Bipolar Complex Intuitionistic Fuzzy Subgroups: Axiomatic Foundations, Structural Theory, and Multicriteria Team Selection Under Hesitation and Phase Uncertainty

Existing algebraic extensions of Rosenfeld’s fuzzy subgroup encode at most two of the three representational dimensions: hesitation, bipolarity, and phase periodicity. We close this gap by introducing bipolar complex intuitionistic fuzzy (BCIF) subgroups, in which each group element carries four Cartesian-complex components constrained by intuitionistic-type inequalities in both polarities, yielding a sixteen-condition axiomatization. We then analyze the axiom system itself: the 16 real conditions are equivalent to four complex ones; the eight product conditions are mutually independent; and on a finite group each inverse condition follows from its product condition, so the system reduces to eight, while all 16 remain independent for infinite groups. We prove an eight-fold complement duality, show that lower-level subsets of anti-BCIF subgroups are crisp subgroups together with a complete converse, and obtain a necessary and sufficient condition for the fuzzification of a lower-level subgroup to remain an anti-BCIF subgroup; here the negative channels behave differently from the positive ones, a phenomenon impossible in unipolar structures. Intersection and homomorphism theorems are proved with full verification of the inherent constraints. All results are verified on Z6 and on the Klein four-group and applied to two multicriteria selection problems.