Classical criteria for tensor decomposition uniqueness, such as Kruskal's theorem, rely primarily on linear-algebraic dimension bounds, largely bypassing the structural rigidity enforced by entrywise nonnegativity. This paper introduces a positive scattering term that leverages non-cancellation and support constraints to certify identifiability. By establishing a positive splitting inequality, the authors prove two distinct component thresholds (2|S|-2 for rank minimality and 2|S|-1 for uniqueness) reducible to graph connectivity checks, successfully certifying sparse nonnegative tensor decompositions where generalized Kruskal conditions fail.
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