Hierarchical symmetry selects log-Poisson cascades: classification, uniqueness, and stability

Within i.i.d. multiplicative cascades, a single axiom – the hierarchical symmetry, a linear contraction on incremental scaling exponents – is shown to be necessary and sufficient for the cascade multiplier to be log-Poisson. We establish three results: (1) a characterization theorem proving that the hierarchical symmetry uniquely determines the log-Poisson distribution with explicit parameters; (2) a classification theorem proving that the hierarchical symmetry selects exactly the log-Poisson class from the full log-infinitely-divisible family, excluding log-normal, log-stable, and all intermediate generators; and (3) a stability theorem proving that approximate hierarchical symmetry implies approximate log-Poisson, with an explicit $O(\sqrt{\epsilon })$ Wasserstein bound. The proofs reduce the problem to the Hausdorff moment problem on $[0,1]$ via the change of variables $u=e^{kx}$, where determinacy and stability follow from classical results.