GPT 6 Astra Ultra attempt on Math-929

Erdős Problem #773: independent continuation

Date: 2026-09-23. Research attempt; no claim of a new published result.

1) TARGET AND SOURCE AUDIT

Estimate the largest size $f(N)$ of a standard Sidon subset of $\{1^2,\ldots,N^2\}$, where sums allowing a repeated summand must also be distinct.

All supplied versions considered: 773.tex. The source repairs weak versus standard Sidonicity using a sharp half-size conversion theorem. I give a direct alteration proof that handles three- and four-vertex collisions together, so that the extraction step no longer depends on that conversion theorem or on an unavailable earlier round.

2) WORK

Direct standard-Sidon extraction.

Let $A$ be a set of $n$ real numbers such that every real number has at most $k\ge1$ representations as a sum of two distinct elements of $A$. Then $A$ has a standard Sidon subset of size at least \[ {3\over4}k^{-1/3}n^{2/3}-1. \tag{1} \] For each sum $s$, write $r(s)$ for its number of unordered distinct-pair representations. Distinct representations of the same sum cannot share a vertex. Thus the number of four-vertex collision configurations is at most \[ \sum_s {r(s)\choose2}\le {k-1\over2}\sum_s r(s) \le {kn^2\over4}. \] Three-vertex collisions are precisely $x+z=2y$ with $x<y<z$. Each midpoint $y$ admits at most $k$ endpoint pairs, so there are at most $kn$ such configurations.

Select each element independently with probability $p=(kn)^{-1/3}$. Delete at most one element from every surviving collision configuration. No deletion creates another collision. The expected surviving size after these deletions is at least \[ np-{kn^2\over4}p^4-knp^3 ={3\over4}k^{-1/3}n^{2/3}-1. \] Some realization attains this bound, proving $1$. The list of configurations is exhaustive: equal sums of distinct pairs either involve four distinct elements, or cancellation identifies the pair; one repeated pair gives a three-term progression; two repeated pairs force the same midpoint.

Applying the lemma to the representation bound $k\le N^{3\epsilon}$ quoted for squares in the supplied source recovers $f(N)\gg_\epsilon N^{2/3-\epsilon}$. That application is conditional on that representation estimate; the extraction lemma itself is entirely proved above. It does not improve the stronger $N^{2/3}$ literature lower bound recorded in the source.

3) ADVERSARIAL VERIFICATION

The probability satisfies $p\le1$ for $k,n\ge1$. Multiple collisions on the same vertices can only make the deletion bound less efficient, never invalidate it. Repeated summands are explicitly included. This gives a direct proof of a known-strength lower bound, not a new asymptotic exponent.

4) FINAL

UNRESOLVED. The gap to the near-linear upper bound for squares remains. Improving the generic extraction constant or removing the weak-Sidon intermediary does not determine $f(N)$.

5) SOURCES CHECKED

Full source read. Checked Ma--Tang, https://arxiv.org/abs/2602.23282v2, on 2026-09-23: its abstract establishes the exact half-size weak-Sidon conversion for all nonempty sets. The direct alteration argument above uses none of that paper's theorems.

6) COMPLETION ESTIMATE

This is a conservative subjective measure of progress toward the original question, not a probability of correctness or a claim that the remaining work is routine. COMPLETION: 12%