introduced a comprehensive framework for neutrosophic statistical manifolds, bridging the gap between information geometry and neutrosophic logic. Our main contributions can be summarized as follows: A. Malkawi, A. Rabaiah / Eur. J. Pure Appl. Math, 18 (4) (2025), 7120 13 of 16 • We defined a novel neutrosophic MR-metric structure on statistical manifolds, incorporating truth (T ), indeterminacy (I), and falsity (F) membership functions to quantify distributional similarity, epistemic uncertainty, and dissimilarity. • We proved that the triplet (T , I,F) satisfies all axioms of a Neutrosophic MR- Metric Space, with particular attention to the corrected asymptotic behavior where limγ→∞ T (p, q, γ) = 0 for p 6= q and 1 for p = q. • We established explicit relations between the contraction constant R and the curva- ture of the underlying statistical manifold, demonstrating how geometric properties influence the metric structure. • We provided detailed applications across multiple domains including Gaussian and categorical statistical manifolds, hypothesis testing, model selection, geometric machine learning, and quantum information geometry. • The proposed NBIC (Neutrosophic Bayesian Information Criterion) offers a novel approach to model selection that incorporates epistemic uncertainty quantification. 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