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Claim. Robert Joseph George posted a partial proof claim to the proof-claims thread of [[problems/discrete_geometry/E0634/_index|Problem 634]] on 17 July 2026, with a write-up dated 16 July 2026 (Candidate partial solution to Erdős Problem 634, whose byline reads Robert Joseph George, GPT-5.6 Pro and Inkling) and a Lean repository, both linked above. The claim is that no triangle can be cut into $19$ congruent triangles (Theorem 1.2), that none can be cut into $46$ (Theorem 1.5), and more generally that no prime $p>3$ with $p\equiv3\pmod4$ is a possible number of congruent pieces (Theorem 1.3), so that a prime $p$ occurs exactly when $p=2$, $p=3$ or $p\equiv1\pmod4$ (Corollary 1.4). For $19$ the classification results reduce the question to five configurations whose tile has an angle of $120^\circ$; the rationality theorem of Beeson and Zhang scales the tile to primitive integer sides, and each configuration is eliminated through the side vectors of the large triangle, the integrality of their common scale, and divisibility, congruence and square arguments. The general prime case applies a scalar form of Laczkovich's boundary invariant, in which every counterclockwise $120^\circ$ tile contributes $\pm(a+c-b)$, to the isosceles configuration and derives an integrality and parity contradiction with the area equation; for $46$ the classification leaves a single tile, with sides $(7,8,13)$, which the invariant excludes by parity. The manuscript presents its results as proposed partial results. Its byline names the AI systems GPT-5.6 Pro and Inkling, the latter credited with a modular simplification in the proof for $19$; the site's tab spells them "GPT5.6" and "inking (Thinking Machine)". The claim was posted under the account robertboy18.

Submission note. Posted to erdosproblems.com as a proof claim by Robert Joseph George (account robertboy18) on 17 July 2026, giving "GPT5.6 and inking (Thinking Machine)" as the AI used:

I claim that $N=19$ and $N=46$ are impossible, and more generally that no prime $p\equiv 3 \pmod 4$, $p>3$, is admissible. For $N=19$, existing classification results reduce the problem to five cases involving a $120^\circ$ tile, all of which are eliminated using integer side parametrizations and area equations. The general prime result uses Laczkovich's boundary invariant to obtain an integrality and divisibility contradiction in the remaining isosceles case. For $N=46$, the classification leaves only the tile $(7,8,13)$, and the invariant gives a parity contradiction. Notes: The proof relies on recent Beeson–Zhang rationality results and Beeson’s isosceles classification. It is partial progress, not a complete solution of Problem 634.

Covers. The exclusion of every prime $p>3$ with $p\equiv3\pmod4$, in particular of $19$, and the exclusion of $46$. Nothing is claimed about other values of $n$, and the characterization asked for by the problem is not claimed.

Depends on. [[problems/discrete_geometry/E0634/claims/1991_08_01_snover_waiveris_williams|Snover, Waiveris and Williams's claim page]], for the rep-tiling theorem the manuscript imports.

Inputs. The manuscript's inputs are external theorems, none of them carded: the rationality theorem of Beeson and Zhang (arXiv:2604.01314, Theorem 1.1 and Section 4.2), Beeson's isosceles paper in its May 2026 revision (arXiv:1206.1974v7, for the isosceles formulas and the small row-I candidates), Beeson's paper on seven and eleven (arXiv:1811.09723v5, for Laczkovich's list of commensurable-angle counts), Beeson's equilateral paper (arXiv:1812.07014), Beeson's papers on the case $3\alpha+2\beta=\pi$ (arXiv:1206.2229v3) and on tiles similar to the triangle or right-angled (arXiv:1206.2231), Laczkovich's classifications of 1995 and 2012, and the rep-tiling theorem of Snover, Waiveris and Williams. It cites neither the solution of Problem 633 nor Zhang's $2\pi/3$ preprint, and it states that it uses no result of the withdrawn arXiv:1206.2228. For the branch $3\alpha+2\beta=\pi$ it imports Theorems 8, 12, 15, 18 and 20 of arXiv:1206.2229; Bonfioli's manuscript (claim page) disputes the printed proofs of that paper's Theorems 14, 18, 19 and 20.

Standing. Claimed. The write-up is hosted on a file-sharing service and has no arXiv or journal record. The Lean repository, pinned above at its last commit before the posting, is a third-party development that this corpus has not built or audited, so it gives no formalized evidence; it is linked as the claimant's own formalization of the claimant's result. Its README states that the global theorems nineteen_not_admissible, fortySix_not_admissible and prime_admissible_iff take the structure PublishedTilingResults as an explicit hypothesis, which collects the imported literature results, among them the exhaustive five-case reduction for $19$ and the reductions for $46$ and for the primes; only the arithmetic after those reductions is proved without hypotheses. The site's thread shows no comment on the claim and the site's label and remarks are unchanged (page last edited 30 December 2025), so no outside acceptance is recorded. Harries's second manuscript cites this write-up as a corroborating independent prime proof and a separate audit of $46$. The same prime exclusion was claimed a week later, with a different proof, on [[problems/discrete_geometry/E0634/claims/2026_07_24_harries|Harries's claim page]], and on 26 July 2026 by Beeson (claim page).