## 1 Introduction
Such an arrangement affords the decision-makers flexibility in assigning any MD, NMD, or \(\
eg\)MD value inside \([0,1]\). For instance, if we take \(n=3\), and \(\text{MD}=0.7\), NMD = 0.4, and \(\
eg\text{MD}=0.6\), we obtain \((0.7)^{3}+(0.4)^{3}+(0.6)^{3}\leq 1\). [PERSON] et al. [11] introduced and defined the term 'correlation coefficients of T-SFS', where levels of membership degrees of the terms should meet the constraints, i.e., \((\text{MD})^{n}+(\text{NMD})^{n}+(\
eg\text{MD})^{n}\leq 1\). This assumes homogeneity in the levels of MD, NMD, and \(\
eg\)MD. In actual scenarios, these levels might differ. So, if a decision-maker sets \(\text{MD}=0.9\), NMD = 0.5, and \(\
eg\text{MD}=0.7\), for instance, we see that \((0.9)^{3}+(0.5)^{3}+(0.7)^{3}=1.12\) implies this violate linearly aggregation rule. So, a more nuanced approach is needed. Enabling different term levels for MD, NMD, and \(\
eg\)MD makes it possible to reflect the multi-factorial reality. [PERSON] and [PERSON] [12] proposed a new extension of T-SFS, called r,s,t-spherical fuzzy sets (strt-SFSs), which lift all restrictions inflicted by classical sets. Equivalently, [PERSON] et al. [13] introduced the concept of pqr-spherical fuzzy sets (pqr-SFSs) with little restriction. We follow the latter notation in this work. In pqr-SFSs, the triplet (MD, NMD, and \(\
eg\)MD) is constrained according to \((\text{MD})^{p}+(\text{NMD})^{q}+(\
eg\text{MD})^{r}\leq 1\), where \(p,r\) are \(+\)ve integers with \(q=LCM(p,r)\). The pqr-SFSs have been studied in several manners by several researchers; see [14, 15]. [PERSON] and [PERSON] [16] also extended this concept to interval-valued pqr-spherical fuzzy sets. Bipolarity is the existence of two opposing or contrasting components, powers, or circumstances. In many areas, it refers to the state of balance between two opposing or complementary poles, commonly considered as good/bad or beneficial/non-beneficial angles. Thus, bipolarity has practical and theoretical applications in everything from psychology and political science to physics and decision-making. Bipolar fuzzy sets are a generalization of conventional fuzzy sets, which are very useful in complex real-life situations where both, \(+\)ve, and negative, \(-\)ve, feelings need to be considered. These methods can accommodate dual uncertainties, making them applicable in a variety of fields like decision-making, pattern identification, medical diagnosis, sentiment analysis, risk assessment, and AI. As the demand for higher-level models in ambiguous and uncertain situations develops, extended bipolar fuzzy sets are anticipated to play an increasingly important role in enhancing computer intelligence. The first extension in this form was the bipolar intuitionistic fuzzy set (BIFS) [17] introduced by [PERSON] and [PERSON]. By the same scenario stated in the first paragraph, the bipolar Pythagorean fuzzy set (BPyFS) [18], Fermatean bipolar fuzzy set (FBFS) [19], bipolar q-rung orthopair fuzzy set (Bq-ROFS) [20], bipolar m,n-rung orthopair fuzzy set (Bmn-ROFS) [21], and spherical bipolar fuzzy set (SBFS) [22] are introduced. Lastly, the bipolar T-spherical fuzzy set (BT-FS) [23], is a model that includes both \(+\)ve and \(-\)ve values of MD, NMD, and \(\
eg\)MD. Specifically, \(\text{MD}^{+}\), NMD\({}^{+}\), \(\
eg\text{MD}^{+}\in[0,1]\), and \(\text{MD}^{-}\), NMD\({}^{-}\), \(\
eg\text{MD}^{-}\in[-1,0]\), satisfy the conditions \(0\leq(\text{MD}^{+})^{n}+(\text{NMD}^{+})^{n}+(\
eg\text{MD}^{+})^{n}\leq 1\) and \(-1\leq(\text{MD}^{-})^{n}+(\text{NMD}^{-})^{n}+(\
eg\text{MD}^{-})^{n}\leq 0\), for some \(+\)ve integer, \(n\geq 2\). This requirement forces decision-makers to give an equal \(n\) to MD, NMD, and \(\
eg\)MD. In some cases, decision-makers may have to assign different levels of authority to the MD, NMD, or \(\
eg\)MD. Such requirements cannot be catered to by BT-SFSs.