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Claim. Jan Philipp Harries, New constructions, obstructions, and multiplier structure for Erdős Problem 634, a manuscript in the repository jphme/math-problems, first committed 27 July 2026 and linked above at that commit and at version 0.5, dated 28 August 2026. Writing $S$ for the set of $N>1$ such that some triangle can be cut into $N$ congruent triangles, the question of Problem 634, the manuscript claims: (Theorem 12) a triangle with sides $(21,55,56)$ is tiled by $88$ copies of the tile $(3,5,7)$ and a triangle with sides $(36,36,63)$ by $189$ copies of $(2,3,4)$, both by exact coordinate certificates checked for congruence, containment, disjoint interiors and area, so $88m^2$ and $189m^2$ lie in $S$ for every $m\ge1$; (Corollary 13) on the ray $21n^2$ a tiling exists exactly for $n\ge2$, the multipliers $2$ and $4$ being Beeson's known $84$- and $336$-tilings, $3$ the new certificate and larger multipliers an addition theorem, while $21$ itself is excluded by a computer-assisted, independently certified exhaustion; and (Theorem 16) $33\notin S$, by reducing a hypothetical $33$-tiling to the single instance of the isosceles triangle $(21,21,33)$ tiled by $(5,3,7)$ and exhausting it, the complete search tree being exported as a refutation certificate accepted by a checker written from the certificate specification alone. The paper's remaining content, a Laurent-polynomial boundary invariant for tiles with a $2\pi/3$ angle, the finite generation of the multiplier sets, trapezoid constructions and a provisional programme below $200$, settles no further value and is not part of the claim. The manuscript's disclaimer says that OpenAI GPT-5.6 Pro agents and Anthropic Claude Fable agents proposed lemmas, derived formulae, wrote and audited the exact-search code and helped draft it, and that agreement among models is an error filter, not verification.

Covers. The values $88$ and $189$, and with them $88m^2$ and $189m^2$ for every $m\ge1$ and $21n^2$ for every $n\ge2$, occur; the values $21$ and $33$ do not. Nothing is claimed about other values of $n$, and the characterization asked for by the problem is not claimed; the manuscript says the full problem remains open.

Depends on. [[problems/discrete_geometry/E0634/claims/1991_08_01_snover_waiveris_williams|Snover, Waiveris and Williams's claim page]], for the reptile counts its ledger lists; [[problems/discrete_geometry/E0634/claims/2025_12_27_zhang|Zhang's claim page]], for the trapezoid lemmas and the equilateral construction.

Inputs. The manuscript's dependency ledger (its Appendix E.1) lists its imports with the version read. The branch completeness behind the exclusions of $33$ and $21$ rests on Laczkovich's classifications of 1995 and 2012, on Beeson's tabulation of them (arXiv:1811.09723v5), on Beeson's isosceles paper (arXiv:1206.1974v7), on Beeson's paper on the case $3\alpha+2\beta=\pi$ (arXiv:1206.2229v3) and, for the isosceles non-equilateral targets, on Theorem 3 of Beeson's preprint No prime tiling of an isosceles triangle, arXiv:2607.19572v1, marked load-bearing; that preprint was withdrawn on 24 September 2026 with the note that its Lemma 9 is not correct as stated, so the exclusions rest in part on a withdrawn source. The constructions import the rationality theorem of Beeson and Zhang (arXiv:2604.01314v1) and Zhang's trapezoid lemmas and equilateral construction (claim page); the solution of Problem 633 is cited as corroboration only. None of these is carded as the manuscript uses it. The ledger also notes that the manuscript does not use the clause $g\mid M$ of Theorems 14 and 19 of arXiv:1206.2229, the clause Bonfioli's manuscript refutes.

Standing. Claimed. The manuscript is not on arXiv and has no journal record; its version history records an independent referee report of 27 July 2026 and a referee query of 6 August 2026, both answered in later versions, without naming the referee. The site's thread carries no claim for these values, and the site's label and remarks are unchanged (page last edited 30 December 2025), so no outside acceptance is recorded. The same manuscript revises Harries's earlier prime-case proof (claim page) in its treatment of the imported statements. Bonfioli's manuscript (claim page) also excludes $21$ and $33$, by exhaustive search, and lists $88$ as open.