The Poisson boundary of Thompson’s group T is not the circle

The uploaded paper proves that for minimal, proximal, topologically nonfree actions on S1 and any nondegenerate μ with finite entropy, the circle (S1, ν) is not the Poisson boundary, via Kaimanovich’s conditional entropy criterion and a conditional Erschler method, supported by new dynamical inputs (dominating intervals) and an “invisibility” lemma tailored to boundary conditioning. This matches Theorem A and its proof sketch in the paper and appears sound. By contrast, the model’s argument misuses an erasure map E: it asserts ℬ ⊂ σ(E) (boundary measurability through E) using a false “eventual equality” of state sequences after infinitely many block deletions, and it then incorrectly claims H(Yi|E)=HB(p) although Yi is determined by E (so H(Yi|E)=0). Hence the model’s conditional-entropy lower bound fails, and its proof is invalid. See Theorem A and the proof architecture in the paper, including the conditional entropy criterion (Theorem 2.6), the Erschler-style conditional method, and key lemmas/propositions addressing invisibility and frequency bounds.