The limit point in the Jante’s law process has a continuous distribution

Edward Crane, Stanislav Volkov

correcthigh confidence

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

Audit review

The paper rigorously proves continuity of the limit by (i) precise inner/outer ball control of Keep(S;B), (ii) supermartingale and compactness arguments ensuring all original points are removed, and (iii) a coupling/mixture-of-continuous-distributions scheme (with a boundary shell estimate), culminating in Theorem 2. The candidate outline matches the paper on the Keep-geometry but makes two critical errors: an unjustified pathwise “locality” bound (used to control the future by the current radius) and an invalid step replacing P(ξ∈A|S) by a uniform-in-Keep volume ratio bound. These gaps prevent the candidate argument from establishing absolute continuity, while the paper’s proof is complete and correct (see Theorem 2 and its construction via Theorem 1 and Section 6).

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript settles a natural and previously conjectured property in a well-studied rank-driven process. The proof architecture is elegant—reducing to a scale-free core, establishing continuity there via a translation-invariance/mixture argument, and lifting to general convex bodies with a coupling and an inner-shell bound. While technically dense, the exposition is well organized. A few minor clarifications (restating core geometric inclusions when used, motivating the inner-shell lemma) would enhance readability.