GPT 6 Astra Ultra attempt on Math-146

OBJECTIVE AND SOURCE REVIEW

Erdős #52; independent attempt, 2026-09-23. The two-fold sum-product target is $\max(|A+A|,|AA|)\gg_\varepsilon |A|^{2-\varepsilon}$ for every finite integer set $A$. I read the entire supplied 52.tex. Cushman's Theorem 1.3 does state the quoted $4/3+10/4407-\varepsilon$ exponent. Requoting it is not an independent improvement. Instead I isolate a regime where an elementary divisor argument reaches the conjectured exponent.

WORK: SETS OF POLYNOMIAL HEIGHT

Fix $C>0$ and $\varepsilon>0$. If $|A|=n\ge2$ and every $a\in A$ satisfies $|a|\le n^C$, then \[ |AA|\gg_{C,\varepsilon}n^{2-\varepsilon}. \] First, for every $\delta>0$ there is $K_\delta$ such that $d(m)\le K_\delta m^\delta$ for all positive integers $m$. To see this, factor $m=\prod p^{e_p}$. For $p^\delta\ge2$, $(e+1)\le2^e\le p^{\delta e}$. For each of the finitely many smaller primes the supremum of $(e+1)/p^{\delta e}$ over $e\ge0$ is finite. Multiplying these finitely many constants proves the assertion.

Remove zero from $A$ and write $B=A\setminus\{0\}$. For nonzero $t$, the number of ordered pairs $(a,b)\in B^2$ with $ab=t$ is at most $2d(|t|)$, because the sign and the absolute value of the first factor determine the pair. Since $|t|\le n^{2C}$, choosing $\delta=\varepsilon/(2C)$ gives \[ |B|^2=\sum_{t\in BB}r(t)\le |BB|\,2K_\delta n^\varepsilon. \] Now $|B|\ge n/2$ yields the result with constant $1/(8K_\delta)$.

ADVERSARIAL VERIFICATION AND REMAINING GAP

This proof includes negative elements and removes the exceptional zero product before counting divisors. Constants depend on the fixed height exponent $C$; taking $C=C(A)$ destroys the uniform assertion needed in the problem. Additive translation can reduce the diameter but does not preserve product-set cardinality, so one cannot normalize arbitrary sets into this regime. The proof is a standard divisor-bound special case, not a new unrestricted sum-product theorem. It leaves sets with superpolynomial integer height uncontrolled. The multi-fold variants mentioned in the official entry are also not settled here.

SOURCES CHECKED

Complete local source: gpt_pro_5.4/52.tex. Cushman, Theorem 1.3, https://arxiv.org/html/2512.13849v2; official indexed entry https://www.erdosproblems.com/52, checked 2026-09-23. The displayed special-case proof is self-contained.

FINAL

UNRESOLVED. Polynomial-height sets satisfy the desired exponent, but no uniform bound for all integer sets is proved.

COMPLETION ESTIMATE

Conservative subjective progress toward the unrestricted problem. COMPLETION: 15%