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Jeff Pickhardt, New Bounds for Sequences with Distinct Consecutive Sums in Erdős Problem 357, a manuscript of five pages whose title page names the Paratelligent Research Agent and Jeff Pickhardt as its authors and is dated 14 July 2026; it was published on paratelligent.com on 31 August 2026 and linked the same day in the discussion thread of Problem 357 by the account JPickhardt, whose comment says that the bounds were found while trying, without success, to prove the problem. The acknowledgment says that manuscript development used the Paratelligent Research Agent. With $f(n)$ the problem's function, the largest $k$ for which some $1\le a_1<\cdots<a_k\le n$ has all sums of consecutive terms distinct, Theorem 1.1 states that, as $n\to\infty$,

$$\Big(\frac4{\sqrt3}-o(1)\Big)\sqrt n\le f(n)\le\frac n2 +\Big(\frac32\,2^{-2/3}+o(1)\Big)n^{2/3}.$$

Lower bound (Theorem 2.1 and Corollary 2.2). Fix an integer $B\ge1$ and let $0\le y_1<\cdots<y_k$ be the first $k$ nonnegative integers not congruent to $-B$ modulo $3$; if $B>3k^2/16+k/8+1$ then $a_i=B+y_i$ has all consecutive sums distinct. The congruences exclude collisions between intervals whose lengths differ by one, and a size estimate excludes every larger difference. Taking $B=N^2-4N$ and $k=\lfloor cN\rfloor$ for $c<4/\sqrt3$ keeps the terms in $[1,N^2]$, which gives the lower bound. Upper bound (Lemma 3.1 and Theorem 3.2). For every integer $R\ge1$,

$$f(n)\le\Big\lceil\frac n2\Big\rceil+\Big\lfloor\frac n{2R}\Big\rfloor +\frac{R(R-1)}2,$$

because the terms in $(n/(2r),n/r]$, for $1\le r\le R$, form a contiguous block of indices, so their sums over $r$ consecutive terms are distinct integers in $(n/2,n]$; the choice $R=\lceil(n/2)^{1/3}\rceil$ gives the stated second-order term, with $(3/2)2^{-2/3}\approx0.945$. The manuscript says that the increasing order is used at exactly this point, which is why the bound improves on the one inherited from the unrestricted form, and that whether $f(n)=o(n)$ remains open.

Covers. Both bounds. The lower bound improves the constant $2$ of the construction the site records (from Problem 874) to $4/\sqrt3\approx2.309$. The upper bound has leading constant $1/2$ and second-order constant $(3/2)2^{-2/3}$, below the $9\cdot2^{-7/3}\approx1.786$ of Lenthall-Cleary's claim of 27 July 2026, which proves an upper bound of the same shape by a different packing argument; the manuscript's printed date precedes that claim and its posting follows it. It does not answer whether $f(n)=o(n)$, the problem's question.

Standing. Claimed: the manuscript is published on the Paratelligent site, not on a preprint server, has no refereed version and no Lean development, and was linked as a thread comment rather than filed on the proof-claims tab; the site's label was OPEN on 2026-10-07 (page last edited 12 January 2026), with no reply to the comment on the thread, and no independent review was found on 2026-10-07. This corpus has not reviewed the proofs.