OBJECTIVE AND REVIEW OF ALL VERSIONS
Erdős #30; independent attempt, 2026-09-23. The target is $h(N)=\sqrt N+O_\varepsilon(N^\varepsilon)$ for maximum Sidon subsets of $[N]$. I read
30.tex and
30_v2.tex completely. The former proves the classical quarter-power error; the latter gives a valid distinct-gap refinement of a Singer embedding, conditional on the standard Singer theorem. Here I sharpen the finite short-difference inequality rather than claim another constant-size subsequence gain solves the asymptotic problem.
WORK: KEEPING THE ENDPOINT LOSSES
For $1\le a_1<\cdots<a_m\le N$ with distinct positive differences and $1\le u<m$, put $T=um-u(u+1)/2$. The supplied telescoping argument can retain the spacing of the endpoints: \[
\sum_{i=1}^{m-k}(a_{i+k}-a_i)
=\sum_{j=m-k+1}^m a_j-\sum_{j=1}^k a_j
\le kN-k^2.
\] Indeed the first sum is at most $kN-k(k-1)/2$ and the second at least $k(k+1)/2$. Distinctness gives a lower bound $T(T+1)/2$ on the combined difference sum. Consequently \[
T(T+1)\le u(u+1)\left(N-\frac{2u+1}{3}\right).
\] Equivalently, every such $m$ satisfies the explicit finite bound \[
m\le\frac{u+1}{2}+
\frac{\sqrt{1+4u(u+1)(N-(2u+1)/3)}-1}{2u}
\quad(1\le u<m).
\] This gives a direct integer certificate for ruling out proposed small Sidon configurations, retaining information discarded by the coarser $kN$ estimate.
ADVERSARIAL VERIFICATION AND REMAINING GAP
The telescoping identity remains valid even when its two endpoint blocks overlap; shared terms cancel. The inequality uses the strong Sidon convention including doubled summands, exactly the convention that makes positive differences distinct. Optimizing still takes $u$ of order $N^{1/4}$ and leaves the same error exponent. No improved exponent, new literature record, or new global lower bound has been established. A constant improvement of a lower-bound subsequence is consistent with every $O_\varepsilon(N^\varepsilon)$ conjecture.
SOURCES CHECKED
Both local versions listed above. Official indexed entry
https://www.erdosproblems.com/30, checked 2026-09-23, records stronger known quarter-power constants; the elementary finite refinement here is not advertised as surpassing them.
FINAL
UNRESOLVED. A stronger finite necessary inequality is proved, while the requested subpower error remains out of reach.
COMPLETION ESTIMATE
Conservative subjective progress toward the full error term.
COMPLETION: 12%