RELATIVE DIFFERENTIAL CLOSURE IN HARDY FIELDS

The paper explicitly proves Boshernitzan’s conjecture, showing that the global intersections over maximal, maximal smooth, and maximal analytic Hardy fields all coincide (E = E∞(Q) = Eω(Q) = D(Q)), using relative differential closure and a boundedness argument specialized at H = Q; see the statement of Theorem A in the introduction and Corollary 6.8 derived from Corollary 6.7, together with the definitions of E, E∞, Eω via E(H) and Er(H) (r ∈ {∞, ω}) as intersections over maximal extensions . By contrast, the candidate solution relies on a key false equivalence (†): for Hardy fields K ⊂ L with L having DIVP, “K is differentially closed in L iff K = L.” The paper’s correct criterion is subtler (e.g., H is d-closed in E iff CE = CH and E† ∩ H = H†, or in the Hardy setting with DIVP iff E ∩ exp(H) ⊆ H), and d-closedness certainly does not force equality K = L in general . The model also confuses algebraic with differentially algebraic at several points. While the overall high-level route (reduce to a bounded base and use relative differential-closure) is close to the paper’s, the model’s proof as written is not sound.