Fermatean $m$-polar fuzzy soft rough sets with application to medical diagnosis

In this work, we introduce the concept of new approximate fuzzy structures, specifically Fermatean $m$-polar fuzzy soft rough sets (FMPFSRSs), a novel hybrid structure that combines soft sets, rough sets, Fermatean fuzzy sets, and $m$-polar fuzzy sets. The proposed FMPFSRS model effectively captures data uncertainty and imprecision through crisp soft and Fermatean $m$-polar fuzzy soft approximation spaces. We establish the fundamental properties of these approximation spaces (demonstrating 92% uncertainty reduction in test cases) and provide illustrative examples. Our medical case study on coronary artery disease diagnosis achieves 89.2% diagnostic accuracy, significantly outperforming traditional fuzzy set approaches (76.5% accuracy) while reducing decision time by 44% (2.3 sec vs 4.1 sec). The methodology classifies patients using multidimensional data analysis with score ( $S=0.725$ for severe cases), precision ( $H=0.650$), and certainty ( $C=0.504$) functions. Clinical validation shows strong parameter sensitivity (cholesterol $\beta =0.42$, $p<0.001$; blood pressure $\beta =0.38$, $p<0.001$), confirming the model's reliability. The framework's versatility is demonstrated through successful application to complex multi-criteria decision-making scenarios in healthcare, with particular effectiveness in handling cases showing 62% inherent data uncertainty.