Fully Coupled One-Factor Jaccard Paths for Interval-Valued Fuzzy Soft Profiles: Exact Extrema and Ranking Transitions

This paper studies Jaccard similarity under a fully coupled one-factor scoring policy in which all coordinates of an interval-valued fuzzy soft profile are governed by the same parameter. Each interval runs from confidence-discounted evidence to the value obtained when that evidence is accepted in full. In contrast to the usual independent coordinate envelope, the coordinates move together. We prove that the exact extrema can be found from finitely many coordinate crossings and that every possible change in pairwise ranking is determined by breakpoint evaluations and polynomial equations of degree at most two. The method is illustrated in a single-corpus six-character proof of concept based on a traceable positive evidence analysis of Elif Shafak’s The Saint of Incipient Insanities. The case study yields seven positive pair similarities and two exact ranking transitions. It also shows that some similarity values and rankings change when normalization, weighting, parameter selection, or coding decisions change. Thus, this paper provides an exact way to study similarity and ranking stability under a common scoring policy while keeping the literary conclusions within the limits of the analyzed corpus.