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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

Status follow-up 87: 11 status corrections (records from problem lists on visibility graphs and on Grünbaum's problems in many dimensions)

• Each item was checked against the current dataset (commit d371ee9, release v1.8.0), against its source (S. K. Ghosh, P. P. Goswami, Unsolved Problems in Visibility Graphs of Points, Segments, and Polygons, ACM Comput. Surv. 46 (2013) (arXiv:1012.5187), records AMR-131-*; M. Kahle, Branko Grünbaum in Many Dimensions: Open Problems, 2019 (arXiv:1901.08622), AMR-142-*; cited by problem number) and against the cited papers, read on arXiv or in an open journal or author version; publication data are from Crossref. Theorem numbers follow the versions read. Suggested changes are in bold.
• Following the policy of release v1.8.0, answers that exist only in preprints are listed separately.

Solved in published papers (labels → solved)

• AMR-131-0012 and AMR-131-0013 → solved (yes, unless all segments lie on one line); the dataset currently labels both open. S. K. Ghosh, P. P. Goswami, Unsolved problems in visibility graphs of points, segments, and polygons, ACM Comput. Surv. 46 (2013), no. 2 (arXiv:1012.5187; numbers and pages of v4, February 2012), Section 3.2, Conjectures 7 and 8, p. 16 (the records' Conjectures 14 and 15), both due to A. Mirzaian: the endpoint visibility graph G of a set of disjoint segments contains a Hamiltonian cycle (Conjecture 7), and even one that bounds a simple polygon when drawn on the segments, a Hamiltonian circuit (Conjecture 8). The list itself records the answer: in the paragraph after the conjectures it states that Conjecture 8 was proved for all sets of segments by M. Hoffmann, C. D. Tóth, Segment endpoint visibility graphs are Hamiltonian, Comput. Geom. 26 (2003) 47–68, who also give an O(n log n) algorithm, and that Conjecture 7 follows. The result needs that not all segments lie on one line; for collinear segments G is a path (our remark). In this form it is recalled by H. A. Akitaya, M. Korman, O. Korten, M. Rudoy, D. L. Souvaine, C. D. Tóth, Discrete Comput. Geom. 68 (2022) 218–254 (arXiv:1903.07019, p. 2): every set of disjoint segments, unless all are collinear, has a simple polygon through all endpoints whose sides cross no segment. Such a polygon has no endpoint inside a side, so its sides are edges of G also under the list's definition, which forbids passing through endpoints (our remark). The statement of AMR-131-0012 begins with the caption of the list's Figure 12. Read: Section 3.2 and the definitions of the list; Theorem 1 of the conference version of Hoffmann–Tóth (Proc. CCCG 2001, for segments with no three collinear endpoints); p. 2 of Akitaya et al. The journal version was not accessible to us.
• AMR-131-0014 → solved (no); the dataset currently labels it open. Ghosh–Goswami, Conjecture 9, p. 16 (the record's Conjecture 16), also due to Mirzaian: G contains a Hamiltonian cycle that bounds a circumscribing polygon, a simple polygon through all endpoints in which every segment is a side or an internal chord. The list itself records the answer in the same paragraph: M. Urabe, M. Watanabe, On a counterexample to a conjecture of Mirzaian, Comput. Geom. 2 (1992) 51–53, gave a counterexample. The same is recalled in Section 1 of the conference version of Hoffmann–Tóth and by Akitaya et al. (p. 2), who give its size, 16 segments, and recall a 6-segment example of B. Grünbaum (Geombinatorics 3 (1994)). Read: the conjecture and the paragraph after it; Section 1 of the CCCG version of Hoffmann–Tóth; p. 2 of Akitaya et al. The paper of Urabe–Watanabe was not accessible to us.
• AMR-131-0009 → solved (no); the dataset currently labels it open. Ghosh–Goswami, Section 2.5, Open Problem 11, p. 13 (the record's Problem 10), a question of H. Alpert, C. Koch, J. D. Laison (Discrete Comput. Geom. 44 (2010)): does every planar graph have obstacle number 1? No: L. W. Berman, G. G. Chappell, J. R. Faudree, J. Gimbel, C. Hartman, G. I. Williams, Graphs with obstacle number greater than one, J. Graph Algorithms Appl. 21 (2017) 1107–1119 (arXiv:1606.03782), Proposition 3: the icosahedron, and also the 10-vertex skeleton of the gyroelongated square bipyramid, have obstacle number 2; the paper presents this as a negative answer to the question of Alpert, Koch and Laison. The lower bounds are computer-assisted: SAT instances that encode a representation with one obstacle are unsatisfiable (checked with three solvers). The paper allows any connected sets as obstacles, so its lower bound also holds for the simple polygons in general position used by the list (our remark). The record's statement also carries Open Problem 10 of the list, in LaTeX markup, before the question. Read: Section 2.5 of the list; the abstract, Sections 1–2 and Proposition 3 of the journal version.
• AMR-131-0001 → solved (the recognition problem is ∃ℝ-complete); the dataset currently labels it open. Ghosh–Goswami, Section 2.1, Open Problem 1, p. 4: given a graph G in adjacency matrix form, determine whether G is the visibility graph of a set of points in the plane. J. Cardinal, U. Hoffmann, Recognition and complexity of point visibility graphs, Discrete Comput. Geom. 57 (2017) 164–178 (arXiv:1503.07082), Theorem 7: the recognition of point visibility graphs is complete for the existential theory of the reals. The problem is therefore decidable (it lies in ∃ℝ ⊆ PSPACE; the paper credits membership in ∃ℝ to S. K. Ghosh and B. Roy) and NP-hard (shown earlier by B. Roy, as the paper recalls), so a polynomial-time algorithm would imply P = NP. The paper states the problem as the record does, notes that it is the first open problem of this list and presents its theorem as the solution. If the item is read as asking for an efficient algorithm, the answer is negative unless P = NP. Read: the problem; the abstract, Section 1 and Theorem 7 (Section 4).
• AMR-131-0007 → solved (improved; now Ω(n/log log n)); the dataset currently labels it open. Ghosh–Goswami, Open Problem 9, p. 13 (the record's Problem 8): improve the lower bound Ω(n/log² n) of P. Mukkamala, J. Pach, D. Sarıöz (WG 2010) on the largest obstacle number of n-vertex graphs (the list writes O for these lower bounds). Improved first by P. Mukkamala, J. Pach, D. Pálvölgyi, Lower bounds on the obstacle number of graphs, Electron. J. Combin. 19(2) (2012) P32 (arXiv:1103.2724), Theorem 2: for every n there is an n-vertex graph with obstacle number Ω(n/log n), for closed polygonal obstacles in general position as in the list; then by V. Dujmović, P. Morin, On obstacle numbers, Electron. J. Combin. 22(3) (2015) P3.1, Theorem 1: Ω(n/(log log n)²); and by M. Balko, S. Chaplick, R. Ganian, S. Gupta, M. Hoffmann, P. Valtr, A. Wolff, Bounding and computing obstacle numbers of graphs, SIAM J. Discrete Math. 38 (2024) 1537–1565 (arXiv:2206.15414), Theorem 1: Ω(n/log log n) (Theorem 2: Ω(n) for convex obstacles). The item asks only for an improvement. The order of growth is still open; the best upper bound is O(n log n) (see AMR-131-0006). Read: Section 2.5 of the list; the abstracts, introductions and cited theorems of the three papers (journal versions of the first two, arXiv v3 of the third).
• AMR-142-0002 → solved (false: counterexamples on every orientable surface of genus ≥ 5); the dataset currently labels it open. M. Kahle, Branko Grünbaum in many dimensions (Geombinatorics 2019; arXiv:1901.08622, v2 of 14 July 2019), Conjecture 2.2 (the record's Conjecture 2), PDF p. 2, Grünbaum's conjecture of 1968: if G is a simple loopless triangulation of an orientable surface, then the dual graph of G is 3-edge-colourable. The list itself records the answer in the paragraph after the conjecture: for every genus g ≥ 5 there is a triangulation whose dual graph is not 3-edge-colourable. M. Kochol, Polyhedral embeddings of snarks in orientable surfaces, Proc. Amer. Math. Soc. 137 (2009) 1613–1619 (conference version in Graph Drawing 2008, Lecture Notes in Comput. Sci. 5417 (2009) 319–323, which the list cites): for each orientable surface of genus at least 5 there are infinitely many 3-regular graphs without a 3-edge-colouring that have a polyhedral embedding in the surface, i.e. an embedding whose dual is a simple graph. The duals of these embeddings are simple loopless triangulations whose dual graphs are not 3-edge-colourable, as the list says. The torus is AMR-142-0003 below; genus 2 to 4 are open as far as we found. Follow-up 55 cited the same paper for the related record OWR-784-006. Read: the conjecture and the paragraph after it; the abstract of Kochol's paper (Crossref, the same text as the zbMATH summary; the paper itself not read).
• AMR-142-0003 → solved (yes); the dataset currently labels it open. Kahle's list, Conjecture 2.3 (the record's Conjecture 3), PDF p. 2: if G is a simple loopless triangulation of a torus, then the dual graph of G is 3-edge-colourable, the torus case of Conjecture 2.2, which the list calls open. Y. Inoue, K. Kawarabayashi, A. Miyashita, B. Mohar, T. Sonobe, Three-edge-coloring (Tait coloring) cubic graphs and nowhere-zero 4-flow for graphs on the torus, Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM (2026) 6133–6165 (full version arXiv:2505.07002, whose title calls it a proof of Grünbaum's conjecture), Theorem 1.3 of the arXiv version: the only snarks embeddable in the torus are the Petersen graph and certain dot products of copies of it (the Blanuša snark and the families Blanuša-H_k and Blanuša-V_k). By the abstract of the arXiv version, a 2-connected cubic graph in the torus is not 3-edge-colourable if and only if it arises from such a dot product by replacing vertices with planar cubic pieces. The introduction concludes that "the dual graph of a triangulation in the torus is always 3-edge-colorable", that is, every triangulation of the torus has a Grünbaum colouring, which is equivalent to a 3-edge-colouring of the dual. The abstract of the proceedings version says that the result proves a strengthening of Grünbaum's 1968 conjecture. The proof relies on extensive computer verification (more than 14000 reducible configurations; the code is public). Earlier, M. O. Albertson, H. Alpert, s.-m. belcastro, R. Haas, Grünbaum colorings of toroidal triangulations, J. Graph Theory 63 (2010) 68–81 (arXiv:0805.0394), Theorem 1.1, had proved the statement for every toroidal triangulation G with chromatic number χ(G) ≠ 5. Read: the conjecture; Sections 1 and 2 of the arXiv version; the abstract of the proceedings version (the proceedings text itself not read); the abstract and Sections 1–2 of Albertson et al.
• AMR-142-0004 → solved (yes); the dataset currently labels it open. Kahle's list, Question 3.1 (the record's Question 4), PDF p. 3: is every triangulation of the 2-dimensional torus polyhedral, i.e. does it embed in ℝ³ with straight edges and flat triangles? The list itself records the answer in its Update paragraph on the same page: yes, by Archdeacon, Bonnington and Ellis-Monaghan (pointed out by G. Ziegler). D. Archdeacon, C. P. Bonnington, J. A. Ellis-Monaghan, How to exhibit toroidal maps in space, Discrete Comput. Geom. 38 (2007) 573–594, Theorem 1.1: every polyhedral map on the torus can be exhibited; Corollary 1.2: every toroidal triangulation has a geometric realization (vertices as points, edges as straight segments, flat faces, distinct faces meeting only along common boundaries; by Proposition 2.2 an exhibition of a triangulation is such a realization). In the paper a triangulation is a map of a simple graph whose faces are bounded by 3-cycles (Section 2), which covers every triangulation of the torus as a simplicial complex, and Corollary 1.2 is presented as an affirmative answer to Grünbaum's conjecture for toroidal triangulations. Read: the update; the abstract and Sections 1–2 of the journal version.

Answered in preprints (not refereed; under the dataset policy of release v1.8.0: partially solved, evidence unrefereed_claim)

• AMR-131-0032 → solved (yes, with the polygon as input; extended abstract, not refereed); the dataset currently labels it open. Ghosh–Goswami, Section 4.6, Open Problem 37, p. 27: design an approximation algorithm for computing the maximum hidden set of a visibility graph (hidden vertex sets of a simple polygon are the independent sets of its visibility graph). C. Alegría, P. Bhattacharya, S. K. Ghosh, A 1/4-approximation algorithm for the maximum hidden vertex set problem in simple polygons, extended abstract, EuroCG 2019 (Utrecht), Theorem 4.1: for a simple polygon P with n vertices, a hidden vertex set of at least 1/4 of the maximum size can be computed in O(n²) time (the abstract assumes, for simplicity, that no three vertices of P are collinear). The problem is APX-hard (S. Eidenbenz, as recalled there), so a constant ratio is the strongest kind of guarantee to expect. The abstract asks to be regarded as a preprint rather than a formally reviewed paper, and we found no later version (Crossref, arXiv). As for AMR-131-0030, the algorithm uses the polygon, while the item speaks of the visibility graph. The record's statement also carries Open Problem 36 of the list, in LaTeX markup, before the question. Read: Section 4.6 of the list; the whole extended abstract.

Partial results in print (labels → partially solved)

• AMR-131-0006 → partially solved (linear for graphs of bounded subchromatic number, e.g. bipartite and split graphs and graphs of bounded chromatic number; O(n log n) in general); the dataset currently labels it open. Ghosh–Goswami, Section 2.5, Open Problem 8, p. 12 (the record's Problem 7), a question of Alpert, Koch and Laison: is the obstacle number of an n-vertex graph bounded above by a linear function of n? The list mentions only the trivial bound n(n − 1)/2. M. Balko, J. Cibulka, P. Valtr, Drawing graphs using a small number of obstacles, Discrete Comput. Geom. 59 (2018) 143–164 (arXiv:1610.04741), Theorem 3: every n-vertex graph G can be represented with at most (n − 1)(⌈log χ_s(G)⌉ + 1) convex obstacles, where the subchromatic number χ_s(G), the least number of classes in a partition of the vertices into sets that induce disjoint unions of cliques, is at most the chromatic number of G and of its complement; Theorem 1: at most n − 1 for bipartite and split graphs and their complements; Theorem 2: at most n⌈log n⌉ − n + 1 for every graph. The obstacles there are polygons and the points are in general position, as in the list. The paper says that a linear bound for all graphs remains open, and so do Balko et al. (SIAM J. Discrete Math. 38 (2024); arXiv:2206.15414, p. 4), whose Theorem 1 gives n-vertex graphs with obstacle number Ω(n/log log n). Read: Section 2.5 of the list; the abstract and Section 1 (Theorems 1–3) of Balko–Cibulka–Valtr; the abstract and Section 1 of Balko et al.

These are AI-assisted, self-audited checks, not independent human peer review: for each item we read the source and the cited statements and took publication data from Crossref; we did not check the proofs. The suggested labels are suggestions. Corrections are welcome. — Alper Ferudun, Mercury Software GmbH

Status follow-up 88: 7 status corrections (records from the problem list on arithmetic dynamics)

• Each item was checked against the current dataset (commit d371ee9, release v1.8.0), against its source (R. Benedetto, L. DeMarco, P. Ingram, R. Jones, M. Manes, J. H. Silverman, T. J. Tucker, Current Trends and Open Problems in Arithmetic Dynamics, Bull. Amer. Math. Soc. 56 (2019) (arXiv:1806.04980), records AMR-126-*; cited by problem number) and against the cited papers, read on arXiv or in an open journal or author version; publication data are from Crossref. Theorem numbers follow the versions read. Suggested changes are in bold.
• Following the policy of release v1.8.0, answers that exist only in preprints are listed separately.

Solved in published papers (labels → solved)

• AMR-126-0008 → solved (no); the dataset currently labels it open. R. Benedetto, L. DeMarco, P. Ingram, R. Jones, M. Manes, J. H. Silverman, T. J. Tucker, Current trends and open problems in arithmetic dynamics, Bull. Amer. Math. Soc. 56 (2019) 611–685 (arXiv:1806.04980; item numbers and pages of arXiv v2, which in places differ from the journal version), Conjecture 5.4 (Odoni), p. 10: for every Hilbertian field K and every d ≥ 2 there are a monic f ∈ K[X] of degree d and a point P ∈ K with Odoni index ι_K(f, P) = 1, i.e. the arboreal Galois representation on the complete preimage tree of P is surjective (Section 5 works in characteristic 0). No: P. Dittmann, B. Kadets, Odoni's conjecture on arboreal Galois representations is false, Proc. Amer. Math. Soc. 150 (2022) 3335–3343 (arXiv:2012.03076), Theorem 1.2: every countable Hilbertian field k of characteristic 0 has a Hilbertian algebraic extension F such that for every f ∈ F[x] of degree d ≥ 2 the image of the arboreal representation on the preimage tree of 0 has infinite index in the automorphism group of the tree; their Conjecture 1.1 is Odoni's conjecture with base point 0. For P ∈ F, translation by P carries the tree of f over P to the tree of f(x + P) − P over 0, again monic of degree d, so ι_F(f, P) is infinite whenever the tree is complete (our remark): the conjecture fails for K = F and every d ≥ 2. For number fields the statement holds, as the list records (Looper; Specter). Read: Section 5 of the list up to Conjecture 5.5; Section 1, Definition 2.1 and Remark 2.2 of the paper.
• AMR-126-0026 → solved (no); the dataset currently labels it open. The list, Conjecture 14.8 (Integrality Conjecture of Bellon–Viallet), p. 33: for every dominant rational map f: ℙ^N ⇢ ℙ^N the dynamical degree δ_f = lim deg(fⁿ)^{1/n} is an algebraic integer; the list records the cases of birational maps and of polynomial maps of ℙ² (Diller–Favre; Favre–Jonsson). No: J. P. Bell, J. Diller, M. Jonsson, A transcendental dynamical degree, Acta Math. 225 (2020) 193–225 (arXiv:1907.00675), Main Theorem: for every field k of characteristic ≠ 2 there is a dominant rational map f: ℙ²_k ⇢ ℙ²_k whose dynamical degree, defined as in the list, is transcendental, hence not an algebraic integer. The paper quotes the journal version of the list ("Conjecture 13.17") for the question whether the dynamical degree is always an algebraic integer and answers it negatively. For birational maps, J. Bell, J. Diller, M. Jonsson, H. Krieger, Birational maps with transcendental dynamical degree, Proc. Lond. Math. Soc. 128 (2024) e12573 (arXiv:2107.04113), Theorem 1.1, give maps of ℙ^N with transcendental dynamical degree for every N ≥ 3, presented there as a negative answer to Bellon and Viallet's conjecture. Read: Section 14 of the list up to Question 14.17; the introduction of each paper (Main Theorem; Theorem 1.1).
• AMR-126-0028 → solved (no); the dataset currently labels it open. The list, Question 14.17, p. 35: for a smooth projective variety X of dimension N and a dominant rational map f: X ⇢ X, are the dynamical degrees δ_f^{(i)} = lim (((fⁿ)^L)^i · L^{N−i})^{1/n} (L ample) algebraic integers for all 1 ≤ i ≤ N? No: the Main Theorem of Bell–Diller–Jonsson (see AMR-126-0026) gives, for X = ℙ² and i = 1, a dominant rational map whose first dynamical degree δ_f^{(1)} = lim ((fⁿ)^H · H)^{1/n} is transcendental. Read: as for AMR-126-0026, with the definition of δ_f^{(i)} before Question 14.17.
• AMR-126-0052 → solved (no; the limit statement fails); the dataset currently labels it open. The list, Conjecture 21.4 (Dynamical Lang–Siegel Conjecture for ℙ^N), p. 60: let f: ℙ^N → ℙ^N be a morphism of degree d ≥ 2 over ℚ such that for 1 ≤ n ≤ N + 1 the first coordinate function of fⁿ is not a constant multiple of X_0^{dⁿ}; then every orbit has only finitely many points integral with respect to x_0 = 0, and "more precisely", for wandering P ∈ ℙ^N(ℚ) with fⁿ(P) = [A_0(n), …, A_N(n)] in coprime integers, log|A_0(n)| / log max_i |A_i(n)| → 1. Y. Matsuzawa, Growth of local height functions along orbits of self-morphisms on projective varieties, Int. Math. Res. Not. IMRN 2023, no. 4, 3533–3575 (arXiv:2005.08093), Example 6.2: for f = σ⁻¹ ∘ (X³ : Y³ : Z³) ∘ σ on ℙ²_ℚ with σ = (X + Y + Z : 2X + Y + Z : X − Y + Z) and P = (2 : 3 : −4), which has infinite orbit, the ratio tends to log 3 / log 5, and the first coordinates of fⁿ for n = 1, 2, 3 are not constant multiples of powers of X; Remark 1.24 says that the hypothesis of Conjecture 21.4 is not sufficient and refers to this counterexample. We recomputed fⁿ(P) for n ≤ 6 in exact arithmetic (ratios 0.728, 0.716, 0.694, 0.686, 0.684, 0.683, against log 3 / log 5 = 0.6826; the logarithms agree with the paper's table) and checked the hypothesis: the first coordinate of fⁿ is a constant multiple of (2X + Y + Z)^D − (X + Y + Z)^D, D = 3ⁿ, never a multiple of X^D. In the example |A_0(n)| → ∞, so it does not contradict the first sentence (finitely many integral points). Corollary 1.23 derives a corrected form from Vojta's conjecture: if moreover e_{f,+}(Z) < d_1(f) for every f-periodic subvariety Z meeting {x_0 = 0}, then for every infinite orbit the points with ratio below 1 − ε are not Zariski dense, and the limit is 1 for generic orbits (those meeting every proper closed subset in finitely many points); the example violates this condition at the periodic point (0 : 1 : −1). Read: Section 21 of the list; Section 1 (Corollary 1.23, Remark 1.24) and Section 6 (Proposition 6.1, Example 6.2) of the paper.

Answered in preprints (not refereed; under the dataset policy of release v1.8.0: partially solved, evidence unrefereed_claim)

• AMR-126-0018 → solved (yes; preprint); the dataset currently labels it open. The list, Conjecture 11.6 (Dynamical André–Oort Conjecture for 1-dimensional families, Baker–DeMarco), p. 25: for a family T → M_d^crit, t ↦ (f_t, P_{1,t}, …, P_{2d−2,t}), of rational maps with marked critical points over a non-singular curve T, f_t is PCF for infinitely many t if and only if every two marked critical points P_i, P_j are f-dynamically related (Definition 11.2: some algebraic curve in ℙ¹ × ℙ¹ over the algebraic closure of k(T) contains (P_i, P_j) and is mapped into itself by f × f); the list records the polynomial case, announced by Favre and Gauthier, and calls the conjecture open for rational maps. Z. Ji, J. Xie, DAO for curves, preprint arXiv:2302.02583 (v2, September 2023), Theorem 1.2 proves their Conjecture 1.1: for a non-isotrivial algebraic family of rational maps of degree d ≥ 2 parametrized by an algebraic curve over ℂ, f_t is PCF for infinitely many t if and only if any two critical points are dynamically related, which they define (after DeMarco) by a curve through the pair that is preperiodic under f × f. The forward images of such a curve form a curve mapped into itself, so the two definitions agree (our remark). Non-isotrivial means that the induced map to the moduli space is not constant, so that the family is one-dimensional in moduli; for a constant map the equivalence can fail (our remark). We found no journal version (arXiv journal-ref field and Crossref); the result is presented, with a sketch of the proof, in J. Xie, Rigidity in complex dynamics: multiplier spectrum and dynamical André–Oort conjecture, Proc. ICM 2026, Vol. 3, SIAM, pp. 674–689 (arXiv:2511.12111). Read: Section 11 of the list from Definition 11.2 to the end of the section; Section 1 of the preprint (Conjecture 1.1, Theorems 1.2 and 1.3, previous results); Section 2 of the ICM article.

Partial results in print (labels → partially solved)

• AMR-126-0044 → partially solved (part (a) for separable maps); the dataset currently labels it open. The list, Conjecture 18.10, p. 54: for a dominant rational map f: ℙ^N ⇢ ℙ^N over 𝔽_q, with ℙ^N(K)_f the set of points whose forward orbit never meets the indeterminacy locus I(f), (a) ℙ^N(𝔽̄_q)f is Zariski dense in ℙ^N, and (b) #ℙ^N(𝔽{q^k})f / #ℙ^N(𝔽{q^k}) → 1 as k → ∞. E. Amerik, Existence of non-preperiodic algebraic points for a rational self-map of infinite order, Math. Res. Lett. 18 (2011) 251–256 (arXiv:1007.1635), Corollary 2, derived from a theorem of Hrushovski (Theorem 1 there): for a separable dominant rational self-map f of an irreducible variety over 𝔽_q, the f-periodic points over 𝔽̄_q with no iterate in the indeterminacy or ramification locus are Zariski dense. For ℙ^N this is part (a) for separable f. The list cites this paper for its theorem over ℚ̄ (Theorem 13.1) and among the known cases of Conjecture 7.6, not for this corollary. M. H. Weinreich, The algebraic dynamics of the pentagram map, Ergodic Theory Dynam. Systems 43 (2023) 3460–3505 (arXiv:2104.06211), Section 8, states the density of the dynamical domain without the separability condition, "by work of Hrushovski", citing Corollary 2; in Corollary 2 separability serves to exclude the ramification locus, which part (a) does not involve (our remark). We found no result on part (b); Weinreich conjectures (Conjecture 8.3) that for the pentagram map the proportion tends to 0, which would give counterexamples on ℙ^{2n−1}. Read: Theorem 13.1 and Section 18 of the list from Question 18.9 to Remark 18.12; Amerik's paper up to Corollary 2; Section 8 of Weinreich's paper up to Conjecture 8.3.

Partial results in preprints (not refereed; under the dataset policy of release v1.8.0: partially solved, evidence unrefereed_claim)

• AMR-126-0016 → partially solved (yes to the first question; preprint); the dataset currently labels it open. The list, Question 10.10, p. 22: with FOM/FOD(N, d) the supremum over morphisms f ∈ End_d^N(ℚ̄) of the least degree [L : K_f] of a field of definition over the field of moduli (Definitions 10.7–10.8), is sup_{d≥2} FOM/FOD(N, d) < ∞ for each N ≥ 1, and if so, what is its value as a function of N? The list's Theorem 10.9 gives FOM/FOD(1, d) = 1 for even d and 2 for odd d, and finiteness for each pair (N, d) (Doyle–Silverman). G. Bresciani, Uniform bounds for fields of definition in projective spaces, preprint arXiv:2405.03621 (May 2024), Theorem 1: over a field of characteristic 0, for each n there is a constant C_n such that every algebraic structure on ℙ^n with finite automorphism group, among them dynamical systems of any degree d ≥ 2, is defined over an extension of degree at most C_n of its field of moduli. Hence sup_d FOM/FOD(N, d) ≤ C_N, and the paper presents this as a positive answer to Doyle and Silverman's question. The value as a function of N is not determined beyond N = 1: the bound from the proof is very large (Remark 10), and for N = 2 the author remarks, referring to the proof of a theorem in his work on plane curves, that the optimal constant C_2 is 3 or 4. The paper cites the author's withdrawn preprint arXiv:2405.03612 only as an example of the notion of algebraic structure. We found no journal version (arXiv journal-ref field and Crossref). Read: Section 10 of the list from Definition 10.7 to Question 10.10; the introduction (Theorems 1 and 2), Section 1 and Remark 10 of the preprint.

These are AI-assisted, self-audited checks, not independent human peer review: for each item we read the source and the cited statements and took publication data from Crossref; we did not check the proofs. The suggested labels are suggestions. Corrections are welcome. — Alper Ferudun, Mercury Software GmbH

Status follow-up 89: 8 status corrections (records from problem lists on holomorphic foliations and on pluripotential theory)

• Each item was checked against the current dataset (commit d371ee9, release v1.8.0), against its source (T.-C. Dinh, N. Sibony, Some Open Problems on Holomorphic Foliation Theory, Acta Math. Vietnam. 45 (2020) (arXiv:1807.05640), records AMR-135-*; S. Dinew, V. Guedj, A. Zeriahi, Open Problems in Pluripotential Theory, Complex Var. Elliptic Equ. 61 (2016) (arXiv:1511.00705), AMR-147-*, whose record labels differ from the question numbers of the paper (see the notes in the next follow-up); cited by problem number) and against the cited papers, read on arXiv or in an open journal or author version; publication data are from Crossref. Theorem numbers follow the versions read. Suggested changes are in bold.
• Following the policy of release v1.8.0, answers that exist only in preprints are listed separately.

Solved in published papers (labels → solved)

• AMR-135-0011 → solved (T_h(r) ≤ ψ(r) log r is possible for every ψ → ∞, while T_h(r)/log r → ∞ always); the dataset currently labels it open. T.-C. Dinh, N. Sibony, Some open problems on holomorphic foliation theory, Acta Math. Vietnam. 45 (2020) 103–112 (arXiv:1807.05640), Problem 9.1 (the record's "Problem 11"): let H be the space of entire maps ℂ → ℂ (resp. ℂ → ℙ¹) and U the set of universal h, those for which every f ∈ H is a limit of translates h(z + aₙ) uniformly on compact sets; find the minimal growth of T_h(r) = ∫₀^r (dt/t) ∫_{𝔻(t)} h*ω, ω the Fubini–Study form. Z. Chen, D. T. Huynh, S.-Y. Xie, Universal entire curves in projective spaces with slow growth, J. Geom. Anal. 33 (2023), no. 9, Art. 308 (arXiv:2304.04929), Theorem 1.2: for every n ≥ 1 and every positive continuous φ with φ(r) → ∞ there is a universal entire curve h: ℂ → ℙⁿ with T_h(r) ≤ φ(r) log r for all r ≥ 1, while every universal curve is transcendental, so T_h(r)/log r → ∞ (remark before the theorem). The paper presents this as an optimal answer to Problem 9.1; n = 1 is the case of maps to ℙ¹. Maps ℂ → ℂ: B. Guo, S.-Y. Xie, Universal holomorphic maps with slow growth I: an algorithm, Math. Ann. 389 (2024) 3349–3378 (arXiv:2306.11193), Theorem A with n = m = 1: for every growth function φ(r) = Σ Aᵢ rⁱ with Aᵢ > 0 and Aᵢ₊₁/Aᵢ → 0 there are entire functions F, hypercyclic for a translation (hence universal), with |F(z)| ≤ φ(|z|). Their comparison lemma (Section 2.1) puts such a φ below any continuous function growing faster than every polynomial, such as r^{ψ(r)}, and Cartan's formula T_F(r) = (1/2π) ∫₀^{2π} log⁺|F(re^{iθ})| dθ + O(1) then gives universal entire functions with T_F(r) ≤ ψ(r) log r + O(1) (our combination); the paper draws the same conclusion from a 1983 theorem of S. M. Duĭos Ruis that it recalls, and says the holomorphic part of Problem 9.1 thus "has a clear answer". Both papers integrate from t = 1 instead of 0, which changes T_h by a constant. Read: Section 9 of the list; the abstract and Section 1 of Chen–Huynh–Xie (Question 1.1, Theorems 1.2 and 1.3); Section 1 (Theorems 1.1 and 1.2, Theorems A and B) and Section 2.1 of Guo–Xie.
• AMR-147-0007 → solved (no); the dataset currently labels it open. S. Dinew, V. Guedj, A. Zeriahi, Open problems in pluripotential theory, Complex Var. Elliptic Equ. 61 (2016) 902–930 (arXiv:1511.00705), Question 8 of the paper (corpus label Question 7), Demailly: for φ plurisubharmonic in the unit ball of ℂⁿ with well-defined (dd^c φ)ⁿ (for instance locally bounded off 0), let φⱼ = (1/2j) log Σₖ |f_{k,j}|², where (f_{k,j}) is an orthonormal basis of the holomorphic functions f with ∫ |f|² e^{−2jφ} dV < ∞; does (dd^c φⱼ)ⁿ(0) converge to (dd^c φ)ⁿ(0)? No: C. Li, Analytical approximations and Monge–Ampère masses of plurisubharmonic singularities, Int. Math. Res. Not. IMRN 2024, no. 1, 359–381 (arXiv:2012.15599), Theorem 1.5: there is φ = max{log|z|² + π*u, 2 log|z|²} on ℂ², plurisubharmonic and locally bounded off 0, with residual mass e₂(φ) = (dd^c φ)²({0}) = 2 and e₂(φₘ) = ((m − 1)/m)² for every m ≥ 1, where φₘ is built in the same way from the weight e^{−mφ} on the unit ball (Section 2). The source's φⱼ is φ₂ⱼ, so its masses tend to 1, not 2. The paper states Demailly's conjecture with the reference "[DGZ16, Problem 8]" and says it answers it negatively; Section 5 sketches examples in higher dimensions, and the paper leaves the Guedj–Rashkovskii question (Question 7 of the paper) open. Read: the question; the abstract, Section 1 (Conjecture 1.2, Theorem 1.5), Section 2 (Corollary 2.2) and Section 5.
• AMR-147-0009 → solved (yes); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 10 of the paper (corpus label Question 9), Boucksom–Eyssidieux–Guedj–Zeriahi: on a compact Kähler manifold, if T ∼ S are cohomologous closed positive (1,1)-currents and T is more singular than S, is ⟨Tⁿ⟩ ≤ ⟨Sⁿ⟩, without the small-unbounded-locus assumption under which these authors proved it (their Theorem 1.16)? The comparison is between total non-pluripolar masses, as in that theorem; as measures the inequality already fails for two bounded potentials (our remark). Yes: D. Witt Nyström, Monotonicity of non-pluripolar Monge–Ampère masses, Indiana Univ. Math. J. 68 (2019) 579–591 (arXiv:1703.01950), Theorem 1.2: on a compact Kähler manifold, if φ and ψ are θ-psh and φ is less singular than ψ (φ ≥ ψ + O(1)), then ∫_X MA_θ(φ) ≥ ∫_X MA_θ(ψ); the paper says this proves the conjecture of Boucksom–Eyssidieux–Guedj–Zeriahi. T. Darvas, E. Di Nezza, C. H. Lu, Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity, Anal. PDE 11 (2018) 2049–2087 (arXiv:1705.05796), Theorem 1.1, extend it to mixed products of currents from different classes. Read: the question; the abstract, Section 1 (Definition 1.1, Theorem 1.2) and Theorem 2.2 (the statement of Theorem 1.16 of Boucksom–Eyssidieux–Guedj–Zeriahi) of Witt Nyström; Section 1 of Darvas–Di Nezza–Lu up to Theorem 1.1.
• AMR-147-0017 → solved (a PDE proof is given); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 18 of the paper (corpus label Question 17), Błocki–Kołodziej: find a proof based on PDE techniques of Kołodziej's estimate quoted just before the question (the record leaves it out): if φ is smooth and plurisubharmonic in the unit ball B of ℂⁿ, φ = 0 on ∂B and det(φ_{jk̄}) = f, then ‖φ‖_{L^∞(B)} ≤ C ‖f‖_{L^p(B)}^{1/n} for p > 1, with C depending only on n and p. J. Wang, X.-J. Wang, B. Zhou, A priori estimate for the complex Monge–Ampère equation, Peking Math. J. 4 (2021) 143–157 (arXiv:2003.06059), Theorem 1.2: on a bounded smooth strictly pseudoconvex domain Ω, a smooth plurisubharmonic solution of (dd^c u)ⁿ = f dμ with f ∈ L^p, p > 1, and smooth boundary values φ satisfies |inf u| ≤ |inf_{∂Ω} φ| + C ‖f‖_{L^p}^{1/n} |Ω|^δ for every δ ∈ (0, 1/(np*)), p* = p/(p − 1), with C depending on n, p, δ and diam Ω; for the ball and zero boundary values this is the quoted estimate. The proof rests on the authors' Sobolev-type inequality for plurisubharmonic functions (Theorem 1.1 there; J. Funct. Anal. 279 (2020) 108765), proved with a gradient flow, and the abstract says it uses "PDE techniques but not the pluri-potential theory". The paper cites the source among those that asked for a PDE approach and presents its theorems as an affirmative answer; Theorems 1.3 and 1.4 treat stability and Hölder continuity in the same way. Read: the question with the theorem before it; the abstract, Section 1 and the beginning of Section 2.
• AMR-147-0028 → solved (exactly the metrics with dd^c ω = 0 and dω ∧ d^c ω = 0); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 29 of the paper (corpus label Question 28), Dinew–Kołodziej: on a compact Hermitian manifold (X, ω) of dimension n ≥ 3, characterize the forms ω with ∫_X (ω + dd^c φ)ⁿ = ∫_X ωⁿ for every bounded ω-psh φ; the source proposes the Guan–Li condition dd^c ω = 0, dω ∧ d^c ω = 0 as a candidate. I. Chiose, On the invariance of the total Monge–Ampère volume of Hermitian metrics, Ann. Fac. Sci. Toulouse Math. (6) 33 (2024) 575–579 (arXiv:1609.05945), Theorem 1: if the volume is invariant for all smooth ω-psh functions, then i∂∂̄ωᵏ = 0 for k = 1, …, n − 1; Theorem 2: for smooth ω-psh functions, invariance is equivalent to i∂∂̄ω = i∂ω ∧ ∂̄ω = 0 (the Guan–Li condition), to i∂∂̄ωᵏ = 0 for all k ≤ n − 1 and to the comparison principle (numbering of the journal version; Theorems 0.1 and 0.2 on arXiv). The paper says it answers Question 29. For bounded φ, as the question asks, necessity follows from Theorem 1 because smooth functions are bounded, and sufficiency from the result cited in the proof of Theorem 2: S. Dinew, S. Kołodziej, Pluripotential estimates on compact Hermitian manifolds (Adv. Lect. Math. 21 (2012) 69–86, as cited there; arXiv:0910.3937), Proposition 3.2, the comparison principle for bounded ω-psh functions under the Guan–Li condition (even under one-sided inequalities), which gives the invariance by comparing φ − C < 0 < φ + C. Read: the question; the whole note (arXiv and the open journal version); Section 3 of the Dinew–Kołodziej preprint.
• AMR-147-0034 → solved (yes); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 36 of the paper (corpus label Question 34), Berman–Boucksom–Eyssidieux–Guedj–Zeriahi: X compact Kähler (or in the Fujiki class), ω a smooth semi-positive (1,1)-form with ∫_X ωⁿ > 0; does every φ ∈ E(X, ω), the ω-psh functions of full non-pluripolar Monge–Ampère mass, have zero Lelong number at every point? T. Darvas, E. Di Nezza, C. H. Lu, On the singularity type of full mass currents in big cohomology classes, Compos. Math. 154 (2018) 380–409 (arXiv:1606.01527), Theorem 1.1(i): on a compact Kähler manifold, for a big class {θ} and φ ∈ E(X, θ), ν(φ, x) = ν(V_θ, x) at every x, where V_θ is the envelope with minimal singularities. For θ = ω ≥ 0 one has V_ω = 0, so all Lelong numbers vanish (Theorem 1.1(ii) also gives this for big and nef classes), and the paper states that this answers [DGZ15, Question 36] affirmatively. The theorem is stated for compact Kähler X; for X in the Fujiki class, pulling back to a Kähler modification keeps full mass and does not decrease Lelong numbers (our remark). Read: the question; the abstract, Section 1 (Theorem 1.1 and the paragraph after it) and Section 2.1.
• AMR-147-0041 → solved (yes, with singular segments over the points of positive Lelong number); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 44 of the paper (corpus label Question 42), Guedj: H.-S. Do's theorem, quoted before the question, runs the flow u̇ = log det(u_{ij̄}) on Ω × (0, T), Ω a bounded smooth strictly pseudoconvex domain, with smooth boundary values φ, from a bounded plurisubharmonic u₀ defined near Ω̄; can one run it from an arbitrary plurisubharmonic function, in view of the global results of Guedj–Zeriahi and Di Nezza–Lu (smooth flows on time-increasing Zariski open sets when the initial current has positive Lelong numbers)? H.-S. Do, Weak solution of parabolic complex Monge–Ampère equation II, Internat. J. Math. 27 (2016), no. 12, 1650098 (arXiv:1605.01879), Theorem 0.2: for u̇ = log det(u_{αβ̄}) − Au + f (A ≥ 0, f smooth) and any plurisubharmonic u₀ on a neighbourhood of Ω̄ with u₀ = φ(·, 0) on ∂Ω, the weak solution (the decreasing limit of smooth solutions) equals φ on ∂Ω × [0, T) and tends to u₀ pointwise on Ω (and in L¹) as t → 0; for each m > 1/T, on Ω̄ × (1/m, T) it is smooth and solves the equation outside finitely many segments {a} × (1/m, εₐ], a running over the points with ν(u₀, a) ≥ 2/m, where εₐ is bounded above and below in terms of ν(u₀, a) and A, and ν(u(·, t), a) > 0 for t < εₐ. With A = 0 and f = 0 this is the flow of the question; the paper refers to Di Nezza–Lu for the corresponding problem on compact Kähler manifolds and does not cite the source. Read: the question with Do's theorem quoted before it; the abstract and the introduction (Definition 0.1, Theorem 0.2).

Answered in preprints (not refereed; under the dataset policy of release v1.8.0: partially solved, evidence unrefereed_claim)

• AMR-147-0040 → solved (no; preprint); the dataset currently labels it open. S. Dinew, V. Guedj, A. Zeriahi, Open problems in pluripotential theory, Complex Var. Elliptic Equ. 61 (2016) 902–930 (arXiv:1511.00705), Question 43 of the paper (corpus label Question 41), posed by Di Nezza and Lu, PDF p. 22: on a compact Kähler manifold, Di Nezza and Lu run the Kähler–Ricci flow from a positive closed current S₀ with some positive Lelong numbers and obtain currents that are Kähler forms on increasing Zariski open sets; by analogy with Demailly's regularization these are expected to have analytic singularities; does the flow produce approximants with analytic singularities? No: E. Di Nezza, V. Guedj, C. H. Lu, Kähler–Ricci flow: from divisors to cusps, preprint arXiv:2606.15816 (June 2026), Main Theorem (Theorem 3.6 in the body): let D₁, …, D_r be smooth ample hypersurfaces with simple normal crossings, D_j = {s_j = 0}, m_j ≥ 0 and T₀ = ω + dd^c Σ_j m_j log|s_j|² ≥ 0; the solution of the flow ∂ω_t/∂t = −Ric(ω_t) + Ric(dV_X) starting at T₀ is ω_t = Σ_j max(m_j − t, 0)[D_j] + β_t, where β_t is a Kähler metric of Poincaré type along the D_j with m_j > t, and ω_t is a Kähler form on X once t > max m_j. A metric of Poincaré type has potential −2 Σ log(−log|s_j|²) + O(1) (Definitions 2.2–2.3), so for 0 < t < max m_j the potentials do not have analytic singularities in the sense of Definition 1.2 (our remark); the authors state in the introduction that this disproves the expectation of Question 43. The theorem is stated for this twisted normalization of the flow, in which the cohomology class stays fixed, while the source writes ∂ω_t/∂t = −Ric(ω_t). Read: the question and the paragraph before it; the introduction, Definitions 1.2 and 2.1–2.3 and Theorem 3.6 of the preprint.

These are AI-assisted, self-audited checks, not independent human peer review: for each item we read the source and the cited statements and took publication data from Crossref; we did not check the proofs. The suggested labels are suggestions. Corrections are welcome. — Alper Ferudun, Mercury Software GmbH

Status follow-up 90: 7 status corrections and 3 notes on records (records from problem lists on holomorphic foliations and on pluripotential theory, continued)

• Each item was checked against the current dataset (commit d371ee9, release v1.8.0), against its source (T.-C. Dinh, N. Sibony, Some Open Problems on Holomorphic Foliation Theory, Acta Math. Vietnam. 45 (2020) (arXiv:1807.05640), records AMR-135-*; S. Dinew, V. Guedj, A. Zeriahi, Open Problems in Pluripotential Theory, Complex Var. Elliptic Equ. 61 (2016) (arXiv:1511.00705), AMR-147-*, whose record labels differ from the question numbers of the paper (see the notes at the end); cited by problem number) and against the cited papers, read on arXiv or in an open journal or author version; publication data are from Crossref. Theorem numbers follow the versions read. Suggested changes are in bold.
• Following the policy of release v1.8.0, answers that exist only in preprints are listed separately.

Partial results in print (labels → partially solved)

• AMR-147-0016 → partially solved (yes for bounded strictly pseudoconvex domains); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 17 of the paper (corpus label Question 16), Zeriahi: μ a probability measure on a bounded pseudoconvex domain Ω ⊂ ℂⁿ with a Hölder continuous plurisubharmonic subsolution u (u = 0 on ∂Ω, μ ≤ (dd^c u)ⁿ); is there a Hölder continuous plurisubharmonic φ with φ = 0 on ∂Ω and (dd^c φ)ⁿ = μ? N. C. Nguyen, On the Hölder continuous subsolution problem for the complex Monge–Ampère equation, II, Anal. PDE 13 (2020) 435–453 (arXiv:1803.02510), Theorem B: on a bounded strictly pseudoconvex domain, for every positive Borel measure μ ≤ (dd^c u)ⁿ, u plurisubharmonic and Hölder continuous on Ω̄ with u = 0 on ∂Ω, and every Hölder continuous boundary datum, the Dirichlet problem has a Hölder continuous solution; the abstract says that the answer to Zeriahi's question "is always affirmative". Part I (Calc. Var. Partial Differential Equations 57 (2018), Art. 8, arXiv:1703.01549) had covered, as recalled there, subsolutions of finite total mass. A. Benali and A. Zeriahi (J. Éc. polytech. Math. 7 (2020) 981–1007, Section 1) recall the problem from the source on a bounded domain and say that for the complex Monge–Ampère equation it "has been solved" by Nguyen in these two papers; their own theorems, for Hessian equations, are on strongly m-pseudoconvex domains. The printed theorems need strict pseudoconvexity, while the question allows any bounded pseudoconvex domain; for domains that are not strictly pseudoconvex we found no result. Read: the question; the abstract, Section 1 and the setting of Section 2 of Nguyen's second paper; Section 1 of Benali–Zeriahi.
• AMR-147-0013 → partially solved (yes on strictly pseudoconvex domains when the modulus of continuity of the subsolution satisfies a Dini-type condition); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 14 of the paper (corpus label Question 13), posed by Kołodziej, PDF p. 9: in a bounded hyperconvex domain Ω ⊂ ℂⁿ, if μ ≤ (dd^c u)ⁿ for a continuous psh u with u = 0 on ∂Ω, is there a continuous psh φ with φ = 0 on ∂Ω and (dd^c φ)ⁿ = μ? S. Kołodziej, N. C. Nguyen, A remark on the continuous subsolution problem for the complex Monge–Ampère equation, Acta Math. Vietnam. 45 (2020) 83–91 (arXiv:1808.07197), Theorem 1: on a strictly pseudoconvex domain, if the modulus of continuity ϖ of the subsolution satisfies ∫₀¹ ϖ(t)^{1/n} / (t |log t|) dt < ∞, the Dirichlet problem with any continuous boundary data has a unique continuous solution; the authors add that the case of an arbitrary continuous subsolution is still open. M. Charabati, A. Zeriahi, The continuous subsolution problem for complex Hessian equations, Indiana Univ. Math. J. 73 (2024) 1639–1688 (arXiv:2007.10194), Theorem 4: the same conclusion on bounded strictly pseudoconvex domains for any positive Borel measure, with an estimate of the modulus of continuity of the solution when μ has finite mass (Theorem 3 treats complex Hessian equations). Hölder continuous subsolutions, which satisfy the condition, had been treated by N. C. Nguyen, Anal. PDE 13 (2020) 435–453 (arXiv:1803.02510), Theorem A, who notes that Question 14 is still open in general. Open as far as we found: arbitrary continuous subsolutions, and hyperconvex domains that are not strictly pseudoconvex. Read: the question and the text around it; Sections 1–2 of Kołodziej–Nguyen; the introduction (Theorems 1–4) of Charabati–Zeriahi; the introduction (Theorems A and B) of Nguyen.
• AMR-147-0029 → partially solved (T(X,ω) > 0 on manifolds of the Fujiki class, on manifolds with a Guan–Li metric, which include all compact complex surfaces, and on threefolds with a Hermitian metric satisfying dd^c ω ≤ 0); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 30 of the paper (corpus label Question 29), posed by Tosatti, PDF p. 15: for a compact Hermitian manifold (X,ω), let T(X,ω) be the infimum of ∫_X (ω + dd^c φ)ⁿ over smooth ω-psh φ; can T(X,ω) = 0 for some manifold? V. Guedj, C. H. Lu, Quasi-plurisubharmonic envelopes 2: Bounds on Monge–Ampère volumes, Algebraic Geom. 9 (2022) 688–713 (arXiv:2106.04272): their v₋(ω) (Definition 3.1, an infimum over bounded ω-psh functions) equals T(X,ω) for a Hermitian ω, since, as they note there, the infimum may be taken over smooth functions. Theorem A: the condition v₋(ω) > 0 does not depend on the Hermitian metric, is invariant under bimeromorphic maps and holds for every manifold of the Fujiki class (Theorem 3.7, Corollary 3.8); Proposition 3.3(4) gives v₋(ω) = ∫_X ωⁿ > 0 for a Guan–Li metric (dd^c ω = 0 and dd^c ω² = 0), and the text after it notes that every compact complex surface has one (a Gauduchon metric). D. Angella, V. Guedj, C. H. Lu, Plurisigned hermitian metrics, Trans. Amer. Math. Soc. 376 (2023) 4631–4659 (arXiv:2207.04705), Theorem B(2): v₋(ω) > 0 on a threefold with a Hermitian metric satisfying dd^c ω ≤ 0, for instance a pluriclosed one. On all these manifolds T(X,ω) > 0 for every Hermitian ω. Guedj and Lu know of no manifold with v₋ = 0 and call this an important open problem. Read: the question; the introduction, Definition 3.1, Propositions 3.2–3.3, Theorem 3.7 and Corollary 3.8 of Guedj–Lu; the introduction of Angella–Guedj–Lu.
• AMR-147-0031 → partially solved (true when the class of ω contains a smooth closed semi-positive form); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 33 of the paper (corpus label Question 31), posed by Tosatti and Weinkove, PDF p. 16: for points z_j and numbers τ_j > 0 with Σ τ_jⁿ < ∫_X ωⁿ, construct an ω-psh function with φ ≤ τ_j log‖z − z_j‖ + O(1) near z_j; the source says that Tosatti and Weinkove treated dimension 2 and, under assumptions, dimension 3, and that the general case is open. The cited paper, V. Tosatti, B. Weinkove, Plurisubharmonic functions and nef classes on complex manifolds, Proc. Amer. Math. Soc. 140 (2012) 4003–4010 (arXiv:1102.1676), poses the problem on a compact complex manifold for a closed real (1,1)-form whose Bott–Chern class is nef with positive volume, proves it for n = 2, 3 and for rational classes on Moishezon manifolds, and conjectures it in general. N. C. Nguyen, The complex Monge–Ampère type equation on compact Hermitian manifolds and applications, Adv. Math. 286 (2016) 240–285 (arXiv:1501.00891, January 2015, not cited in the list), Theorem 0.6 (= Theorem 4.1): the statement holds in every dimension when the class contains a smooth closed semi-positive form β with ∫_X βⁿ > 0, a hypothesis the paper notes is stronger than nef. Open as far as we found: nef classes without a semi-positive representative, outside the cases of Tosatti–Weinkove. If ω is read as a non-closed Hermitian metric, as in the surrounding section of the list, this theorem does not apply. Read: the question and the text before and after it; the introduction and Main Theorem of Tosatti–Weinkove; the introduction, Theorem 4.1 and Remark 4.2 of Nguyen.
• AMR-147-0035 → partially solved (yes when {ω} lies in the real Néron–Severi space; yes for Kähler currents when Y has a holomorphic retraction or e^φ is Hölder continuous); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 37 of the paper (corpus label Question 35), posed by Coman, Guedj and Zeriahi, PDF p. 18: for a compact Kähler manifold (X,ω) and a complex submanifold Y, is PSH(Y, ω|_Y) = PSH(X, ω)|_Y? The source records the cases of Hodge forms, of smooth strictly ω|_Y-psh functions and of Kähler currents with analytic singularities (Collins–Tosatti). D. Coman, V. Guedj, A. Zeriahi, On the extension of quasiplurisubharmonic functions, Anal. Math. 48 (2022) 411–426 (arXiv:2202.01325), Theorem 1.1: equality holds whenever {ω} lies in NS_ℝ(X), for every analytic subvariety Y; by Section 3 there this includes every Kähler class on Fano and on rational manifolds. For Kähler currents, i.e. φ with ω|_Y + dd^c φ ≥ εω|_Y for some ε > 0: J. Ning, Z. Wang, X. Zhou, On the extension of Kähler currents on compact Kähler manifolds: holomorphic retraction case, Ann. Fac. Sci. Toulouse Math. (6) 33 (2024) 183–195 (earlier version arXiv:2002.11381), Theorem 1.2: such φ extends to Φ with ω + dd^c Φ ≥ ε′ω when a neighbourhood of Y retracts holomorphically onto Y; T. Darvas, R. Reboulet, D. Witt Nyström, M. Xia, K. Zhang, Transcendental Okounkov bodies, J. Differential Geom. 132 (2026) 135–178 (arXiv:2309.07584), Theorem 1.3: for every connected compact submanifold Y of positive dimension, such φ extends to a Kähler current on X when e^φ is Hölder continuous. Open as far as we found: arbitrary ω|_Y-psh functions for Kähler classes outside NS_ℝ(X). Read: the question and the paragraph after it; the introduction and Section 3 of Coman–Guedj–Zeriahi; Section 1 of the journal version of Ning–Wang–Zhou; the introduction of Darvas et al.
• AMR-147-0042 → partially solved (yes on compact Kähler manifolds when μ does not depend on t, on the ample locus of the lower bound θ); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 45 of the paper (corpus label Question 43), posed by Eyssidieux, Guedj and Zeriahi, PDF p. 23: for the flows (ω_t + dd^c φ_t)ⁿ = e^{φ̇t + F(t,x,φ_t)} μ(t,x) on a compact Kähler space X, with ω_t ≥ θ for a fixed semi-positive and big form θ, F continuous and non-decreasing in φ and μ ≥ 0 bounded and continuous, do a viscosity subsolution u and supersolution v with u ≤ v at t = 0 satisfy u ≤ v on [0,T] × X? The source says this had been proved in particular cases. T. D. Tô, Convergence of the weak Kähler–Ricci flow on manifolds of general type, Int. Math. Res. Not. IMRN 2021, no. 8, 6373–6404 (arXiv:1905.01276), Theorem B (= Theorem 2.10): on a compact Kähler manifold, for smooth closed forms θ_t ≥ θ with θ in a big class α, F as above and μ = f dV with f ≥ 0 continuous and independent of t, a subsolution φ and a supersolution ψ satisfy φ − ψ ≤ max(sup{t=0} (φ − ψ)*, 0) on [0,T) × Amp(α); the paper says this removes conditions on ∂_t φ or θ_t needed in earlier comparison principles of Eyssidieux–Guedj–Zeriahi. When θ is a Kähler form, Amp(α) = X and the answer is yes (our remark). Not covered as far as we found: singular X, time-dependent μ, and points outside Amp(α) when θ is only semi-positive and big. Read: the question and the setting before it; the introduction, Section 2 (definitions) and Theorem 2.10 of the arXiv version.

Partial results in preprints (not refereed; under the dataset policy of release v1.8.0: partially solved, evidence unrefereed_claim)

• AMR-147-0003 → partially solved (yes for functions with locally analytic singularities; preprint); the dataset currently labels it open. Dinew–Guedj–Zeriahi, Question 3 of the paper (corpus label Question 3), posed by Błocki, PDF p. 4: the source recalls that maximality is a local notion for functions in the domain of definition DMA(Ω) of the Monge–Ampère operator; is it a local notion for arbitrary psh functions? Thai Duong Do, Pham Hoang Hiep, Local-to-global maximality for plurisubharmonic functions with locally analytic singularities, preprint arXiv:2610.05937 (v2, October 2026), Theorem 1.1: a psh function on a domain in ℂⁿ, n ≥ 2, that near each point equals c log|f| + O(1) (f a holomorphic tuple, c > 0, the remainder bounded on both sides) and is locally maximal is maximal; Corollary 1.3 gives a criterion in terms of the analytic spread of the defining ideals. The paper notes that such functions need not belong to the domain of definition of the operator. A second preprint of the same authors, arXiv:2610.05948, Theorem 1.5, reaches the same conclusion on domains in ℂ² for functions u with φ + ψ ≤ u and u + ψ ≤ φ near each point, where φ = α log‖f‖ and ψ is in Cegrell's class ℰ of a small hyperconvex neighbourhood. Neither preprint cites the list. The general case is open as far as we found. Read: the question and the text before it; Section 1 of both preprints.

Notes on the records (labels unchanged)

• Dinew–Guedj–Zeriahi, records AMR-147-*: The corpus labels of the AMR-147 records do not follow the numbering of the source. The arXiv version of Dinew–Guedj–Zeriahi (v1, the only version, 27 pages) numbers 45 questions consecutively. Records AMR-147-0001 to -0005 (labels Question 1–5) are Questions 1–5; -0006 to -0030 (labels Question 6–30) are Questions 7–31; -0031 to -0039 (labels Question 31–39) are Questions 33–41; -0040 to -0042 (labels Question 41–43; no record has the label Question 40) are Questions 43–45. Questions 6 (is DMA(X,ω) convex?), 32 (Kołodziej–Nguyen: can the condition f ≥ c₀ > 0 be removed?) and 42 (does P(α) coincide with NB(α)?) have no record. Records AMR-147-0043 and -0044 (labels Question 44 and 45: Dinh's question on the analyticity of the sets E_c(T) for positive dd^c-closed currents, and the Bloom–Levenberg question on regular subsets of large capacity) come from text that follows \end{document} in the arXiv source file and is not part of the arXiv paper; that text also contains a question of Błocki on an optimal constant C(n,p), which has no record. We could not check whether these three questions appear in the journal version.
• Dinh–Sibony, records AMR-135-*: The labels Problem 1–17 of AMR-135 count the problems of Dinh–Sibony (arXiv:1807.05640; Acta Math. Vietnam. 45 (2020) 103–112) consecutively; they are Problems 2.3, 3.2, 3.3, 4.1, 5.1, 5.2, 6.1, 7.1, 8.1, 8.2, 9.1, 9.2, 9.3, 9.4, 10.1, 10.2 and 11.1 of the list. Problems 2.2 (i)–(ii) and 2.4 have no record.
• AMR-135-0006: The statement of AMR-135-0006 (Problem 5.2: does ℙ² contain a smooth or real-analytic Levi-flat hypersurface?) begins with the remark that follows Problem 5.1 in the list, "The question is quite old and is due to Camacho-Lins Neto-Sad", together with the argument after it; that remark concerns Problem 5.1, the record AMR-135-0005, and not the Levi-flat question.

These are AI-assisted, self-audited checks, not independent human peer review: for each item we read the source and the cited statements and took publication data from Crossref; we did not check the proofs. The suggested labels are suggestions. Corrections are welcome. — Alper Ferudun, Mercury Software GmbH

Status follow-up 91: 11 status corrections and 1 note on records (records from the problem list on geodesics)

• Each item was checked against the current dataset (commit d371ee9, release v1.8.0), against its source (K. Burns, V. S. Matveev, Open Problems and Questions about Geodesics, Ergodic Theory Dynam. Systems 41 (2021) (arXiv:1308.5417), records AMR-149-*; cited by problem number) and against the cited papers, read on arXiv or in an open journal or author version; publication data are from Crossref. Theorem numbers follow the versions read. Suggested changes are in bold.

Solved in published papers (labels → solved)

• AMR-149-0001 → solved (yes); the dataset currently labels it open. K. Burns, V. S. Matveev, Open problems and questions about geodesics, Ergodic Theory Dynam. Systems 41 (2021) 641–684 (arXiv:1308.5417; numbers and pages of v3, September 2019, the numbering used by the citing papers), Conjecture 2.2.1 (Long, Bangert), PDF p. 3 (the record's "Conjecture 1"): every irreversible Finsler metric on S² has either exactly two or infinitely many distinct prime closed geodesics; the list recalls Katok's examples with exactly two and the lower bound two of Bangert and Long. D. Cristofaro-Gardiner, U. Hryniewicz, M. Hutchings, H. Liu, Proof of Hofer–Wysocki–Zehnder's two or infinity conjecture, J. Amer. Math. Soc. 39 (2026) 915–983 (arXiv:2310.07636), Corollary 1.4: every Finsler metric on S², reversible or not, has either two or infinitely many prime closed geodesics. It follows from Theorem 1.1 (a contact form on a closed connected 3-manifold whose contact structure has torsion first Chern class has two or infinitely many simple Reeb orbits, with no nondegeneracy assumption), since prime closed geodesics correspond to simple Reeb orbits on the unit tangent bundle; the paper cites the list's Conjecture 2.2.1 for this statement. Follow-up 26 cited the same corollary for the related record AMR-067-0028. Read: Section 2.2 of the list; Section 1.1 of the paper (Theorem 1.1, Corollaries 1.4, 1.7 and 1.8 and the remarks between them).
• AMR-149-0002 → solved (yes); the dataset currently labels it open. Burns–Matveev, Conjecture 2.2.2 (Long), PDF p. 4 (the record's "Conjecture 2"): the existence of one hyperbolic prime closed geodesic on a Finsler S² implies the existence of infinitely many distinct prime closed geodesics. The same paper, Corollary 1.8: a Finsler metric on S² with a closed geodesic that is not irrationally elliptic has infinitely many prime closed geodesics; it is derived from Corollary 1.7(b) and the authors' earlier result that, when there are exactly two simple Reeb orbits, both are irrationally elliptic. In the paper's definitions (Section 2.1) a hyperbolic orbit has real eigenvalues off the unit circle and an irrationally elliptic one has eigenvalues e^{±2πiα} with α irrational, so every hyperbolic closed geodesic qualifies; the paper calls Corollary 1.8 slightly stronger than Long's conjecture, which it cites as Conjecture 2.2.2 of the list. Read: as for AMR-149-0001, and the definitions in Section 2.1 of the paper.
• AMR-149-0051 → solved (no); the dataset currently labels it open. Burns–Matveev, Section 9.1, Question 9.1.1, PDF p. 25 (the record's "Question 52"): does every closed Lorentzian manifold have at least one closed geodesic? The list defines a closed geodesic as an embedding γ: S¹ → M with ∇_γ̇ γ̇ = 0, so light-like geodesics that return with a proportional but different velocity are not counted; the record omits this definition. S. Allout, A. Belkacem, A. Zeghib, On closed geodesics in Lorentz manifolds, Geom. Funct. Anal. 34 (2024) 1297–1309 (arXiv:2307.16810), Theorem 1.2: for G = SL(2, ℝ) or Sol there is a left-invariant Lorentzian metric on G such that no compact quotient Γ\G has a closed geodesic (γ(s) = γ(t) and γ̇(s) = γ̇(t) for some s ≠ t); these quotients have countably many weakly closed null geodesics (velocities proportional but not equal), which are not closed geodesics in the list's sense. The paper cites Question 9.1.1 of the list and presents these examples as a negative answer; its Theorem 1.1 gives a compact quotient with a left-invariant metric of signature (2,4) without closed or weakly closed geodesics. Read: Section 9.1 of the list; Section 1 of the paper (definitions, Theorems 1.1–1.3).

Partial results in print (labels → partially solved)

• AMR-149-0007 → partially solved (yes when the metric has a periodic Killing field preserving ω, under the conditions below; e.g. left-invariant ω on a compact Lie group with a bi-invariant metric); the dataset currently labels it open. Burns–Matveev, Section 2.5, Question 2.5.1, PDF p. 5 (the record's "Question 7"): for Kropina metrics F(ξ) = g(ξ, ξ)/ω(ξ) with ω ∧ dω nowhere zero (the setting of the section, which the record omits), does every Kropina metric on a closed manifold have a closed geodesic? The list notes that the answer is yes when the fundamental group is non-trivial and proposes S³ = Spin(3) with an arbitrary metric and a left-invariant ω as a test case. E. Caponio, F. Giannoni, A. Masiello, S. Suhr, Connecting and closed geodesics of a Kropina metric, Adv. Nonlinear Stud. 21 (2021) 683–695 (arXiv:2103.16912; numbers of v2), cite the question and prove, for a compact Riemannian manifold (S, g₀) and the Kropina metric K = −g₀/(2ω) (their normalisation): Theorem 5.6, if g₀ has a non-trivial periodic Killing field Y with L_Y ω = 0 and ω(Y) a negative constant, there are at least two closed geodesics; Theorem 5.8, the same if instead Y has constant length and ω(Y) < 0 everywhere; Corollary 5.9, every compact Lie group with a bi-invariant metric and a non-trivial left-invariant 1-form has at least two. These results do not need ω ∧ dω ≠ 0 but include such cases: on S³ = SU(2) every non-zero left-invariant 1-form is a contact form (our remark), so Corollary 5.9 covers the list's test case when the metric is the round one. Their Theorem 5.1 and Corollary 5.2 treat non-trivial fundamental groups, the case the list already records. Simply connected manifolds without such symmetry are open as far as we found. Read: Section 2.5 of the list; Sections 1 and 5 of the paper.
• AMR-149-0016 → partially solved (yes for compact quotients of Fuchsian buildings and for simple thick surface amalgams, within the classes of metrics below); the dataset currently labels it open. Burns–Matveev, Section 5.1, Question 5.1.2 (B. Schmidt), PDF p. 11 (the record's "Question 16"): for a 2-dimensional simplicial complex whose simplices are hyperbolic with geodesic edges, which satisfies the CAT(−1) condition at every vertex and has every edge in at least two simplices, does the marked length spectrum determine the metric, among all locally CAT(−1) metrics on the space or among all such complexes homeomorphic to it? The list records the positive answer for manifolds (Hersonsky–Paulin, 1997) and calls the case of edges in at least three simplices an easier version. D. Constantine, J.-F. Lafont, Marked length rigidity for Fuchsian buildings, Ergodic Theory Dynam. Systems 39 (2019) 3262–3291 (arXiv:1503.01321): let X be the quotient of a thick Fuchsian building (every edge in at least three chambers, the chambers hyperbolic polygons) by a group of combinatorial automorphisms acting freely and cocompactly, and g₀ a piecewise hyperbolic metric on X with non-singular vertices (angle sum 2π in every apartment); by Theorem 1, a locally negatively curved piecewise Riemannian metric g₁ on X (geodesic chamber sides) with curvature ≤ −1 has the same marked length spectrum as g₀ only if (X, g₁) is isometric to (X, g₀), and by the Main Theorem the same holds for g₁ with curvature in [−1, 0) and non-singular vertices. Y. Wu, Marked length spectrum rigidity for surface amalgams, Trans. Amer. Math. Soc. 378 (2025) 7075–7106 (arXiv:2310.09968), Theorem 1.3: on a simple, thick two-dimensional P-manifold (compact surfaces with boundary glued along disjoint closed geodesics, each gluing curve attached to at least three boundary components), two metrics of the class M_≤ (curvature ≤ −1 and geodesic boundary on each surface, isometric gluing, smooth with curvature ≤ −1 across any two sheets at a gluing curve) with the same marked length spectrum differ by an isometry isotopic to the identity. Piecewise hyperbolic metrics lie in these classes, and subdividing the pieces into geodesic triangles gives complexes of the list's type (our remark). Neither paper is cited in the list. Other complexes, and comparison with all locally CAT(−1) metrics, are open as far as we found. Read: Section 5.1 of the list; Sections 1–2 of both papers.
• AMR-149-0019 → partially solved (true near real and complex hyperbolic metrics, in dimension ≥ 3); the dataset currently labels it open. Burns–Matveev, Section 5.3, Conjecture 5.3.1 (Katok), PDF p. 12 (the record's "Conjecture 19"): for a metric of negative curvature on a compact manifold, h_top = h_Liou if and only if the metric is locally symmetric. The list records Katok's proof for surfaces, its extension to metrics conformal to a locally symmetric one, and L. Flaminio's result for deformations of constant curvature metrics. T. Humbert, Katok's entropy conjecture near real and complex hyperbolic metrics, Duke Math. J. 175 (2026) 135–184 (arXiv:2409.11197), Theorem 1: if g₀ is a real or complex hyperbolic metric on a closed manifold of dimension n ≥ 3, there are N(n) and ε > 0 such that every negatively curved metric g with ‖g − g₀‖_{C^N} < ε and equal Liouville and topological entropies is homothetic to g₀, hence locally symmetric. The paper describes Flaminio's theorem as an infinitesimal version near real hyperbolic metrics (positivity of the Hessian of h_top − h_Liou), so the neighbourhood statement and the complex hyperbolic case are new. Metrics away from these neighbourhoods are open as far as we found. Read: Section 5.3 of the list; Section 1 of the paper (the conjecture, Theorems 1–2 and the account of earlier results).
• AMR-149-0023 → partially solved (yes for orientable cusped manifolds; the closed and the non-orientable cusped cases only in preprints); the dataset currently labels it open. Burns–Matveev, Section 5.6, Question 5.6.1 (Miller, Reid), PDF p. 14 (the record's "Question 23"): are there infinitely many simple closed geodesics in every complete hyperbolic 3-manifold of finite volume? The list records the answer for surfaces and arithmetic examples in which every closed geodesic is simple. S. M. Kuhlmann, Geodesic knots in cusped hyperbolic 3-manifolds, Algebr. Geom. Topol. 6 (2006) 2151–2162 (arXiv:0906.5469), Theorem 1.1: every cusped (non-compact, finite-volume) orientable hyperbolic 3-manifold contains infinitely many simple closed geodesics; the list does not mention this paper. The other cases are claimed in preprints, not refereed: Q. Luo, V. Marković, Simple geodesics in closed hyperbolic manifolds, preprint arXiv:2608.29761 (August 2026), Theorem 1.4: in a closed hyperbolic n-manifold, n ≥ 3, the proportion of closed geodesics with length in (r − ε, r + ε) that are not r^(−κ)-collared tends to 0 for κ > 3/(n − 2); collared geodesics are simple, so every closed hyperbolic manifold of dimension ≥ 3 contains infinitely many simple closed geodesics (Theorem 1.5); F. Xia, Simple closed geodesics in cusped hyperbolic 3-manifolds, preprint arXiv:2110.14376 (2021), Theorem 1.2: every non-elementary hyperbolic 3-manifold with a virtually rank-2 cusp, in particular every cusped one of finite volume, orientable or not, contains infinitely many simple closed geodesics. We found no journal versions of the two preprints (arXiv, Crossref). Follow-up 36 reported the same question for the record KP-3.16. Read: Section 5.6 of the list; the introductions of the three papers (Theorem 1.1; Theorems 1.4–1.5; Theorems 1.1–1.2).
• AMR-149-0025 → partially solved (no, for metrics without focal points whose set of negative curvature has finitely many components); the dataset currently labels it open. Burns–Matveev, Section 6.2, Question 6.2.1 (Burns), PDF p. 15 (the record's "Question 25"): on a closed surface of genus ≥ 2, is there a C^∞ (or at least C^k, k ≥ 3) metric with a non-closed geodesic along which the Gaussian curvature is everywhere zero? The list records W. Wu's negative answer for non-positive curvature when {K < 0} has finitely many components (Ann. Fac. Sci. Toulouse Math. 24 (2015)). W. Wu, F. Liu, F. Wang, On the ergodicity of geodesic flows on surfaces without focal points, Ergodic Theory Dynam. Systems 43 (2023) 4226–4248 (arXiv:1812.04409), Theorem 1.9: for a C^∞ metric without focal points on a closed surface of genus ≥ 2 such that {p : K(p) < 0} has at most finitely many components, every geodesic along which K ≡ 0 is closed (and the geodesic flow is ergodic); the paper presents this as a negative answer to Question 6.2.1 for such surfaces. Metrics of non-positive curvature have no focal points, so this extends the case in the list to metrics with some positive curvature. The question is open in general as far as we found. Read: Section 6.2 of the list; Section 1 of the paper (arXiv v1, Theorems 1.5–1.9) and the abstract of the journal version.
• AMR-149-0029 → partially solved (yes when the total curvature exists, i.e. ∫K⁺ dA < ∞); the dataset currently labels it open. Burns–Matveev, Section 7.2, Question 7.2.2 (from K. Burns, G. Knieper, J. Differential Geom. 34 (1991)), PDF p. 18 (the record's "Question 29"): if a complete Riemannian metric on ℝ² has, for every geodesic and every point not on it, exactly one geodesic through the point that does not meet the given one, must the metric be flat? J. Ge, L. Guijarro, P. Solórzano, Riemannian rigidity of the parallel postulate in total curvature, Math. Ann. 376 (2020) 177–185 (arXiv:1611.07761), Theorem 1.3: a Riemannian plane that satisfies this parallel axiom (their Definition 1.1, the same condition) and admits total curvature is isometric to the flat Euclidean plane; admitting total curvature means that ∫K⁺ dA or ∫K⁻ dA is finite, which for planes is equivalent to ∫K⁺ dA < ∞ (Section 2). The paper presents this as an answer, in this case, to the question of Burns and Knieper. Planes with ∫K⁺ dA = ∞ are open as far as we found. Read: Section 7.2 of the list; Sections 1–2 of the paper.
• AMR-149-0038 → partially solved (yes for C^∞-generic metrics on closed surfaces); the dataset currently labels it open. Burns–Matveev, Section 8.4, Question 8.4.1, PDF p. 20 (the record's "Question 38"): is the union of the closed geodesics always dense in the manifold? The list states that metrics are not assumed generic in this section. K. Irie, Dense existence of periodic Reeb orbits and ECH spectral invariants, J. Mod. Dyn. 9 (2015) 357–363 (arXiv:1508.07542), Theorem 1.5: for every closed Riemannian surface (Σ, g), the conformal factors f for which the union of closed geodesics of fg is dense in Σ form a residual subset of C^∞(Σ, ℝ_{>0}); by Remark 1.6 the same holds for a residual set of all C^∞ metrics on Σ. So the answer is yes for generic metrics on surfaces; the paper is not cited in the list. For every metric, and in higher dimensions, we found no such result. Read: Sections 8.3–8.4 of the list; Section 1 of the paper.
• AMR-149-0045 → partially solved (yes for C²-close metrics, as the list itself records, and for all spheres of revolution in ℝ³); the dataset currently labels it open. Burns–Matveev, Section 8.7, Question 8.7.2 (Babenko), PDF p. 23 (the record's "Question 46"): does every Riemannian metric on S² of area 4π that is close enough to the round metric have a closed geodesic of length ≤ 2π, i.e. σ = area/sys² ≥ 1/π near the round metric? The list itself answers yes for C²-close metrics, by A. Abbondandolo, B. Bramham, U. Hryniewicz, P. Salomão, Math. Ann. 367 (2017) 701–753 (and for C³-close Finsler metrics by the same authors, Invent. Math. 211 (2018) 687–778), and asks about weaker topologies such as C¹ or Gromov–Hausdorff. New: the same authors, Sharp systolic inequalities for Riemannian and Finsler spheres of revolution, Trans. Amer. Math. Soc. 374 (2021) 1815–1845 (arXiv:1808.06995), Theorem 1: a sphere of revolution (a smooth surface in ℝ³ diffeomorphic to S² and invariant under the rotations about an axis) has ℓ_min²/area ≤ π, with equality only for Zoll spheres. So the answer is yes for all such spheres with no closeness assumption, in particular for those close to the round sphere in the C⁰, C¹ or Gromov–Hausdorff sense; the paper is not cited in the list. Other metrics close to the round one in topologies weaker than C² are open as far as we found. Note on the record: it omits the list's next paragraph, which says that the answer depends on the meaning of "close", records the C² case and asks about weaker topologies. Read: Section 8.7 of the list; the introduction of the paper (Theorems 1–2).

Notes on the records (labels unchanged)

• AMR-149-0006 (Burns–Matveev, Section 2.4, Question 2.4.1 (Paternain), PDF p. 5, the record's "Question 6": on a closed surface, is there at least one closed magnetic geodesic in every positive energy level?). Note on the record: it omits the list's next sentence, which gives a negative case (ω the area form of a hyperbolic surface, with the horocycle flow as magnetic flow) as the only one known, followed by positive cases; as worded, the question therefore already has a negative instance in its source, and the open part concerns the other cases.

These are AI-assisted, self-audited checks, not independent human peer review: for each item we read the source and the cited statements and took publication data from Crossref; we did not check the proofs. The suggested labels are suggestions. Corrections are welcome. — Alper Ferudun, Mercury Software GmbH

AIM-ARITHMETIC_GEOMETRY-0079: sharp interpolation for the equidegree Hilbert–Burch family

New unrefereed preprint: Sharp Interpolation for Equidegree Hilbert–Burch Curves, Alper Ferudun, Mercury Software GmbH (7 pages, version 1.0).

• DOI and files: https://doi.org/10.5281/zenodo.23435412
• Paper page: https://eulersolve.org/papers/aim-arithmetic_geometry-0079/
• Original source: AIM, Components of Hilbert Schemes (2010), Problem 32, PDF p. 4: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf

For every pair of positive integers t,a over an algebraically closed field of characteristic zero, let H(t,a) be the Hilbert component defined by maximal minors of a (t+1)-by-t matrix of degree-a forms. The paper proves that its interpolation number is

iota(H(t,a)) = t(t+1)(binom(a+3,3)-2)/2 = dim(H(t,a))/2.

Smooth integral curves attain this bound; the upper bound applies to every member of the component, including its boundary. The proof uses a bidiagonal degeneration and two triangular normal-evaluation systems. Normal-section evaluation has maximal rank for every number of general markings. For a >= 4 the normal bundle is special, so this is not a claim of nonspecial normal-bundle interpolation.

Scope: the source remains partially solved. This is a complete theorem for the specified infinite equidegree family, not a solution for arbitrary ACM degree matrices or arbitrary Hilbert components. The original AIM wording is broader than the locally Cohen–Macaulay interpretation in the dataset's research record. The complete-intersection and linear cases, and a small quadratic example, are credited to earlier work (Perrin; Ellia, with the attribution discussed in Kleppe–Miró-Roig). The paper also uses Ellingsrud's component theorem.

The source archive contains the full proof, references, and replayable exact checks. These checks support the written proof; they do not replace it. AI-assisted and self-audited, not independently refereed or formally verified. The literature comparison was bounded; no absolute-priority certification or maintainer acceptance is claimed. Corrections are welcome.

Status follow-up 92: 6 status corrections and 4 notes on records (records from problem lists on geometric hydrodynamics, on spaces with lower Ricci curvature and on Hénon maps, and an Oberwolfach problem on ReLU networks)

• Each item was checked against the current dataset (commit d371ee9, release v1.8.0), against its source (B. Khesin, G. Misiołek, A. Shnirelman, Geometric Hydrodynamics in Open Problems, Arch. Ration. Mech. Anal. 247 (2023) (arXiv:2205.01143), records AMR-143-*; A. Naber, Conjectures and Open Questions on Spaces with Lower Ricci Curvature, SIGMA 16 (2020) 104 (arXiv:2010.10031), AMR-128-*; Hénon Maps: A List of Open Problems, Arnold Math. J. 10 (2024) (arXiv:2312.03907), AMR-152-*; the Oberwolfach report 55/2021 Applied Harmonic Analysis and Data Science, OWR-8415360-010; cited by problem number) and against the cited papers, read on arXiv or in an open journal or author version; publication data are from Crossref. Theorem numbers follow the versions read. Suggested changes are in bold.
• Following the policy of release v1.8.0, answers that exist only in preprints are listed separately.

Solved in published papers (labels → solved)

• AMR-152-0042 → solved (S̄(g) ≅ S̄(f) exactly when g is conjugate to f in Aut(ℂ²)); the dataset currently labels it open. Hénon maps: a list of open problems (author line "Julia Xénelkis de Hénon"; on arXiv P. Berger, E. Bedford and 14 further authors), Arnold Math. J. 10 (2024) 585–620 (arXiv:2312.03907), Question 42, posed by C. Favre (§10.2). For a generalized Hénon map f, i.e. a composition of maps (x, y) ↦ (ay + P(x), x) with a ≠ 0 and deg P ≥ 2, let S(f) be the quotient by f of the basin Ω(f) of the super-attracting point at infinity, and S̄(f) the minimal resolution of its compactification by one normal singular point, a non-Kähler surface of class VII₀. The question: describe all generalized Hénon maps g with S̄(f) biholomorphic to S̄(g). The list recalls partial results of R. Pal and of Bonnot–Radu–Tanase. F. Bacher, Hénon maps with biholomorphic Kato surfaces, Comm. Amer. Math. Soc. 5 (2025) 457–474 (arXiv:2410.19547), Theorem 1.1: let F = F_N ∘ ⋯ ∘ F_1 with Hénon maps F_i, and let G be a generalized Hénon map. Then the Kato surfaces S_F and S_G are biholomorphic if and only if G is affinely conjugate to one of the cyclic reorderings F_k ∘ ⋯ ∘ F_1 ∘ F_N ∘ ⋯ ∘ F_{k+1}, 1 ≤ k ≤ N. By Friedland–Milnor, as the paper notes, this means that F and G are conjugate in Aut(ℂ²). S_F is the Kato surface of the germ of F at infinity, which compactifies Ω⁺/F. The paper states Favre's question in this form, and its abstract says that it answers it. Read: §10.2 of the list (arXiv version and journal HTML); the abstract, Section 1 and the definitions of Section 2 of the paper (v2).

Answered in a doctoral dissertation (not a journal publication; under the dataset policy of release v1.8.0: partially solved, evidence unrefereed_claim)

• OWR-8415360-010 → partially solved (no, for every s ≥ 2; answered in a doctoral dissertation); the dataset currently labels it open. F. Krahmer (joint with S. Bamberger and R. Heckel), abstract in the Oberwolfach report 55/2021, Applied Harmonic Analysis and Data Science, Oberwolfach Rep. 18 (2021), no. 4 (issued 2022), pp. 3053–3055 (DOI 10.4171/owr/2021/55), Theorem 2.3 and the question after it: for s ≥ 2 and A ∈ ℝ^{m×n} with Ax ≠ 0 for every nonzero 2s-sparse x, an unbiased ReLU network with ⌈log₂ s⌉ + 2 hidden layers recovers every s-sparse x from Ax exactly; is this number of layers optimal? No: S. Bamberger, Representing and Recovering Structured High-Dimensional Data: Fast Dimension Reduction, Recovery Guarantees, and Neural Network Representation, doctoral dissertation, Technische Universität München (accepted 2023; mediaTUM 1659564), Theorem 3.30 (p. 109): ⌈log₂(s) + 1⌉ hidden layers suffice for the same matrices, with an unbiased network. The theorem is stated without the hypothesis s ≥ 2; for s = 1 it would give one hidden layer, which Corollary 3.3 of the same thesis rules out when m < n (the gluing step, Lemma 3.38, needs at least one layer of its own), so it is to be read for s ≥ 2, where the proof goes through after a small repair of Lemma 3.38 for 0-dimensional pieces (our reading). We also checked the bound ⌈log₂ s⌉ + 1 for every s ≥ 2 by an independent argument (a generic projection to ℝ^{2s}, a positively homogeneous piecewise-linear extension of the decoder and refereed depth bounds for piecewise-linear functions). For s = 2 two hidden layers are therefore optimal when m < n, since one hidden layer cannot recover all 1-sparse vectors (Corollary 3.3 of the thesis; S. Bamberger, R. Heckel, F. Krahmer, J. Approx. Theory 311 (2025) 106177, arXiv:2308.02836, Corollary 2.1). For s ≥ 3 the exact optimum is not determined, and no lower bound above two hidden layers is known. Read: Krahmer's abstract; Chapter 3 of the dissertation (Corollary 3.3, Theorems 3.30, 3.34 and 3.41, Lemmas 3.35–3.40 and the discussion on p. 124); Section 2 of the arXiv version of Bamberger–Heckel–Krahmer.

Partial results in print (labels → partially solved)

• AMR-143-0026 → partially solved (non-transitivity and non-mixing in C^k for every k ≥ 1, with any helicity and large energy; the local non-mixing near steady flows is not covered); the dataset currently labels it open. B. Khesin, G. Misiołek, A. Shnirelman, Geometric hydrodynamics in open problems, Arch. Ration. Mech. Anal. 247 (2023), Art. 15 (arXiv:2205.01143), Problem 31, §11 (the record's Problem 26): relax the restrictions on the smoothness index k of the C^k spaces, which come from the use of KAM, and prove non-transitivity and non-mixing of the 3D Euler equation "in full generality". The text before the problem recalls two results of Khesin, Kuksin and Peralta-Salas. The first is non-mixing in C^k (k > 4, non-integer) on every closed Riemannian 3-manifold, for any prescribed helicity and sufficiently large energy. The second is local non-mixing near "typical" steady flows on S³ in C^k (k > 10). B. Khesin, The Euler non-mixing made easy, Nonlinearity 37 (2024) 095025 (arXiv:2402.08836), Theorem 1: (i) on a compact 3-manifold the Euler equation is non-transitive, hence non-mixing, in the C^k phase space of velocities for every k ≥ 1. There are two open sets, which can be chosen among null-homologous fields with any helicity and sufficiently high energy, such that the flow image of one never meets the other. (ii) The same holds in every dimension. In Remark 5, Khesin, a co-author of the list, writes that without extra constraints Problem #31 "becomes rather straightforward". The note lists local non-mixing near steady solutions among the variations that need more subtle arguments and does not prove it. R. Cardona, F. Torres de Lizaur, preprint arXiv:2312.03514 (v2, January 2024), Theorem 1: on every closed Riemannian 3-manifold the Euler equation is non-mixing in the C¹ topology among fields with fixed helicity h ≠ 0, fixed energy e ≥ e₀ and a fixed homotopy class of non-vanishing vorticity. The authors also present this as solving the problem. Open as far as we found: local non-mixing near steady flows in low smoothness. Read: §11 of the list (v2); the whole note of Khesin (v2); the abstract and Section 1 of the preprint.
• AMR-128-0003 → partially solved (dimension 4); the dataset currently labels it open. A. Naber, Conjectures and open questions on the structure and regularity of spaces with lower Ricci curvature bounds, SIGMA 16 (2020) 104 (arXiv:2010.10031), Conjecture 2.9 (the record's Conjecture 3): if (M_i^n, g_i, p_i) → (X, d, p) with Ric ≥ −(n − 1) and Vol(B_1(p_i)) > v > 0, then at a fixed point the tangent cones of X are all homeomorphic, away from a set of codimension five. The list states it after the Colding–Naber example of a 5-dimensional limit with non-homeomorphic tangent cones at one point. It is Conjecture 1.2 of T. H. Colding, A. Naber, Geom. Funct. Anal. 23 (2013) 134–148 (arXiv:1108.3244), who add that for a four-dimensional limit the tangent cones at each point should be homeomorphic. E. Bruè, A. Pigati, D. Semola, Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below, Duke Math. J., advance publication 2026, doi:10.1215/00127094-2025-0078 (arXiv:2405.03839), Theorem 1.1: for a 4-dimensional noncollapsed Ricci limit and each x ∈ X there is a 3-manifold Σ_x with universal cover S³ such that every tangent cone at x is a cone over a space homeomorphic to Σ_x. So all tangent cones at x are homeomorphic, which is the conjecture for n = 4; the paper says that this confirms the Colding–Naber conjecture in dimension 4. For n ≥ 5, Theorem 1.4(ii) gives the same conclusion only at points where all tangent cones are (n − 4)-symmetric. Remark 1.5 says that Naber's Conjecture 2.16 together with Theorem 1.4(ii) would give the full conjecture. Open for n ≥ 5 as far as we found. Read: Section 2 of the list; Section 1 of the paper; Section 1 of Colding–Naber.
• AMR-152-0029 → partially solved (no for all maps that are not unstably connected, e.g. when J is disconnected or |Jac f| > 1, and no for global foliations); the dataset currently labels it open. Hénon list, Question 31, posed by R. Dujardin (§8; the record's Question 29): is there a generalized complex Hénon map f, an open set U ⊂ ℂ² meeting J⁺ and a holomorphic foliation of U such that J⁺ ∩ U is a union of leaves? The list conjectures that the answer is no. A negative answer would imply that f preserves no holomorphic foliation of ℂ², and Brunella had excluded algebraic ones. S. Cantat, R. Dujardin, Some rigidity results for polynomial automorphisms of ℂ², Cambridge J. Math. 14 (2026) 539–601 (arXiv:2411.10339), work with automorphisms of positive entropy (the conjugates of generalized Hénon maps) and allow foliations with isolated singularities. Proposition 4.7: if J⁺ is a union of leaves of a holomorphic foliation in some open set U meeting it, then f is unstably connected in the sense of Bedford–Smillie, hence J is connected and |Jac f| ≤ 1. So the answer is no for every other map, for instance horseshoes and maps with |Jac f| > 1. Theorem B (Theorem 5.1): J⁺ is never subordinate to a global real-analytic, in particular holomorphic, foliation; the authors call this "a partial answer to Question 31". Proposition 4.10: if |Jac f| < 1 and the interior of K⁺ is empty or a union of sink and parabolic basins, U cannot be a neighbourhood of J*. For the local question the paper says it has only partial answers. Open as far as we found: unstably connected maps outside these cases. Read: §8 of the list; the introduction (Theorem B and the text after it), Section 4 and Theorem 5.1 of the paper (numbers of the arXiv version; the journal's numbering was not checked).

Partial results in preprints (not refereed; under the dataset policy of release v1.8.0: partially solved, evidence unrefereed_claim)

• AMR-152-0046 → partially solved (yes in odd degrees, one map in each degree; preprint); the dataset currently labels it open. Hénon list, Question 45 of the arXiv version, Question 46 of the journal version (the record's Question 46), posed by P. Ingram (§11): over a number field K, can one construct infinite families of generalized Hénon maps of degree d with K-rational cycles of length at least d + 3? Can one construct maps with N_d periodic points where N_d − d → ∞, or even N_d/d → ∞, as d → ∞? The list notes that interpolation gives cycles of length d + 2. H. Kim, H. Krieger, M.-I. Postolache, V. Szeto, Hénon maps with many rational periodic points, preprint arXiv:2412.01668 (v2, July 2025). Theorem A: for each odd d > 2, the Hénon map h_d(x, y) = (y, −x + s_d(y)) of degree d over ℚ, with an explicit polynomial s_d, has at least (d − 4)² rational periodic points. Theorem B: for d ≡ 1 mod 6, h_d has a cycle of integer points of length (8d + 10)/3, which exceeds d + 3. The authors state that this answers "the first two parts" of the question in the affirmative. Not covered: even degrees; cycles of length at least d + 3 in odd degrees d ≢ 1 mod 6, for which the paper only remarks that a shift of s_d gives similar results; and several maps in one fixed degree. We found no journal version (arXiv journal-ref field, Crossref), so under the policy of release v1.8.0 this is an unrefereed claim. Read: §11 of the list (arXiv version and journal HTML); the abstract, Section 1 and the definition of s_d in Section 2 of the preprint.

Notes on the records (labels unchanged)

• Khesin–Misiołek–Shnirelman, records AMR-143-*: The background of the AMR-143-* records gives the source as "Khesin, Misiolek, Modin". The authors of Geometric hydrodynamics in open problems (arXiv:2205.01143; Arch. Ration. Mech. Anal. 247 (2023), Art. 15) are B. Khesin, G. Misiołek and A. Shnirelman. The paper numbers its problems 1–40 consecutively. The 35 records, labelled Problem 1–35, follow them in order but skip five paper problems, which have no record: 7 (ill-posedness in the Besov spaces B¹_{∞,q}), 9 (Palais–Smale condition for the L² energy in 2D), 15 (curl u = const in B₀ for GM-flows), 21 (complexity of a typical geodesic ≃ t) and 25 (entropy decrease). So labels 1–6 are Problems 1–6, label 7 is Problem 8, labels 8–12 are Problems 10–14, labels 13–17 are Problems 16–20, labels 18–20 are Problems 22–24 and labels 21–35 are Problems 26–40. Papers citing the list use the paper's numbers (Problem 31 for AMR-143-0026).
• Naber, records AMR-128-*: The labels Conjecture 1–9 of the AMR-128-* records count Naber's conjectures consecutively. In the paper (SIGMA 16 (2020) 104, arXiv:2010.10031), where conjectures share one counter with theorems, open problems, definitions and examples, they are Conjectures 2.4, 2.7, 2.9, 2.15, 2.16, 2.18, 2.20, 3.3 and 3.5. Open Problems 2.2, 3.2 and 3.4 have no record.
• OWR-8415360-010 (source): the record's literature link names "Thesing–Hansen". The question is in F. Krahmer's abstract (joint with S. Bamberger and R. Heckel), pp. 3053–3055 of the report; the abstract of L. Thesing (joint with A. C. Hansen) on pp. 3030–3032 of the same report concerns a different question.
• OWR-8415360-010 (statement): the statement speaks of "the known depth bound ⌈log₂s+2⌉", while the source counts hidden layers (⌈log₂ s⌉ + 2 hidden layers, Theorem 2.3). With the common convention depth = hidden layers + 1 the question would change (for s = 2 the answer would then be yes), so "the bound ⌈log₂ s⌉ + 2 on the number of hidden layers" would match the source.

These are AI-assisted, self-audited checks, not independent human peer review: for each item we read the source and the cited statements and took publication data from Crossref; we did not check the proofs. The suggested labels are suggestions. Corrections are welcome. — Alper Ferudun, Mercury Software GmbH