---

Claim. Vico Bonfioli, A signed-direction invariant for triangle tilings, and the exclusion of primes congruent to 3 modulo 4, a manuscript in the repository ElVec1o/erdos_634_proof, which was created on 27 June 2026, the day the paper says its proof was made public and communicated to Beeson; the paper is dated 1 September 2026 and is linked above at the repository's commit of 13 September 2026, with its Lean development. The paper introduces a translation-invariant signed-direction functional on tilings and proves with it that no isosceles, non-equilateral triangle can be cut into a prime number of congruent copies of a tile with angles $(\alpha,\beta,2\pi/3)$ and $\alpha/\pi$ irrational (Theorem 42). With a reduction of the scalene shapes, its Theorem 1 excludes every prime $N>3$ with $N\equiv3\pmod4$ that is not of the form $3f^2-e^2$ with $e,f$ coprime, and Theorem 4 shows those excepted primes to be exactly the primes $N\equiv11\pmod{12}$, so that (Corollary 7) no prime $N\equiv7\pmod{12}$ occurs, in particular not $19$, with no hypothesis beyond the cited classification of the branches. The exclusion of the primes $N\equiv11\pmod{12}$ in general (Theorem 2) is stated under a complete-corner-wall hypothesis from a companion note, which the paper labels a conjecture and shows to be equivalent to its own conclusion at $e=1$; thirteen of those primes and composite candidates, $11$, $23$, $26$, $47$, $59$, $66$, $71$, $107$, $191$, $431$, $587$, $971$ and $1451$, are excluded individually by certified exhaustive search. Beyond the primes the paper determines the admissible spectrum of each sporadic $2\pi/3$ branch, proves membership in the set of tile counts decidable, and settles every $N\le80$: no triangle can be cut into $14$, $15$, $21$, $22$, $30$, $33$, $35$, $38$, $39$, $42$, $51$, $55$, $56$, $57$, $60$, $69$ or $76$ congruent triangles, nor into $46$, $62$ or $78$, while some triangle can be cut into $28$, $44$ (the isosceles triangle $(16,16,22)$ by the tile $(2,3,4)$), $63$ (correcting an exclusion asserted in an earlier version of the paper), $77$, $80$ and $99$ (the triangle $(24,24,33)$ by $(2,3,4)$); the values $66$ and $70$, previously excluded through theorems the paper disputes, are re-excluded by exhaustive search; past $80$ it excludes $86$ and $87$, realizes $81$, $82$, $85$, $89$ and $90$, and leaves $84$ and $88$ under search. The paper and its citation file say the work was developed with AI assistance from Anthropic's Claude.

Covers. No prime $N\equiv7\pmod{12}$ occurs, so in particular $19$ does not; the values $11$, $14$, $15$, $21$, $22$, $23$, $26$, $30$, $33$, $35$, $38$, $39$, $42$, $46$, $47$, $51$, $55$, $56$, $57$, $59$, $60$, $62$, $66$, $69$, $70$, $71$, $76$, $78$, $86$, $87$, $107$, $191$, $431$, $587$, $971$ and $1451$ do not occur; the values $28$, $44$, $63$, $77$, $80$, $81$, $82$, $85$, $89$, $90$ and $99$ occur. The exclusion of the primes $N\equiv11\pmod{12}$ in general is conditional on the unproved hypothesis above and is not covered; the characterization asked for by the problem is not claimed.

Depends on. No page of this wiki.

Inputs. The paper rests on Laczkovich's classification and on Beeson's branch theorems for the shapes it does not re-derive, and it disputes several of them: it shows the divisibility $g\mid M$ in Theorem 14 of Beeson's paper on the case $3\alpha+2\beta=\pi$ (arXiv:1206.2229) false, refuted by its $99$-tiling, finds the printed proofs of that paper's Theorems 18, 19 and 20 unsound (Theorem 19 through a divisibility count of the same faulty kind and the false squarefree claim of that paper's Lemma 8, Theorem 20 through Theorem 19, and Theorem 18 through a dropped factor in a residue computation), and replaces Theorems 18 and 20 by its Propositions 30 and 12, whose arithmetic is machine-checked; the values $66$ and $70$, previously excluded through those theorems, are re-excluded by exhaustive search. It also records that Beeson's preprint No prime tiling of an isosceles triangle, arXiv:2607.19572 (21 July 2026), proves the isosceles $2\pi/3$ case independently by a different method and disposes of the branch $3\alpha+2\beta=\pi$ by citing the disputed Theorem 14; that preprint was withdrawn on 24 September 2026 with the note that the theorem has meantime been proved by Bonfioli. None of the inputs is carded.

Standing. Claimed. The paper is not on arXiv and has no journal record; its README calls it a preprint not yet independently refereed. The Lean development, pinned above, is the author's own: by the README it has no sorry, reports only the three standard axioms, checks the arithmetic and combinatorial layer, the tiling certificates for $28$, $44$, $77$ and $99$ and parts of the forcing chain, while the geometric layer rests on the written proofs; this corpus has not built it, so it gives no formalized evidence. The site's thread carries no claim by this author, and the site's label and remarks are unchanged (page last edited 30 December 2025), so no outside acceptance is recorded. The prime manuscripts of George, Harries and Beeson ([[problems/discrete_geometry/E0634/claims/2026_07_17_george|George's claim page]], [[problems/discrete_geometry/E0634/claims/2026_07_24_harries|Harries's claim page]], [[problems/discrete_geometry/E0634/claims/2026_07_26_beeson|Beeson's claim page]]) claim the full prime classification, which this paper's own labels leave open for the primes $11\pmod{12}$; Harries's second manuscript (claim page) excludes $21$ and $33$ and realizes $88$, which this paper lists as under search.