Topologically conjugate classification of diagonal operators

Every substantive step in the candidate solution matches a numbered result in the uploaded paper and uses the same core ideas. Step 1 (phase elimination) is exactly Lemma 3.1 (the norm-preserving, coordinatewise homeomorphisms fw with fw(|w|z)=w·fw(z)) . Steps 2–3 (uniform expansion to 2I and uniform contraction to 1/2 I) reproduce Theorem 3.3 and Corollary 3.5 via the paper’s Key Lemma h_p^S and the two-step reduction through ρI (with S = (log_ρ|w_n|) and S' = constant log_ρ 2) ; the Key Lemma is stated and proved in Section 2 . Step 4 (non-conjugacy when inf|w_n|=1 vs inf|t_n|>1) is Proposition 3.6, arguing by an invariant circle when some |w_n|=1 and by a small-orbit/expansion contradiction when all |w_n|>1 but inf=1 . Minor issues in the candidate solution: (i) Step 1 omits the explicit w_n=0 subcase (handled in the paper by taking f_0=Id) , and (ii) it briefly (and inconsistently) describes the Key Lemma’s construction as “coordinatewise,” whereas the paper’s h_p^S is nonlocal in coordinates (depends on tail sums) . These do not affect the overall correctness.