  In the study of Ising models on large locally tree-like graphs, in both
rigorous and non-rigorous methods one is often led to understanding the
so-called belief propagation distributional recursions and its fixed points. We
prove that there is at most one non-trivial fixed point for Ising models with
zero or certain random external fields. Previously this was only known for
sufficiently ``low-temperature'' models. Our main innovation is in applying
information-theoretic ideas of channel comparison leading to a new metric
(degradation index) between binary-input-symmetric (BMS) channels under which
the Belief Propagation (BP) operator is a strict contraction (albeit
non-multiplicative). A key ingredient of our proof is a strengthening of the
classical stringy tree lemma of (Evans-Kenyon-Peres-Schulman'00).
  Our result simultaneously closes the following 6 conjectures in the
literature: 1) independence of robust reconstruction accuracy to leaf noise in
broadcasting on trees (Mossel-Neeman-Sly'16); 2) uselessness of global
information for a labeled 2-community stochastic block model, or 2-SBM
(Kanade-Mossel-Schramm'16); 3) optimality of local algorithms for 2-SBM under
noisy side information (Mossel-Xu'16); 4) uniqueness of BP fixed point in
broadcasting on trees in the Gaussian (large degree) limit (ibid); 5) boundary
irrelevance in broadcasting on trees (Abbe-Cornacchia-Gu-Polyanskiy'21); 6)
characterization of entropy (and mutual information) of community labels given
the graph in 2-SBM (ibid).
