OAI.ThreeState.regular_supercritical

The theorem states that, for every natural number b ≥ 2 and every real λ that is admissible, meaning −1/2 ≤ λ ≤ 1, if b·λ² > 1 then the sequence of regular-tree advantages reconstructs. Here a three-state spin (an element of Fin 3) is passed through a symmetric channel that keeps the spin with probability (1+2λ)/3 and moves to each other spin with probability (1−λ)/3. The observation at depth 0 is the root spin, and the observation at depth n+1 is a multiset of depth-n observations, one for each of the offspring of the root, where the offspring count is the constant b (the regular case) and each child spin is obtained by passing the parent spin through the channel and then observed recursively. With a uniformly random root spin, the posterior of each spin given an observation is its conditional probability, defaulting to 1/3 when the observation has probability zero. The advantage of a family of laws indexed by the root spin is the sum over observations y of the marginal probability of y times half the sum over spins i of |posterior(y,i) − 1/3|. The regular advantage at depth n is this advantage for the regular b-ary observation law. Reconstructs means that this sequence in n converges to some limit L > 0. The statement is an admitted theorem, with its proof left as sorry.