On the stochastic bifurcations regarding random iterations of polynomials of the form z^2 + cn

The uploaded paper’s Main Result A (Theorem 4.6) proves the exact dichotomy and equivalences the model targets, under the stated hypothesis that int B̄(c,r) contains a superattracting parameter. The proof chain in the paper is precise: (1)⇒(2) via Lemma 4.4, (2)⇒(3) by a cylinder-set/escape-radius argument plus the Brück–Büger–Reitz connectivity criterion (Theorem 3.1), (3)⇒(4) by showing boundedness of the tail critical orbits and then invoking Lemma 4.5 to identify ⋃g g(0) itself as a planar minimal set, and (4)⇒(1) by Definition 2.17; the “supercritical” (1′)–(4′) block follows from Theorems 2.16, 2.20 and the new Theorem 3.7 that upgrades T{∞}(0)=1 to typical fast escape, hence total disconnectedness almost surely . The model reproduces much of this structure but makes two substantive errors: (i) it asserts that O=⋃g g(0) is compact when bounded and then identifies cl(O) as the planar minimal set via Zorn’s lemma, whereas the paper proves the stronger and correct statement that O itself is the planar minimal set under the superattracting-parameter hypothesis (Lemma 4.5) ; and (ii) it tries to deduce “totally disconnected” from “all tail critical orbits escape almost surely” using only BBR, which at best gives “disconnected,” not total disconnectedness. The paper correctly closes this gap by a separate typical-fast-escaping criterion (Theorem 3.7) . Therefore, the paper’s argument is correct and complete for its claims; the model’s proof is incomplete/incorrect on the minimal-set identification and on the total-disconnectedness step.