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Follow-up (2026-09-27): more stale "open" labels, found by syncing with sources that maintain status data. Again, this is only meant to save others duplicated effort.

1. Erdős problems. All 632 EP-* entries are labelled open in v1.6.0. As of today, erdosproblems.com lists 90 of them as resolved (we fetched each page; many resolutions are recent, several by AI systems, and most are Lean-verified). Grouped by the site's status:

• Proved, Lean-verified (28): EP-38, EP-123, EP-126, EP-152, EP-258, EP-281, EP-283, EP-330, EP-351, EP-358, EP-369, EP-457, EP-469, EP-557, EP-571, EP-610, EP-750, EP-793, EP-825, EP-865, EP-987, EP-997, EP-1014, EP-1022, EP-1051, EP-1071, EP-1096, EP-1129
• Proved (9): EP-380, EP-591, EP-652, EP-851, EP-863, EP-986, EP-1021, EP-1105, EP-1130
• Disproved, Lean-verified (14): EP-1, EP-43, EP-74, EP-90, EP-125, EP-146, EP-180, EP-193, EP-533, EP-846, EP-847, EP-884, EP-990, EP-1092
• Disproved (8): EP-92, EP-543, EP-574, EP-575, EP-705, EP-869, EP-960, EP-992
• Solved otherwise (e.g. determined/answered), Lean-verified (12): EP-42, EP-119, EP-183, EP-190, EP-202, EP-318, EP-619, EP-650, EP-694, EP-696, EP-741, EP-888
• Solved otherwise (e.g. determined/answered) (19): EP-320, EP-321, EP-346, EP-387, EP-421, EP-477, EP-603, EP-625, EP-633, EP-690, EP-730, EP-783, EP-858, EP-896, EP-920, EP-948, EP-1005, EP-1089, EP-1091

The site's machine-readable status file (teorth/erdosproblems, data/problems.yaml) may be the easiest way to keep these entries in sync.

2. Ben Green's 100 open problems. The current version of the list (updated December 2025) marks these as solved. Note that the dataset's GREEN-xxx numbers differ from the numbering in Green's list.

3. OWR-1452-012 (Zhao's Vanishing Conjecture for homogeneous quartics). Zhao proved that this conjecture, over all n, is equivalent to the Jacobian conjecture over all n (Trans. AMS 359 (2007), arXiv:math/0409534). The July 2026 Jacobian counterexample (see SMA-016 above) therefore refutes it for some n; we have not identified the smallest such n.

— Alper Ferudun

Follow-up 2 (2026-09-27): Kourovka Notebook, issue 21 (KOU-21.*). The arXiv version of the notebook updated today (arXiv:1401.0300v46) marks the following issue-21 problems as solved (asterisk), while v1.6.0 still labels them open:

(The other nine starred issue-21 problems, 21.8, 21.12, 21.14, 21.15, 21.18, 21.24, 21.43, 21.58, 21.150, are already marked solved in the dataset.)

— Alper Ferudun

Follow-up 3 (2026-09-27): KOU-21.68 — new counterexample (our own result, unrefereed). Kourovka Notebook Problem 21.68 (M. Kida) conjectures that every finite semi-abelian group is monomial. This is false. There is a semi-abelian group of order 768 = 2^8·3 with a non-monomial irreducible character of degree 8:

• Ĝ = B ⋊ W, where W = E ⋊ A₄ is the index-two subgroup of C₂ ≀ A₄, T ≅ SL(2,3) ≤ W is the binary tetrahedral group acting on the eight quaternion units, and B is the augmentation (even-weight) submodule of the permutation module F₂[W/T].
• General reduction (Clifford theory): if a linear character of the abelian normal subgroup N has stabiliser T in W and T has a non-monomial irreducible character, then N ⋊ W is not an M-group.
• Checked by exact computation, including every subgroup of index 8 and the full character table of Ĝ (exactly the three degree-8 irreducible characters are non-monomial). Scripts are in the source archive.

Preprint: https://doi.org/10.5281/zenodo.23000305 · paper page: https://eulersolve.org/papers/kou-21-68/

Suggested label: solved (answered negatively). It has not been peer-reviewed yet, so independent checks are welcome. Minimality of the order is not claimed; Kida's Magma search covered all orders up to 240.

— Alper Ferudun

Follow-up 4 (2026-09-27): internal status inconsistency. For 188 problem numbers, the literature-triage block inside the record itself ("Literature review (checked 2026-08-17)", Status: solved, Classification: SOLVED-IN-LITERATURE) disagrees with the status field, which still says open (same in v1.6.0 and v1.7.0). 94 of them are already covered in the comments above; the remaining 94 are listed below by source so they can be reconciled in one pass. We have not re-verified each triage conclusion ourselves, and some of them refute a literal wording rather than the intended question, so they are pointers, not claims.

• HIL (1): HIL-009
• SMA (1): SMA-004
• NT (5): NT-008, NT-023, NT-035, NT-037, NT-086
• GREEN (3): GREEN-061, GREEN-064, GREEN-100
• GEO (3): GEO-004, GEO-006, GEO-028
• GEOM (1): GEOM-026
• GT (1): GT-008
• GRAPH (4): GRAPH-006, GRAPH-008, GRAPH-046, GRAPH-049
• TOP (1): TOP-003
• ALG (2): ALG-033, ALG-036
• COMB (2): COMB-012, COMB-004
• HL (1): HL-F
• GUY (3): GUY-A8a, GUY-A11, GUY-A15
• KP (6): KP-1.51, KP-3.14, KP-4.37, KP-4.125, KP-5.9, KP-5.15
• OPG (51): OPG-23298, OPG-50149, OPG-37185, OPG-426, OPG-1797, OPG-37167, OPG-37181, OPG-37230, OPG-692, OPG-37086, OPG-610, OPG-37341, OPG-34908, OPG-37081, OPG-37089, OPG-37218, OPG-37316, OPG-37364, OPG-57613, OPG-59952, OPG-59997, OPG-824, OPG-46606, OPG-47285, OPG-59911, OPG-2242, OPG-59994, OPG-616, OPG-36939, OPG-52200, OPG-47031, OPG-47643, OPG-37305, OPG-47646, OPG-677, OPG-690, OPG-691, OPG-735, OPG-177, OPG-157, OPG-732, OPG-760, OPG-2379, OPG-37444, OPG-37863, OPG-37402, OPG-655, OPG-1783, OPG-37245, OPG-37295, OPG-57401
• EP (9): EP-129, EP-520, EP-524, EP-545, EP-550, EP-612, EP-638, EP-654, EP-996 — note that erdosproblems.com still lists all nine as open (EP-550 as "open (Lean)"), so these triage conclusions deserve a second look before relabelling.

A JSON list (problem number, status field, triage status) is available on request.

— Alper Ferudun

Hello, thank you for these detailed additions! They are now integrated.

Follow-up 5 (2026-09-28): two new results, one problem already solved in the literature, and corrections to Oberwolfach and Kourovka records (stale statuses, transcription errors, merged records, one misattached note). Each item was checked against the current version (commit 2ea030b). Where a page is given, it was also checked in the report itself; literature was checked against arXiv and Crossref. Suggested changes are in bold.

New results (unrefereed preprints with full proofs, verification scripts and independent referee checks)

• OWR-12861-021 → solved (no). Heinig's Question 2 (OWR 01/2014, p. 82) has a negative answer for every odd n ≥ 7: K_{(n+1)/2,(n−1)/2} with a perfect matching (or a matching plus one P₃) inside the larger side has minimum degree ⌈n/2⌉ but no spanning copy of the near-square, under both readings of "periphery"; at n = 9, K_{4,4,1} is a counterexample under every reading. For n = 7 the host is Heinig's own graph X (arXiv:1112.5101, Def. 28). The cycle-space Question 1 is not affected (Hou–Yin). Paper: doi:10.5281/zenodo.23004012 (page).
• KOU-21.76 → solved. The existence question was first answered by Ya. N. Nuzhin, Sib. Math. J. 67 (2026) 840–845, Corollary 1 (examples in every characteristic, over F(y,z)); the notebook (v46) does not record this yet. Our note gives explicit examples with short proofs, including one-variable examples over F₃(t) (which also satisfy the hypotheses of 19.48), and shows that in characteristic 3 such nets exist over K iff K is not algebraic over F₃. Paper: doi:10.5281/zenodo.23004034 (page).

Stale statuses

• OWR-12330-013 → solved (yes). King answers his own Question 1 on the next page: "The answer to this is yes" (OWR 02/2013, p. 105). K_k with k pendant vertices at every vertex has χ_f = k > 3k/4 + 1 for k ≥ 5.
• OWR-1319-022 → solved (no). Two word metrics on H₃(ℤ)×ℤ have ratio → 1 but unbounded difference: Breuillard, Groups Geom. Dyn. (2014) (arXiv:0704.0095), and Breuillard–Le Donne, PNAS 110 (2013), §5, who cite this report.
• OWR-5158-016 → solved (yes). By Deroin–Hurtado (arXiv:2008.10687, Thm 1.3), irreducible lattices in finite-centre semisimple groups of real rank ≥ 2 are not left-orderable, so any cocompact arithmetic lattice in SL(3,ℝ) works. See also Witte Morris's exposition (2026).
• OWR-14213-006 → solved. The statement is the unit-interval (Hessenberg) form of the Stanley–Stembridge conjecture, proved by Hikita, J. Amer. Math. Soc. (2026) (arXiv:2410.12758).
• OWR-1386-016 → solved, if "threshold" has its usual coarse meaning (the constant b is then immaterial). This is the two-graph Kohayakawa–Kreuter conjecture: the 1-statement is due to Mousset–Nenadov–Samotij (CPC 2020), and Christoph–Martinsson–Steiner–Wigderson (Proc. LMS 130, 2025) completed the proof. A sharp threshold at b·n^(−1/m₂) would be a different question; we could not re-read the wording in OWR 48/2006.
• OWR-14299911-005 → solved (disproved). The source itself reports Chalopin–Chepoi's counterexample to the "MSO decidable ⇔ grid-free" conjecture (ACM TOCL 20 (2019)). Please also drop the second sentence: the special-cube-complex result concerns Thiagarajan's other conjecture, and the counterexample itself comes from a virtually special complex.
• OWR-17294-009 → at least partially solved. Part (1), Byott's question, is answered no by Di Matteo–Ferrara–Trombetti, arXiv:2607.22795 (July 2026 preprint). The record bundles Vendramin's Problems 1, 3 and 5 (OWR 51/2019, p. 3222); splitting it would help.
• KOU-21.115 → solved (yes). Sambale, arXiv:2609.09052 (September 2026 preprint), proves |G∖U| ≥ |G|/2ⁿ whenever n left cosets have union U ≠ G, and right cosets are left cosets of conjugates. The notebook (v46) has not starred it yet.
• OWR-4213-002 → solved. The statement is the Feit–Thompson theorem (Pacific J. Math. 13 (1963)). Also, "nontrivial" simple groups should read "non-abelian".
• OWR-1265-024 → not an open problem. The source poses it as a puzzle and gives the answer C(a+b,a) − C(a+b,a−1), which counts standard (b,a)-tableaux (G. James, OWR 15/2006, pp. 965–966).
• Pointer only, OWR-11568-005. Lutowski's Theorem 2 (Publ. Math. Debrecen 99 (2021)) says that the holonomy of a non-torus Kähler flat manifold has at least two distinct irreducible constituents. Since h^{1,1}(A/G) = dim End_G(T₀A), this seems to force b₂ ≥ 2 for free torus quotients of dimension ≥ 2. Worth a check.

Transcription errors

• OWR-12330-010, -011, -012: the ceilings in the source are missing (OWR 02/2013, p. 103, Conj. 3; p. 104, Conj. 4–6). As written, C₅ violates all of them: the right-hand sides are 5/2 or 3 − ε, while χ(C₅) = 3.
• OWR-1189-007: Kohl's Conjecture 2 reads ⌊3d(1−1/n)⌋ + 1, not ⌈·⌉ (OWR 7/2006, p. 416). As written it is false for d = 1 and n ≥ 4, because χ^{1,1}_ℓ(P_n) = ch(P_n²) = 3.
• OWR-5154-009: the source puts the profinite closure on the product [g₁]^F⋯[g_k]^F, not on N(g₁,…,g_k) (OWR 26/2011, p. 1452, Conj. 2). As written, the answer is trivially no.
• OWR-12007-008: with "r, s ∈ ℝ" (the same wording as OWR 35/2012, p. 2173) the question is false for w = a, since R(a,r,s) = r. The source's own results are for rational r, s, so this should read r, s ∈ ℚ.

Merged records

• OWR-4791-001 merges two conjectures that the organisers' introduction (OWR 1/2011, p. 6) reports as proved: Simonovits–Sós (Ellis–Filmus–Friedgut) and Sumner (for large n, Kühn–Mycroft–Osthus). Splitting it would help.
• OWR-9790352-039, OWR-10252930-031, OWR-9790352-036: the statement is already clean, but original_statement still contains a neighbouring item, respectively:
  • the Plummer–Zha conjecture, proved during the workshop (OWR 1/2022, p. 71; Chudnovsky–Seymour, JGT 103 (2023));
  • a weighted Turán conjecture, proved by Bradač (arXiv:2205.08923);
  • Narayanan's permanent inequality (OWR 1/2022, pp. 69–70), which is open and has no record of its own.

Misattached note

• OWR-12861-020: the literature note (Hou–Yin, arXiv:2503.15950) is about Heinig's Question 1 (OWR 01/2014, p. 81), which has no record of its own. It does not concern the Diestel or Friedgut questions in this record, and the record's "partially solved" label rests only on that note.

Correction to Follow-up 2. arXiv:1401.0300v46 (Kourovka Notebook No. 21) was posted on 1 September 2026, not on 27 September as I wrote there; the list of starred problems is unaffected.

— Alper Ferudun

Follow-up 6 (2026-09-28): three new results, and corrections to Oberwolfach combinatorics records (stale statuses, questions answered in their own source, transcription errors, duplicates, merged records, garbled extracts and misattached notes). Each item was checked against the current version (commit 2ea030b). Where a page is given, it was also checked in the report itself; literature was checked against arXiv and Crossref. Suggested changes are in bold.

New results (unrefereed preprints with full proofs, verification scripts and independent referee checks)

• OWR-16164-019 → solved (no). Sportiello's conjecture B_λ ≥ A_λ (OWR 23/2018, pp. 1455–1457) fails for the band λ = {(x,y) ∈ [7]² : −3 ≤ y−x ≤ 4}, where A_λ = 536 and B_λ = 515. Every digitally convex shape of side ≤ 6 satisfies the inequality; among the 5,693,968 shapes of side 7, only this band and its transpose violate it. (The record attaches the red and blue crosses to the opposite colours; that swap is a bijection on colourings and does not change B_λ.) Paper: doi:10.5281/zenodo.23006715 (page).

• OWR-1782-009 and OWR-1386-013 (the same conjecture) → false as stated. Both ask that minimum codegree ⌊(n−k+3)/2⌋ force a tight Hamiltonian cycle for all n: for n ≥ k+1 ≥ 4 in OWR 01/2008 (p. 46), and with no range of n in OWR 48/2006 (p. 2928). This all-n form is Conjecture 1.1 of Rödl–Ruciński–Szemerédi, Adv. Math. 227 (2011), who attribute it to Katona–Kierstead. It fails for (k,n) = (3,7), (3,9), (4,8), (5,9) and (5,10). The smallest counterexample is an apex joined to all 15 pairs of a 6-set W, plus the ten faces of the hemi-icosahedron on W. Its pair-degrees are 3 and 5. But in a cyclic order (apex, w₁, …, w₆) the disjoint windows w₁w₂w₃ and w₄w₅w₆ would both have to be faces, and no two faces are disjoint. The (5,10) example has codegree 4 = (n−k+3)/2, so the version without the floor fails too. The large-n statement is not affected: RRS proved it for k = 3, and Letzter–Lang–Ranganathan–Sanhueza-Matamala have announced it for all k ≥ 3, as reported in arXiv:2609.08613. Suggested status: "false as stated (small counterexamples); the large-n form is proved for k = 3 and announced for all k". Also:

  • OWR-1386-013 should say "tight" Hamiltonian cycle, as its source defines it; under a Berge reading these hypergraphs do have Hamiltonian cycles;
  • mark the two records as duplicates;
  • the literature DOIs on OWR-1782-009 concern other thresholds (10.1016/j.jcta.2015.01.004: Hamilton ℓ-cycles with ℓ < k/2; 10.1112/jlms.12561: minimum d-degree conditions). RRS 2011 and arXiv:2609.08613 are the relevant references.

  Paper: doi:10.5281/zenodo.23006718 (page).

• OWR-14299577-018 → solved (yes, both questions). The problem was proposed by C. Bernert and N. Arala Santos, in the problem session compiled by T. F. Bloom (OWR 51/2025, p. 2756), so proposed_by can be filled in. Let A ⊂ {−n,…,n}∖{0} contain exactly one of k and −k for every k ≤ n. Then at most two elements of [1,n] are missing from A − A. This is sharp: {1,…,a} ∪ {−(a+1),…,−n} misses exactly a and a+1 whenever n/2 ≤ a ≤ n−1. For every B ⊆ [1,n], the number of pairs (a,b) ∈ A² with a − b ∈ B is at least ⌊(|B|−1)²/4⌋, and this is attained for every |B| ≤ ⌊2n/3⌋+1. So for |B| ≥ εn the count is at least (1/4 − o(1))ε²n². The key step is that the representation counts r(d) of any set S of differences of one parity satisfy Σ_{d∈S} r(d) ≥ C(|S|,2), by a double count of the positions where the sign pattern repeats at distance d. The record's original_statement is garbled: it drops the set-up sentence and splices in two sentences from the preceding problem (Assing). Paper: doi:10.5281/zenodo.23006720 (page).

Stale statuses

• OWR-16160-016 → solved as stated (classical). Valette asks whether the congruence subgroups of G ⊂ SL_N(ℤ) depend on the embedding (OWR 19/2018, p. 1151). In general they do. For example, F₂ embeds as ⟨(1 2; 0 1), (1 0; 2 1)⟩ and as ⟨(1 3; 0 1), (1 0; 3 1)⟩. The level-2 subgroup of the second embedding contains no congruence subgroup of the first, and the level-3 subgroup of the first contains none of the second, so the two congruence topologies are incomparable. Classical instances go back to Serre: SLₙ(ℤ) has congruence subgroups whose images under the adjoint map are not congruence subgroups (Lubotzky–Venkataramana, Algebra Number Theory 13 (2019), Prop. 2.1). For the groups of the talk, Z² ⋊_A Z, the answer is the opposite, and the same holds for every solvable G. Solvable groups have the congruence subgroup property (Chahal, Nagoya Math. J. 79 (1980)), and solvable subgroups of GLₙ(ℤ) are polycyclic, so every embedding induces the profinite topology (LV, §1.1). A scope note would help.
• OWR-1265-006 → solved (yes). Olsson's containment question (OWR 15/2006, p. 913) is the Olsson–Stanton conjecture. Vandehey proved it (arXiv:0809.2134, 2008), and Fayers gave another proof (JCTA 118 (2011)).
• OWR-1536-011 → solved (no, both parts). Knox (arXiv:1212.3345) gives a hypergraph on which Breaker, moving second, wins Maker–Breaker, yet Chooser wins Chooser–Picker. Knox notes that the Picker–Chooser part is equivalent to it; the equivalence comes from the transversal-hypergraph duality stated in the source (OWR 20/2007, p. 1095). So the transversal hypergraph of his example refutes the Picker–Chooser part too.
• OWR-4425-020 (duplicate -021) → solved (yes). Shallit's Conjecture 48 (OWR 37/2010, p. 2236) is Theorem 18 of Cassaigne–Currie–Schaeffer–Shallit, J. ACM 61 (2014), for exactly this morphism.
• OWR-2090-021 → partially solved. Adiprasito–Björner (arXiv:1401.7301, Thm 2.1) prove the Mikhalkin–Ziegler conjecture from this problem session:
  • for a generic weight and t ≤ min{0, total weight}, the proper flats of weight > t form a homotopy Cohen–Macaulay poset, hence an (r−3)-connected one;
  • the "non-negative" version follows by a small perturbation;
  • they credit rank 3 to Pinchasi–Ziegler (personal communication, 2008);
  • shellability is still open (their Open Problem 3.2).
• OWR-12481-012 → solved (no). When Friedgut re-posed it for fixed t and large n, the report recorded: "This turns out to be false; a counterexample was found by Gábor Tardos" (OWR 22/2016, p. 1217). The question as posed here already fails for n = 4, t = 2. Pair σ with σ∘(1 2) if σ maps {1,2} onto {1,2} or {3,4}, and with σ∘(3 4) otherwise. The resulting 12 two-cosets refine none of the 8 partitions of S₄ into 1-cosets.
• OWR-16633-025 → solved (no), over every field. We found no written answer, but a 4-element example settles it: T₀ = F², T₁ = ⟨e₁⟩, T₂ = ⟨e₂⟩, T₃ = ⟨e₁+e₂⟩. Then f(0) = 2, f(1) = f(2) = f(3) = 1, and f(S) = 2 for every other nonempty S. Suppose f = Σ c_j r_j with c_j > 0 and matroid ranks r_j on {0,1,2,3}; the r_j need not even be representable.
  • Every matroid has r(0i) ≥ r(0) and r(ij) ≤ r(i) + r(j). Since f attains equality in both, so does every r_j.
  • So in each r_j, the elements 1, 2 and 3 lie in the closure of 0, which has rank ≤ 1, and at most one of them is a non-loop.
  • Hence r_j(123) = r_j(1) + r_j(2) + r_j(3) for every j, and summing gives f(123) = 3. But f(123) = 2.
  • Over GF(2), 2·r(U₂,₄) is another example. Dougherty–Freiling–Zeger (arXiv:0910.0284, §4) represent it over every field. Its only possible matroid summands are copies of U₂,₄, which is not binary.

Answered in the source itself

• OWR-4791-006 → solved. Fox–Lee–Sudakov prove both conjectures in the abstract itself (OWR 01/2011, pp. 11–12: Thm 2, f(m) = ⌊√(4m+1)⌋ − 1, and Thm 4). The paper is Israel J. Math. 191 (2012). The record's own verification note already says so.
• OWR-16931-001 → solved. The record asks only about sufficiently large n. For that case the source says "We answer this affirmatively for all sufficiently large n" (OWR 19/2019, p. 1158). This is Glock–Joos–Kim–Kühn–Osthus, JEMS 23 (2021).
• OWR-14299518-002 → solved. Alon answers both questions in the abstract itself (OWR 42/2025, pp. 2249–2250): (1) no (Thm 3), (2) yes (Thm 4). These are Erdős problems #664 (disproved) and #732 (proved). The note's "structural characterization" is not part of this record.
• OWR-12872-010 → solved, and one formula needs fixing. Stanley (with F. Liu) proves both of Elkies' conjectures in the same abstract (OWR 12/2014, pp. 696–697); the paper is Ramanujan J. 36 (2015). For n = 2m the count of maximum families is 2^((m−1)(m−2))·(2^m − 1), not ·(2m − 1). A brute-force count gives 28 for n = 6 and 960 for n = 8.

Transcription errors and literal readings

• OWR-14298158-004: Claesson's conjecture is monotonicity in the length n for fixed k: |Av_n^k(1324)| ≤ |Av_{n+1}^k(1324)| (OWR 6/2024, p. 284; Claesson–Jelínek–Steingrímsson, JCTA 119 (2012)). The same correction applies to the Av(1324, 231) part. As written (k → k+1), the statement is false for every n, e.g. |Av_{2,1}| = 1 > 0 = |Av_{2,2}|.
• OWR-17135-030, -031, -032: the source prints M_ii = Σ_{S∋i}|S| (OWR 39/2019, p. 2464). This is a typo for Σ_{S∋i} X_S = deg(i).
  • With that diagonal, det M = |X|·|Y|·τ(G)/∏_{y∈Y} deg y. So Conjecture 5 becomes equivalent to Ehrenborg's Conjecture 4, as the source says.
  • As printed, Conjecture 5 is false. Take X = [3], hang 13 leaves on each vertex of X, and add one vertex joined to all three. Then det M = 850 > 512 = det(diag M). It also fails if the sum runs only over the sets S that occur.
  • With the corrected diagonal, Conjecture 5 is Ehrenborg's conjecture, proved by Ho (arXiv:2603.17997; -030 already cites it).
  • The refinement in -031/-032 is false as literally stated, already at n = 3. There det(diag M) − det M is homogeneous of degree 3 with coefficient −1/4 on X₁₂X₁₃X₂₃. No such polynomial equals Σc_μx^μ + Σc_μν(x^μ − x^ν)² with all c ≥ 0, because the square terms would contribute a nonzero part of even degree. If multipliers x^ρ(x^μ − x^ν)² were intended, the record should say so.
• OWR-14299089-007: as written, the question is trivial: (a_n b_n)² ≥ a_{n−1}a_{n+1}b_{n−1}b_{n+1} for nonnegative sequences. The source asks Brändén–Ferroni–Jochemko's Question 6.1 (Trans. AMS 2026, arXiv:2408.12386). Write Σ p(n)xⁿ = W(p)/(1−x)^(deg p+1). If W(p) and W(q) are log-concave with no internal zeros, is W(pq)?
• OWR-11695867-010: as the source itself says (OWR 57/2022, p. 3274), the case ℓ = 0, Σ f_λ² = n!, is "exactly the Robinson-Schensted-Knuth algorithm". Louf's Open problem 1 is a bijective proof of n!·H_{n,ℓ} = Σ_{λ⊢n} f_λ² C_λ^ℓ for all ℓ. Here H_{n,ℓ} counts ℓ-tuples of transpositions with product 1, and C_λ is the content sum.
• OWR-16164-005: the literal question has a classical answer. Put y_i = 1 − x_i for odd i; each constraint then becomes y_i ≤ y_{i+1} or y_i ≥ y_{i+1}. So the polytope is the order polytope of a fence, a poset whose Hasse diagram is the path 1–2–⋯–n. Its Ehrhart polynomial is Ω(P, t+1) (Stanley, DCG 1 (1986)), so h* is the descent polynomial of its linear extensions (natural labelling). We checked this for all 63 sign sequences with n ≤ 6. Mark solved, or restate if another interpretation was intended.

Duplicates

• Across reports: OWR-12481-012 is the Friedgut half of OWR-12861-020 (OWR 18/2013, p. 1118; OWR 01/2014, p. 81). Splitting OWR-12861-020 would leave there only Diestel's k-block question, which is open.
• New: OWR-14299904-003 repeats -002, which is Weigandt's Conjecture 1 (OWR 2/2026, p. 133); -004 is Conjecture 2.
• Already flagged: the following records are marked as repeats in their own statement_verification, but are still published as open or partially solved. So each of these problems is counted twice. A duplicate status, or unpublishing, would help. The records are OWR-734-010, OWR-1183-007, -012, OWR-4425-019, -021, OWR-4791-016, OWR-4798-022, OWR-11136-011, OWR-14604-015, OWR-16633-022, -024, OWR-16763-026, -028, OWR-17135-032 and OWR-1703876-017.

Merged records

• OWR-1703876-016 (duplicate -017) bundles Problems 6–10 of the OWR 30/2020 problem session (p. 1522), posed by five different people. Splitting would help. Two of its notes need fixing:
  • Problem 10 (Welzl, partial triangulations) is solved (yes) by Kupavskii–Volostnov–Yarovikov, Europ. J. Combin. 108 (2023) (arXiv:2104.05855). -017 lists that arXiv paper under the names Aichholzer–Orden–Schnider.
  • Problem 8 (Steiner, bichromatic triangles) is still open. -016 says a 2026 preprint proves it. In fact Radtke–Keszegh–Lauff (arXiv:2601.20574) prove it only for at most 5 red pseudolines (so for n ≤ 11). In general they prove only that a two-coloured triangle or quadrangle exists.
• OWR-4425-003 bundles Currie's Open problems 2–4 (OWR 37/2010, p. 2206): Restivo–Salemi reachability, the curling-number conjecture and the lexicographically least 5/2-power-free word. Splitting would help.
• OWR-2090-006 bundles Linial's open-ended challenges on Latin squares and the Γ function with one precise conjecture (OWR 44/2008, p. 2496): the maximum rank of a real n×n×n tensor is (1+o(1))n²/2. That conjecture deserves its own record.
• OWR-2090-026: the statement is clean (Problem 9, Barvinok–Samorodnitsky), but original_statement also contains Problem 10 (Welzl, spanning trees versus triangulations). That problem is already OWR-2090-027.

Garbled extracts (all already marked "unrecoverable", but still published as open or partially solved)

• OWR-12861-024: the source itself is ill-posed (OWR 01/2014, p. 83). The matrix has rows indexed by S_n but columns only by {σ : LIS(σ) ≥ n−t}, and it is called both M and A, so its determinant is undefined.
• OWR-9790358-007: the opening of Hakopian's abstract plus its title (OWR 7/2022, p. 405), with no question in it. Its only conjecture, Gasca–Maeztu, is OWR-9790358-001.
• OWR-12697684-002, -003: table-of-contents lines (OWR 1/2023, pp. 9–10). -012: background sentences from Bucić's abstract (p. 42) about the Erdős–Hajnal conjecture, which is -011.
• Also in OWR 1/2023: the "original OWR report" link of OWR-12697684-001, -002, -003, -011 and -012 points to 10.4171/owr/2022/57 (Enumerative Combinatorics). It should be 10.4171/owr/2023/1.
• OWR-14298158-001: a table-of-contents entry (OWR 6/2024, p. 277). -016: the closing remark of an abstract on Bevan's conjecture (p. 299); the conjecture itself is -015.
• OWR-723-001: Beck's four circle-discrepancy questions. The source itself reports them as answered, by Schmidt and by Beck (OWR 13/2004, pp. 678–679). The conjecture that remains is OWR-723-002. Mark solved or remove.

Misattached notes

• OWR-2090-021: the note (parametric assignment, rotation matching) belongs to Rote's Problem 2 of the same session, least-squares matching under rotation (OWR 44/2008, pp. 2546–2547). The relevant literature for this record is Adiprasito–Björner (above).
• OWR-4425-012: the note ("sum-square avoidance", Au–Robertson–Shallit) is about the additive-square problem, Problem 47, which is OWR-4425-018. The record itself is Shallit's Problem 38 on pattern characterisations of α-powers (OWR 37/2010, p. 2231).

— Alper Ferudun

Follow-up 13 (2026-09-30): corrections to records from the AMR open-problem lists (AMR-*) in graph theory, number theory and probability.

• Each item was checked against the current version (commit 372682f) and against its source list or paper.
• Literature was checked against arXiv, Crossref and zbMATH. Theorem numbers are those of the versions we read, usually the arXiv ones.
• Each item was re-verified by independent checks of the source, the mathematics and the literature.
• Where others answered first, they are credited. Short arguments of our own are marked as ours; they are unpublished.
• Where a resolution rests only on an unrefereed preprint, partially_solved is the conservative alternative.
• Suggested changes are in bold. Where we call a record's research summary wrong or out of date, please correct it.

Stale statuses

• AMR-011-0015 → solved (yes, for bounded degree). M. Abért ("Some questions", 2010, Question 15) asks whether the moments ∫z^k dμ_n of the chromatic-root distributions converge along every locally convergent graph sequence. Abért–Hubai, Combinatorica 35 (2015), Thm 1.1: yes under a uniform degree bound, as ∫f dμ_n then converges for every f holomorphic near Sokal's disc.
  • The bound (explicit in Question 12) is needed (our example): K_m plus n isolated vertices, m ≈ √n, tends to a single vertex but has second moment ≈ √n/3. Add "of bounded degree".
  • Csikvári–Frenkel, Eur. J. Combin. 52 (2016), extend the full result to other graph polynomials; the summary calls their work partial.
• AMR-011-0019 → solved (no, for all large d); partially_solved if each d counts separately. B. Szegedy asks (Abért, Question 19) whether random d-regular graphs converge in the local-global sense to the weak closure of factor-of-i.i.d. (FIID) processes on the d-regular tree. Hatami–Lovász–Szegedy (HLS), GAFA 24 (2014), note that Gamarnik–Sudan, Ann. Probab. 45 (2017), disprove it for large d. FIID independent sets have density ≤ (1+o(1)) log d/d (Rahman–Virág, Ann. Probab. 45 (2017)), against 2 log d/d in random graphs.
  • We found no answer for small d such as 3 (the threshold is not explicit), nor to HLS's weaker conjecture that the sequence converges.
  • Statement fix: "factor of i.i.d." (the source's "i.i.d.-s"; literally, the record is trivially false). The summary inverts the local limit, which is the tree. Metadata: proposed_by = Balázs Szegedy.
• AMR-011-0030 → solved (no to the first question, yes to the second by our argument). Abért (Question 30, also Question 4 of the Abért–Szegedy BIRS 2009 report) asks whether FIID processes on the 3-regular tree are weakly closed, and whether the weak limit of majorities on n-balls is an FIID.
  • No: Harangi–Virág, Ann. Probab. 43 (2015), Thm 5 (real-valued), answering the BIRS question; {0,1}-valued: Lyons, CPC 26 (2017), Cor. 3.3, the sign of the Gaussian wave, a weak* limit of FIID by Csóka–Gerencsér–Harangi–Virág, RSA 47 (2015).
  • Majorities (ours): by the CLT the limit is sgn G, G Gaussian with covariance 2^{−m} at distance 2m and (2/3)·2^{−m} at 2m+1. Its spectral measure is absolutely continuous w.r.t. Kesten–McKay, so G is a linear FIID (Backhausz–Virág, AIHP 53 (2017), Thm 2).
• AMR-011-0032 → solved (yes). Abért (Question 32) asks whether the Green function at the identity is continuous on the closed set T of transient k-regular Cayley graphs. Tessera–Tointon (TT), arXiv:2001.01467 v2 (2024; unrefereed), Cor. 1.13: if transient vertex-transitive graphs Γ_n converge locally to Γ, their escape probabilities converge. The Green function is 1/(escape probability) (the expected number of returns is that minus 1), and escape probabilities are positive on T, so it is continuous on T, for rooted-ball and marked-group topologies alike.
  • Cor. 1.13 is new in v2 and answers a question of Benjamini; the link to Question 32 is ours. Its proof uses TT's published growth theorem (Combinatorica 41 (2021)) and a heat-kernel bound from Lyons–Peres. The summary's claim of continuity in several classes (Abért–Thom) is unsupported.
• AMR-011-0033 → solved (yes, even with C independent of k). Abért (Question 33) asks whether some C(k) < 1 bounds the return probability of every transient k-regular Cayley graph. TT (above), Cor. 1.3 of v2 (Cor. 1.7 of v1): for a universal c > 0, the walk on every connected, locally finite vertex-transitive graph is recurrent or escapes with probability ≥ c. So C(k) = 1 − c works; TT do not cite Abért, and the link is ours.
  • The uniform c uses the unrefereed arXiv:2403.02485. Fixed k needs only published work (our remark): transient Cayley graphs have balls of size ≥ εn³, ε universal (TT, Discrete Anal. 2018:17), and Coulhon–Saloff-Coste isoperimetry with a standard resistance bound gives C(k) < 1.
• AMR-011-0039 → solved (yes). Abért (Question 39) asks whether lattices in higher-rank semisimple real Lie groups have rank gradient zero (known for non-uniform ones), Conjecture 17 of Abért–Nikolov, J. Eur. Math. Soc. 14 (2012). Frączyk–Mellick–Wilkens, arXiv:2307.01194 (Ann. of Math., to appear), Thm D: such lattices have fixed price 1, which they note confirms Conjecture 17.
  • By Abért–Nikolov (Thm 1, Cor. 11), rank gradient = cost − 1 along Farber chains and for the absolute rank gradient, so both vanish (other chains can differ).
  • The summary's claim that the full statement is open is out of date; Abért–Gelander–Nikolov, Duke Math. J. 166 (2017), is earlier partial progress. Metadata: category = group_theory.
• AMR-099-0016 → solved (no). I. Benjamini, "Coarse geometry and randomness" (Saint-Flour 2011 notes, author's PDF of 30 Oct 2013), Open problem 2.4 (with A. Georgakopoulos), asks whether every Cayley graph G covers, for some r = r(G), every graph whose r-balls are all isomorphic to those of G. De la Salle–Tessera, J. Topol. 12 (2019), call this LG-rigidity and, citing the notes, ask it for finitely presented groups (Question 1.2). Thm B: F₂×F₂×Z/2 and SL₄(Z) have Cayley graphs that are not LG-rigid.
  • Groups that are not finitely presented fail trivially (Prop. 1.4); Thm A gives positive cases. The summary's date, 2024, misreads the file name ("aug24").
• AMR-099-0027 → solved (yes, for vertex-transitive G_n). The Saint-Flour notes (Open problem 5.10, PDF p. 40) ask whether p_c(G_n) → p_c(G) for expanders G_n → G, an "easier" case of Schramm's locality conjecture (Conj. 5.9), though they define expanders as finite graphs. Hutchcroft, Ann. Probab. 48 (2020), proves locality under uniform exponential growth, which he notes is new even for uniformly nonamenable graphs; Easo–Hutchcroft, arXiv:2310.10983 (unrefereed), in full.
  • The literal finite-graph reading is answered in the source: Thm 5.7, from Benjamini–Nachmias–Peres, PTRF 149 (2011), Thm 1.3.
  • Add "vertex-transitive", or it fails (ours): a radius-n ball of T₃ with 3-ary trees glued to its leaves is uniformly nonamenable and tends to T₃, but has p_c = 1/3, not 1/2. The summary's arXiv:0910.1809 is an unrelated QED paper.
• AMR-099-0045 → solved for monotonicity and continuity; the scaling limit is proved for Bäumler's kernel, but we found no proof for β/k² ∧ 1 (partially_solved if the record means that kernel). The Saint-Flour notes (Open problem 11.21, PDF p. 92): in long-range percolation on Z with s = 2, distances grow like n^{θ(β)} (Ding–Sly, arXiv:1303.3995, unrefereed); whether θ is continuous or monotone was unknown, and a nontrivial scaling limit should exist.
  • J. Bäumler, EJP 30 (2025): θ is strictly decreasing (Thm 1.1) and continuous (Thm 1.2); by his CMP 404 (2023), Thm 7.1, θ is the same for Ding–Sly's kernel β/k² ∧ 1.
  • Ding–Fan–Huang, Mem. AMS 318 (2026), Thm 1.1: the rescaled distance converges in law to a unique nontrivial random pseudometric, for Bäumler's kernel 1 − exp(−β∬|x−y|⁻²) and the continuum model.
  • Add nearest-neighbour edges to the statement (else many vertices are isolated when β < 1). The summary calls both questions open.
• AMR-099-0046 → partially_solved (Z⁴: yes; Z³: open). The Saint-Flour notes (Open problem 12.33, PDF p. 104) ask whether two simple random walks on Z³ or Z⁴ started at distance 10 can be coupled so that their paths are disjoint with positive probability. Benjamini–Kozma, arXiv:2412.16600 (2024; unrefereed), construct such a coupling in Z⁴.
  • They pose it for neighbouring starting points, but their construction starts from 0 and an arbitrary x, so it covers distance 10 (our reading).
  • Z³: the authors could not resolve it (d = 2 is impossible, d ≥ 5 trivial), and we found no later work. The summary is out of date.
• AMR-099-0025 → partially_solved. The Saint-Flour notes (Conjecture 4.34, PDF p. 38): isoperimetric dimension I-dim > 1 implies connective constant μ > 1. As p_c ≥ 1/μ, each p_c < 1 theorem gives μ > 1:
  • I-dim > 2, bounded degree: Easo–Severo–Tassion, Forum Math. Pi 13 (2025), Thm 2 with §2, superseding I-dim > 4 of Duminil-Copin–Goswami–Raoufi–Severo–Yadin (DGRSY), Duke Math. J. 169 (2020).
  • Quasi-transitive graphs are settled: Grimmett–Li, Combinatorica 35 (2015), and DGRSY's corollary. Also Kozma, Rev. Mat. Iberoam. 23 (2007) (planar, polynomial growth); Teixeira, PTRF 165 (2016); Candellero–Teixeira, AIHP 54 (2018).
  • We found no answer for general bounded-degree graphs with 1 < I-dim ≤ 2. The summary is out of date.
• AMR-099-0085 → solved (yes, for bond percolation). Benjamini–Schramm (BS), ECP 1 (1996), Conjecture 5: on a connected quasi-transitive graph with a.s. several infinite clusters, each has 2^ℵ₀ ends. Their 1999 update (linked in the record) leaves open only nonunimodular graphs with an infinite cluster at p_c.
  • Unimodular: Häggström–Peres, PTRF 113 (1999); Lyons–Schramm, Ann. Probab. 27 (1999), Prop. 3.10. Nonunimodular, p > p_c: Häggström–Peres–Schonmann, Perplexing Problems in Probability (1999).
  • At p_c, bond percolation on nonunimodular graphs has no infinite cluster (Timár, Ann. Probab. 34 (2006): not infinitely many; Hutchcroft, C. R. Math. 354 (2016): none under exponential growth, which all nonunimodular graphs have).
  • BS use site percolation but say their questions apply to bond too. For site percolation at p_c we found a proof only on transitive graphs (Timár's Remark 5.11 says his proofs carry over, and several infinite clusters means infinitely many), so partially_solved for a site reading. The summary's citations are wrong.
• AMR-099-0083 → solved (yes). I. Benjamini ("Random planar metrics", §4.1, p. 9) conjectures that for every C there is c < 1 such that simple random walk covers an n-vertex simple graph within Cn steps with probability less than cⁿ. Dubroff–Kahn, Ann. Probab. 53 (2025), Thm 1.1, prove it as "a 2009 conjecture of Itai Benjamini".
  • Earlier cases: bounded degree, Benjamini–Gurel-Gurevich–Morris, PTRF 155 (2013); trees, Yehudayoff, Chicago J. Theor. Comput. Sci. 2012.
  • The bound needs n ≥ 3 (on K₂ the walk covers in one step), which the source and Dubroff–Kahn leave implicit. The summary's "not established" is out of date; add the literature source.
• AMR-083-0028 → solved (yes). Adleman–McCurley ("Open problems in number theoretic complexity, II", ANTS I, 1994, O24) ask, as Schoof had in 1985, whether a point on y² = x³ + ax + b mod p can be found in deterministic polynomial time. Shallue–van de Woestijne, ANTS VII (2006), do this for every finite field and Weierstrass equation, extending Skałba, Acta Arith. 117 (2005).
  • They reduce it to a genus-0 curve on Skałba's threefold f(x₁)f(x₂)f(x₃) = y², using a deterministic conic solver and a "selective" square root, deterministic because this product has a known square root (van de Woestijne's thesis, Thm 1.11, Ch. 3).
  • The summary's p ≡ 3 mod 4 trick gives points on the quadratic twist, not on the curve. Metadata: category = number_theory (all 40 AMR-083 records are filed under graph_theory).
• AMR-086-0045 → solved (no). Mahler, Bull. Austral. Math. Soc. 29 (1984), asked whether some transcendental entire function maps Liouville numbers to Liouville numbers (M. Waldschmidt, "Open Diophantine problems", 2004, Question 3.27). D. Marques, arXiv:2609.14202 v2 (Sept. 2026; unrefereed), Thm 1.1: an entire F mapping the Liouville numbers into themselves lies in R[z].
  • Thms 1.1–1.2 are formalized in Lean 4 (MahlerLean). Earlier results for subclasses: Marques–Moreira, Bull. Aust. Math. Soc. 91 (2015); Marques–Schleischitz, J. Aust. Math. Soc. 100 (2016).
  • Metadata: proposed_by = Kurt Mahler, category = number_theory (all 60 AMR-086 records credit the survey's author and are filed under graph_theory).
• AMR-086-0065 → solved (yes, for every n ≥ 2). Bugeaud's question (Waldschmidt's survey, Question 5.2): is the set ZHₙ of reals approximable by algebraic numbers of degree ≤ n exactly to order H(α)^{−n−1} larger than the set of algebraic numbers of degree n+1? Beresnevich, Invent. Math. 202 (2015), Thm 3: ⋂_{k≤n} (W*ₖ ∩ B*ₖ) has Hausdorff dimension 1 in every interval, settling Bugeaud's Problems 24–25 (Zbl 1338.11060).
  • W*ₙ ∩ B*ₙ ⊂ ZHₙ (our check: Beresnevich uses real α only; non-real α meet the lower bound by root separation), so ZHₙ is uncountable and contains transcendental numbers. Degree-(n+1) algebraic numbers lie in it by Liouville and Beresnevich's Appendix B. The summary's n = 1 remark is beside the point. Metadata: proposed_by = Yann Bugeaud, category = number_theory.
• AMR-086-0067 → solved (yes). Loxton–van der Poorten's conjecture (Waldschmidt's survey, Conjecture 5.4): if Σ z^{nᵢ} ∈ F_p[[z]] is algebraic and irrational over F_p(z), then Σ 10^{−nᵢ} is transcendental. By Christol's theorem the 0/1 sequence of the nᵢ is then p-automatic and not eventually periodic, so Σ 10^{−nᵢ} is an irrational automatic number, hence transcendental by Adamczewski–Bugeaud, Ann. of Math. 165 (2007), Thm 2, which they present as this conjecture (Zbl 1195.11094). Announced in Adamczewski–Bugeaud–Luca, C. R. Math. 339 (2004).
  • The summary's "open in full generality" is out of date. Metadata: proposed_by = John H. Loxton and Alfred J. van der Poorten, category = number_theory.
• AMR-090-0001 → solved (no to the first question, yes to the second). S. Ramassamy, Arnold Math. J. 3 (2017), Conjecture 1: for odd p, the Euler zigzag numbers mod p^k have preperiod s(p^k) = k and period d(p^k) = p^{k−1}d(p). B. Güleç, arXiv:2608.27058 v2 (Sept. 2026; unrefereed; its declaration says generative AI wrote up the proofs from the author's arguments), proves the period formula (Thm 5.9) and shows that s(5⁵) = 4 is the smallest counterexample (Thm 6.4).
  • Our check: E₄ ≡ E₂₅₀₄ but E₃ ≢ E₂₅₀₃ mod 3125, while s ≤ 5 and d | 2500 by Knuth–Buckholtz.
  • Metadata for all 3 AMR-090 records: proposed_by = Sanjay Ramassamy, proposed_year = 2017 (not 2020, as the source list and this record's summary say), category = number_theory.
• AMR-093-0064 → solved (yes). The record is the first "conjecture" on Wikipedia's Hardy–Littlewood zeta function conjectures, linked from the pinned list (oldid 1366636767): for T ≥ T₀(ε), (T, T + T^{1/4+ε}] contains a zero of odd order of ζ(1/2+it). It is Theorem 4.41 (p. 184) of Hardy–Littlewood, Acta Math. 41 (1916); the summary calls it open.
  • The page says they "claimed" it, citing their Math. Z. 10 (1921), as does Zbl 1455.11121; that paper proves the density results of AMR-093-0183 instead. Later improvements: T^{1/6} log² T (Mózer, Acta Arith. 31 (1976)), T^{5/32} log² T (Karatsuba, Trudy MIAN 157 (1981)).

Answered in the source

• AMR-093-0183 → solved (yes; answered in the source). The second "conjecture" on the same page: N₀(T+H) − N₀(T) ≥ cH for H = T^{1/2+ε}, where N₀ counts zeros of odd order on the critical line. The page's own Status section says Selberg proved it in 1942 (writing N for N₀).
  • First proved by Hardy–Littlewood, Math. Z. 10 (1921), Theorem B (U = T^a, a > 1/2). Selberg (Skr. Norske Vid.-Akad. Oslo 1942, no. 10; Zbl 0028.11101) proved ≫ H log T, as Ivić recalls in his review Zbl 0545.10026 of Karatsuba (1984), who lowered the exponent to 27/82 + ε. Both Hardy–Littlewood and Selberg count sign changes of Hardy's Z(t), i.e. distinct zeros of odd order.
  • The summary's "major open problem" is wrong.
• AMR-086-0059 → solved (no; answered in the source). Mazur's question in Waldschmidt's survey (§4.4, Question 4.17): for a smooth variety V over Q with V(Q) Zariski dense, is the real closure Z of V(Q) a union of connected components of V(R)? The survey itself then says no: Colliot-Thélène–Skorobogatov–Swinnerton-Dyer, Acta Arith. 79 (1997), §5, disprove it with smooth surfaces over Q (Prop. 5.1; Ex. 5.1.1, 5.2, 5.3).
  • The summary's high-genus counterexamples (Poonen) cannot exist: by Faltings, curves of genus ≥ 2 never have Zariski-dense rational points.
  • Statement fix: define V and Z. The abelian-variety case is AMR-086-0060 (Conj. 4.18, open). Metadata: proposed_by = Barry Mazur, category = number_theory.

AIM-REPRESENTATION_THEORY-0085 source normalization correction

This is a correction to the remark in AIM's 2010 list, Problem 1.1, not a solution of the unrestricted classification and not a new preprint. Suggested status: retain partially_solved, correct the symmetric-group remark, and record the standard algebra-group positive case with attribution.

The question fixes the canonical basis sigma_X = sum_{chi in X} chi(1) chi. Its remark asserts integrality for every minimal symmetric-group theory. With this normalization that assertion is false: the minimal theory of S_n has integral products exactly for n <= 3.

For the smallest counterexample S_4, let chi_3 be the standard character, chi'_3 = sgn * chi_3, and chi_2 the irreducible character of degree two. Then

$${\chi }_3^2=1+{\chi }_2+{\chi }_3+{\chi }_3^{\mathrm{\prime }},{\sigma }_3^2=9{\sigma }_1+\frac{9}{2}{\sigma }_2+3{\sigma }_3+3{\sigma }_3^{\mathrm{\prime }}\mathrm{.}\backslash chi\_3^2=1+\backslash chi\_2+\backslash chi\_3+\backslash chi\text{'}\_3,\backslash qquad\; \backslash sigma\_3^2=9\backslash sigma\_1+\backslash frac92\backslash sigma\_2+3\backslash sigma\_3+3\backslash sigma\text{'}\_3.$$

More generally, the coefficient of sigma_(n-2,2) in sigma_(n-1,1)^2 is 2(n-1)^2/[n(n-3)]: it equals 9/2 for n = 4, 16/5 for n = 5, and lies strictly between two and three for n >= 6. The tensor-square formula is already published in Bessenrodt-Kleshchev (1999), Corollary 4.2(iv), p. 206; this is a normalization-sensitive consequence of classical work, not a priority claim. For S_3, sigma_2^2 = 4 sigma_1 + 4 sigma_sgn + 2 sigma_2, and the smaller groups have only linear characters.

The remark's standard algebra-group clause is affirmative even in the canonical, rather than merely a scaled, basis. For G = 1+J, Diaconis-Isaacs (2008), Theorems 5.5(b) and 5.6 give sigma_O = sum_{lambda in O} hat(lambda) for a two-sided orbit in J*. Its product coefficient for an orbit C counts pairs (alpha,beta) in A x B with alpha+beta = rho, for any fixed rho in C. The common two-sided action is linear, so this nonnegative integer count is constant on C. This is a consequence of those published identities; it concerns the standard algebra-group theory, not every theory on the same abstract group.

The broad "when" classification and the ambiguously worded four-family clause are not resolved here. We checked the normalization directly and corroborated the S_4 identity on all 24 permutations, with exact small algebra-group orbit/Fourier controls. These are AI-assisted, self-audited calculations, not human refereeing or formal certification. Corrections welcome. — Alper Ferudun, Mercury Software GmbH

AMR-021-0015: complete counterexample to the Forsgård–Shapiro coefficient-parity bound (Shapiro 2015, Section 7, Conjecture 10).

For strictly positive coefficients, define c_k = a_k^2 - a_(k-1)a_(k+1), with zero outside coefficients, and count parity changes among all indices with c_k >= 0. An explicit degree-77 polynomial has exactly one such parity change but at least three distinct negative real roots, disproving the original bound. The proof uses reciprocal blocks and an exact rational inequality; the public package includes primitive integer coefficients and reproducible exact-arithmetic checks.

This is an English, AI-assisted, self-audited, unrefereed preprint. It does not settle the neighboring weighted Conjecture 9, and makes no minimum-degree, exact-total-root-count or absolute-priority claim.

Preprint and reproducibility package: https://zenodo.org/records/23050917

DOI: https://doi.org/10.5281/zenodo.23050917

Project page: https://eulersolve.org/papers/amr-021-0015/

Original source: Shapiro, DOI https://doi.org/10.1007/s40598-015-0008-4, Section 7, Conjecture 10.

AMR-096-0015 version 1.1 update

Version 1.1 of Excursion Lengths in a Uniform Eulerian Circuit of the Complete Graph: Aldous's Conjecture Holds if and only if i = o(n^(3/2)) is now available. This is a revision of the previously announced manuscript, not an additional problem solution.

The theorems are unchanged, and the proofs change only in two wording clarifications. A numerical remark in Section 7 about the deviation from the limit at the scale i ~ y n^(3/2) is corrected; the proofs do not use it. The text now says that all circuits were enumerated only for n ≤ 5, and that the n = 6 values come from the exact formula of Remark 2.5. Related work by Hu–Lyons–Tang and Farrell–Levine is cited, bibliographic data are corrected, and the verification report and the reproducibility package are updated.

PDF, source and updated verification report: https://zenodo.org/records/23051216

Version DOI: https://doi.org/10.5281/zenodo.23051216

Paper page: https://eulersolve.org/papers/amr-096-0015/

The original v1.0 remains archived under the same concept DOI, https://doi.org/10.5281/zenodo.23049689. The scope is unchanged: only example (a) of Aldous–Yu, the complete graph, is settled; the torus conjecture, its two-dimensional variant and the Hamming-cube example (b) remain open. This is AI-assisted, self-audited and unrefereed work, not independent human peer review or formal certification. No absolute-priority claim is made. Corrections and prior-work pointers are welcome. — Alper Ferudun, Mercury Software GmbH

OWR-14299911-029 version 1.1 update

Version 1.1 of A Counterexample to a Corner-Peeling Conjecture for Subcomplexes of Z^3 is now available. This is a revision of the previously announced manuscript, not an additional problem solution.

New in v1.1: a smaller counterexample with 49 vertices, a partial subgraph in [0,3]^3 (Remark 6.3). It was found in an exploratory solver search during an independent verification run and was checked by computer. Remark 3.2 is corrected: the 62-vertex graph returned by the vertex-minimising program is the mirror image of G62, not a second example. Lemma 6.1 now includes the torsion step. The abstract now says that both examples are invariant under a cyclic group of order 6; the full symmetry group of H73 has order 12. The description of the solver runs is corrected, a scope statement on isometric subgraphs is added, and the Verification paragraph now states that its "referees" are independent, AI-assisted verification runs, not peer reviews.

PDF, source and updated verification report: https://zenodo.org/records/23051217

Version DOI: https://doi.org/10.5281/zenodo.23051217

Paper page: https://eulersolve.org/papers/owr-14299911-029/

The original v1.0 remains archived under the same concept DOI, https://doi.org/10.5281/zenodo.23049728. The disproof of the conjecture as stated, for partial and for induced subgraphs, is unchanged. The version with the additional condition (3) is not refuted, the variant for isometric subgraphs is not addressed, and the least size of a counterexample remains open (at most 49 vertices, at most 73 for induced subgraphs). This is AI-assisted, self-audited and unrefereed work, not independent human peer review or formal certification. No absolute-priority claim is made. Corrections and prior-work pointers are welcome. — Alper Ferudun, Mercury Software GmbH

AMR-021-0014: complete counterexample to the weighted Forsgård–Shapiro parity bound (Shapiro 2015, Section 7, final-journal Conjecture 9).

For positive coefficients, define tilde_c_k = (k+1)a_k^2 - k a_(k-1)a_(k+1), with zero outside coefficients, and count parity changes among ALL indices with tilde_c_k > 0. An explicit degree-1,000,077 polynomial with positive rational coefficients has one selected-index parity change but at least three distinct negative real roots, disproving the printed bound.

The construction shifts the previously published degree-77 unweighted reciprocal-block seed and fills every missing coefficient with a tiny, strictly log-convex factorial prefix. Exact rational seed margins and analytic prefix bounds verify the counterexample without expanding the million-term prefix.

This is an English, AI-assisted, self-audited, unrefereed preprint. No independent human peer review, minimum-degree, exact-total-root-count or absolute-priority claim is made.

Preprint and reproducibility package: https://zenodo.org/records/23051793

DOI: https://doi.org/10.5281/zenodo.23051793

Project page: https://eulersolve.org/papers/amr-021-0014/

Earlier unweighted seed: https://doi.org/10.5281/zenodo.23050917

Original source: Shapiro, DOI https://doi.org/10.1007/s40598-015-0008-4, Section 7, Conjecture 9.

AMR-021-0016: a finite realizability criterion for Shapiro2015 Problem6 (derivative-root configurations).

This preprint gives a necessary and sufficient finite moment-cone criterion for the prescribed numerical positions of all derivative roots of a real-rooted polynomial-like function. Polynomial-like degree n means that the nth derivative is nowhere zero; it does not mean an ordinary polynomial of degree n. The criterion includes permitted cross-order coincidences and gives a terminating semialgebraic decision procedure for algebraic coordinates. A feasible array also has an ordinary-polynomial realization of unspecified larger degree.

The generic positive-moment and interpolation techniques are classical, with credit to Kakeya (1915), di Dio (2019/2025), and Pinkus–Wulbert (2005). Whether this explicit application is already-known remains uncertain; an inaccessible 1979 Hermite–Birkhoff/monotone-spline paper remains an overlap lead. This is an AI-assisted, self-audited, unrefereed preprint, with no independent human review and no absolute historical-priority claim. It does not claim formal proof-assistant certification, a practical running-time bound, or minimal witness degree.

Source: https://doi.org/10.1007/s40598-015-0008-4
Paper and exact reproducibility artifacts: https://zenodo.org/records/23054600
DOI: https://doi.org/10.5281/zenodo.23054600
Readable paper page: https://eulersolve.org/papers/amr-021-0016/