Conjugating trivial automorphisms of P(N)/Fin

Will Brian, Ilijas Farah

correctmedium confidence

Category: Not specified
Journal tier: Top Field-Leading
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the exact equivalence between: (i) CH implies conjugacy, (ii) same index parity plus elementary equivalence, and (iii) potential conjugacy (Theorem A), via Theorem 4.10 and Corollary 4.13, with the parity obstruction established in Theorem 3.6 . The candidate solution incorrectly assumes that under CH every A_α is countably (ℵ1-)saturated; the paper explicitly shows this is false when Z(f)=s_{Z×Z} (Proposition 4.14 and Theorem 4.16) . Their back-and-forth saturation argument for (2)⇒(3) and (3)⇒(1) relies on this false assumption and on an unspecified forcing, whereas the paper uses a collapse of the continuum that adds no reals plus Theorem 4.10.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} top field-leading

\textbf{Justification:}

The manuscript gives a complete and precise characterization of when CH proves conjugacy of trivial automorphisms, pinning it down to parity and first-order data and showing equivalence with potential conjugacy. The parity obstruction in ZFC and the CH-based saturation/isomorphism for rotary parts are handled deftly, and the connection to uniform Roe coronas is notable. Minor expository enhancements would improve accessibility.