GENERIC TRANSFORMATIONS HAVE ZERO LOWER SLOW ENTROPY AND INFINITE UPPER SLOW ENTROPY

The paper proves three main results: (i) a dense Gδ set of transformations has infinite upper slow entropy for any subexponential rate, (ii) a dense Gδ set has zero lower slow entropy for any diverging rate, and (iii) there exists a family of rigid, weak-mixing transformations with infinite lower slow entropy for any subexponential rate. These are stated and proved in the uploaded preprint (Theorems 1–3; see the abstract and statements around Theorem 1 for the main claims , the Gδ construction for lower slow entropy in Section 3 , the Gδ construction for upper slow entropy in Section 4 including Theorem 5 , and the construction achieving infinite lower slow entropy in Section 5 including Theorem 6 ). The candidate’s solution reaches the same three conclusions via a different route: it uses Shannon–McMillan–Breiman plus Hamming-ball bounds for part (i), periodic approximations for part (ii), and a tower-multiplexing scheme for part (iii). The model’s approach is broadly correct, but it contains a minor misstatement in part (ii): for a periodic map S and arbitrary partition P, SP(S,n,ε,δ;P) need not be bounded by the period q; it is uniformly bounded in n by a constant that depends on P and q (e.g., ≲ r^q), which is sufficient for the argument. With this correction (and a small quantifier reordering when picking N versus q), the model’s proof aligns with the paper’s conclusions. Overall: both are correct, with different proofs; the paper’s proofs are constructive via cutting-and-stacking and combinatorial lemmas (e.g., Lemmas 2.3–2.4 and the open/dense arguments), while the model uses standard entropy and density-of-conjugacy tools. Definitions adopted in the candidate match those in the paper (see Section 2 for SP, slow entropy definitions, and related notation , ).