Pseudo-monodromy and the Mandelbrot set

The paper’s Theorem 4.1 is clearly stated and proved via a careful return-time and lamination argument (using leaves, the sets Rn and Qn, Corollary 5.6, Proposition 5.7, and Proposition 5.9), yielding Π1(H) \ Ξ(H) ⊂ ⋃_{H′▷H}(TK(H′) ∪ T_{K̂(H′)⋆}) for any hyperbolic component H ≠ H♡ . The candidate solution asserts a much stronger direct classification of connected components of Π1(H) \ Ξ(H) into subwakes Π1(H′) and their adjacent intervals, and it invokes, without proof, that the endpoints of each component are exactly the characteristic angles of a unique (conspicuous) H′ and that adjacent pieces have initial blocks equal to K(H′). These steps require nontrivial orbit-portrait/landing and kneading-change facts (e.g., Theorem 2.1, Lemma on kneading change at periodic angles), which the model neither states nor justifies . Consequently, while the model reaches the correct inclusion, the proof is incomplete and relies on unproved claims. The paper’s proof is complete for the inclusion and deliberately avoids claiming equality (cf. Remark 5.8), underscoring where additional structure is conjectural .