Rough <i>I</i> -γτ-Statistical Convergence of Order α in Neutrosophic Normed Spaces

In this paper, we investigate rough <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>I</mml:mi> </mml:math> - <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>γ</mml:mi> <mml:mi>τ</mml:mi> </mml:math> -statistical convergence of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>α</mml:mi> </mml:math> in neutrosophic normed spaces and establish several fundamental structural properties of the associated limit sets. We first show that the family <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> of rough statistical limit sets is monotone with respect to the roughness parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>j</mml:mi> </mml:math> , and each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> is convex under mild monotonicity assumptions on the neutrosophic components. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> , rough convergence reduces to the classical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>I</mml:mi> </mml:math> - <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>γ</mml:mi> <mml:mi>τ</mml:mi> </mml:math> -statistical convergence of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>α</mml:mi> </mml:math> , ensuring that the limit set is a singleton. We further demonstrate that the rough limit set is always neutrosophically closed and neutrosophically convex, highlighting its stability under both topological and geometric operations. A characterization of strong <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>I</mml:mi> </mml:math> - <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>s</mml:mi> <mml:msubsup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>τ</mml:mi> </mml:mrow> <mml:mi>α</mml:mi> </mml:msubsup> </mml:math> -boundedness is obtained via the non-emptiness of the rough limit set. In addition, we introduce the notion of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> <mml:msubsup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>τ</mml:mi> </mml:mrow> <mml:mi>α</mml:mi> </mml:msubsup> </mml:math> -cluster points and prove that every rough limit point is a cluster point, while the cluster set remains neutrosophically closed. Finally, we show that this convergence framework unifies several classical notions of statistical and ideal convergences as particular cases.