THE THEORY OF MAXIMAL HARDY FIELDS

Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper establishes the exact equivalence “H is d-maximal iff H ⊇ R and H is H-closed” as Theorem 11.19, with a complete proof: d-maximal ⇒ (R ⊆ H, Liouville closed, ω-free) via Proposition 2.1 and Theorem 2.3, then 1-linear newtonianity (Corollary 11.17) and finally newtonianity (Lemma 11.18); the converse follows from [ADH, 16.0.3] (H-closed fields are existentially closed), all explicitly cited in the paper . The candidate solution’s (⇒) direction leans on informal “standard extensions” and, critically, cites the same 2024 paper as a dependency for the very equivalence being proved, rendering it circular; it also sketches real closedness via a global C1 root branch without justifying global existence in the Hardy category. The (⇐) direction is essentially correct by [ADH, 16.0.3], but the overall argument is incomplete relative to the paper’s precise chain of lemmas.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper’s theorem is proved cleanly using a precise chain of intermediate results. The model’s solution captures correct high-level ideas for constants and Liouville closure and gives a valid converse proof via existential closedness, but its forward direction both depends on the very paper under review (circular) and omits the critical steps (1-linear newtonianity and the upgrade to newtonianity) that the paper develops. It also sketches a real-closedness argument whose analytic details (global C1 branches) are not justified in the Hardy field setting. These gaps must be addressed for a complete, standalone solution.