Autonomous disproofs of the sum-product conjecture over the real numbers with GPT-5.5 Pro

../

source_digest: Records selected source statements, the two versions read, and verification scope for Huang's arXiv preprint and Zenodo deposit.

theorem_1: States the real-number sum-product counterexample that Huang credits to Bloom, Sawin, Schildkraut and Zhelezov and reports a GPT-5.5 Pro agent proving in seven of eight trials: an absolute c > 0 and arbitrarily large finite real sets A with max of the sumset and product set sizes at most the size of A to the power 2 - c.

---

Yichen Huang, Autonomous disproofs of the sum-product conjecture over $\mathbb{R}$ with GPT-5.5 Pro. arXiv preprint (2026), version 1, arXiv:2607.20525v1. The arXiv record was submitted 2026-07-09 and its PDF is dated 2026-07-24.

For a finite set $A$ in a commutative ring, write $A+A=\{a+b:a,b\in A\}$ and $A\cdot A=\{ab:a,b\in A\}$. The paper's Theorem 1, which it credits to Bloom, Sawin, Schildkraut and Zhelezov (its reference [3], arXiv:2605.28781), states that an absolute $c>0$ and arbitrarily large finite $A\subset\mathbb R$ satisfy

$$\max\{|A+A|,|A\cdot A|\}\le |A|^{2-c}.$$

The paper reports a three-stage GPT-5.5 Pro experiment with correct proofs in seven of eight independent trials and an unresolved gap in trial 2. This is the real-number theorem; it is related to E52 only as a contextual variant. Exact statements, version scopes, and the experiment record are in the source digest.

Released project materials. The paper cites its project repository at yichenhuang/sum-product, which contains released code, intermediate outputs, and generated informal proofs. The filing identifies no proof-assistant formalization or certificate.

Reported verification. The paper reports the three-stage pipeline, the seven-of-eight result, a mean of 132.4k reasoning tokens per trial over all eight runs, and the author's disclosure of human verification and responsibility. These are source reports.

Local verification. The arXiv PDF read for this card and the distinct Zenodo PDF held in this folder are byte-identified below. Rendered arXiv physical pp. 1–5 and 7–8 and Zenodo physical p. 1 were inspected for the theorem, protocol, results, disclosure, and version identity. No released code or proof was run.

Results

Page numbers are those of the arXiv print, arXiv:2607.20525v1 (pp. 1–9).

• Theorem 1 (p. 2), credited by the paper to Bloom, Sawin, Schildkraut and Zhelezov: an absolute $c>0$ and arbitrarily large finite $A\subset\mathbb R$ with $\max\{|A+A|,|A\cdot A|\}\le|A|^{2-c}$; the paper's own contribution is the reported agent experiment proving it, not a new theorem.

Bears on

• Problem 52: context only. The problem concerns finite sets of integers; Theorem 1 gives real sets violating the real-number form of the bound and neither answers the problem nor bounds it.

Source artifacts and versions

The copy read for this card, and the edition it cites, is arXiv:2607.20525v1, nine physical pages, 448,715 bytes; this folder does not hold it.

The distinct alternate, held in this folder, is the Zenodo PDF, record 21286412. Its metadata identifies it as a preprint/deposit with publication_date 2026-07-10. It has nine physical pages and 447,037 bytes. The PDFs are distinct byte versions; their version scopes remain explicit. The metadata supplies no peer-review or named community-acceptance evidence. The source record pins the stable source identity, artifact roles, path alias, and locator limits. For the arXiv edition, the arXiv record names arXiv's non-exclusive distribution license (arXiv:2607.20525), every other right reserved. For the Zenodo PDF, no notice is printed on its nine pages, and the Zenodo record names the Creative Commons Attribution 4.0 license, license id "cc-by-4.0", with the access right "open" (https://zenodo.org/api/records/21286412, read 2026-10-02).

Only the edition under an open license is held; the source's other editions are not, since no license on record permits their redistribution, and the card cites the edition it names above.