Inverse theorems for randomly weighted means: Breiman's conjecture, Lévy endpoint limits, and Λ-coalescents

We study when the law of one randomly weighted mean determines structure in the underlying weights. For centered iid nonconstant integrable marks and countable subprobability weights, two signed Fourier–Mellin identities extract a positive limiting expected power sum from every nondegenerate limit, without assuming convergence of the expected total mass. At the L¹ boundary a uniform subunit power-sum bound is sufficient, while any moment above one removes it. Combined with model-specific Tauberian inversion, this settles Breiman's fixed-mark conjecture under mere integrability and classifies full-sequence endpoint limits for nonzero, unkilled, driftless subordinators with nonconstant integrable marks, assuming infinite activity at zero. The nondegenerate branch is equivalent to regular variation of the Lévy tail with index −α, 0 ≤ α < 1, and yields PD(α, 0)-weighted limits, or the mark itself when α = 0; the constant branch is equivalent to disappearance of the largest normalized jump. At the degenerate boundary this maximal-jump criterion, rather than regular variation with index −1, is exact. The same classification shows that convergence of one nonconstant mean functional in the corresponding fixed-base homogeneous class forces convergence of the entire normalized random measure. A complementary fixed-time inversion for Λ-coalescents uses finite-restriction semigroups and Hausdorff moment determinacy: one asymmetric Bernoulli-marked marginal law determines the collision measure and yields a topological embedding.