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

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

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

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

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

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

— Alper Ferudun

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

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

— Alper Ferudun

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

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

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

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

— Alper Ferudun

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

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

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

— Alper Ferudun

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

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

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

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

Stale statuses

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

Transcription errors

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

Merged records

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

Misattached note

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

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

— Alper Ferudun

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

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

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

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

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

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

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

Stale statuses

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

Answered in the source itself

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

Transcription errors and literal readings

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

Duplicates

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

Merged records

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

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

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

Misattached notes

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

— Alper Ferudun

Follow-up 7 (2026-09-28): corrections to combinatorics records from the EGRES Open list, J. Cooper's "Combinatorial Problems I Like" and TOPP (AMR-*), and to graph-theory records from the Open Problem Garden (OPG-*). They cover stale statuses, questions answered in their own source, literal statements that fail, and errors in notes and research summaries. Each item was checked against the current version (commit 2ea030b) and against the source page; literature was checked against arXiv and Crossref. Suggested changes are in bold.

Stale statuses

• AMR-029-0056 → solved (yes). For fixed k, a minimum k-way cut in a hypergraph with nonnegative hyperedge costs can be found in randomized polynomial time (Chandrasekaran–Xu–Yu, SODA 2018) and in deterministic polynomial time (Chandrasekaran–Chekuri, Math. Oper. Res. 47 (2022)). The EGRES page is outdated: it says no algorithm is known for k = 4.
• AMR-029-0005 → solved (yes). The EGRES page refers to the Bernardi–Kálmán–Postnikov polymatroid Tutte polynomial. Guan–Jin–Kálmán, IMRN 2025 (arXiv:2309.13639) give a deletion–contraction formula for it.
• AMR-029-0084 → solved (no). Hollom–Randall Shaw, Electron. J. Combin. 33(2) (2026) P2.11, Thm 1.2, give a 3-uniform hypergraph with no strongly maximal matching. Their definition is the one on the EGRES page.
• AMR-029-0085 → solved (no). As stated, it already fails for the hypergraph of all finite subsets of ℕ (van der Zypen, arXiv:2205.02296). A cover that is not minimal is never strongly minimal. In a minimal cover K, replacing two members a, b by a ∪ b gives a cover K′ with |K∖K′| = 2 > 1 = |K′∖K|. Hollom–Randall Shaw (above, Thm 1.3) give a 3-uniform example without isolated vertices.
• AMR-030-0037 → solved (yes), with a caveat. Keevash, arXiv:1802.05900, Thm 1.8, shows that the number of d-dimensional permutations of order n is (n/e^d + o(n))^{n^d}. He presents this as the answer to the question of Linial and Luria. It is a preprint, and the details of the lower bound are omitted as similar to those in Counting designs. Please replace the summary's attributions with this reference.
• AMR-030-0078 → solved (yes) for every fixed k and large n. Glock–Joos–Kühn–Osthus, Combinatorica 40 (2020) (arXiv:1808.07720) show that quasirandom k-graphs whose degrees are all divisible by k have tight Euler tours. The complete case is the Chung–Diaconis–Graham conjecture. (The summary already calls it solved, but cites other work.)
• OPG-130 → false as stated; open for k = 1, 2. Han–Li–Wu–Zhang, JCTB 131 (2018) show that for every p ≥ 3 there are 4p-edge-connected graphs with no modulo-(2p+1)-orientation. The case k = 1 is the 3-flow conjecture, OPG-128.
• OPG-135 → solved (false), as the source's own comments (Royle 2009, Šámal 2010) say. Jacobsen–Salas, JCTB 103 (2013) report the disproof by Haggard–Pearce–Royle. They also find real flow roots above 5, e.g. Q ≈ 5.1653 for G(119,7).
• OPG-323 → solved (yes). Šámal, JGT 85 (2017) constructs an infinite set of graphs with no cycle-continuous mapping between any two of them.
• OPG-407 → solved. Brouwer–Haemers, LAA 429 (2008) prove μ_j ≥ d_j − j + 2 for 1 ≤ j ≤ n−1 in every connected graph, which is exactly c_k ≥ d_k. The source's 2008 comment already says so.
• OPG-547 → solved (no, for k = 3). Gebauer–Gundert–Moser–Okamoto, arXiv:1109.3390, cited in the source's 2014 comment, give a saturated 3-forest that is not tight: the 3-graph on {1,…,6} with edges 123, 124, 125, 134, 236, 256, 346, 356. The paper is a preprint, but the example is finite; we re-checked it over all 540 surjective 3-colourings.
• OPG-826 → solved (yes). Scott–Seymour–Spirkl, Pure pairs V, Combinatorica 43 (2023) prove Fox's conjecture: for every c > 0, every perfect graph on n > 1 vertices has a pure pair A, B with |A|, |B| ≥ εn^{1−c}. A complete pair gives K_{A,B} in G, and an anticomplete pair gives K_{A,B} in Ḡ. The source's 2021 comment points to this paper.
• OPG-2226 → solved. Let G be an r-graph and U an odd set. Then 2|E(U)| = r|U| − |δ(U)| ≤ r(|U| − 1), so the density Γ(G) is at most r, and Goldberg–Seymour gives χ′(G) ≤ r + 1. The dataset already records Goldberg–Seymour as solved (OPG-2242, Chen–Jing–Zang, J. Comb. Optim. 50 (2025)).
• OPG-37182 → solved (false). The source's comments (June–July 2010) give a complete counterexample: an odd number n of copies of the Petersen graph minus an edge, joined in a ring. Every 2-factor has only odd cycles, and the oddness is n + 1. We re-checked n = 3 exhaustively: 30 vertices and 72 two-factors, all with only odd cycles, and oddness 4. The record's note asks for a citable source; the argument is short and complete.
• OPG-48583 → solved (yes). Duraj–Gutowski–Kozik, Electron. J. Combin. 23(3) (2016) P3.3 show that ch^OL(K_{N,N}) − ch(K_{N,N}) = Θ(log log N).
• OPG-37217 → solved (yes). The record's own review accepts the MaxEDP part: Ene–Mnich–Pilipczuk–Risteski (SWAT 2016) give an O(r³)-approximation for edge-capacitated MaxEDP on graphs of treewidth r, and k-outerplanar graphs have treewidth O(k).
  • Our remark on MaxIMF: since an integer multiflow splits into paths, MaxIMF is the same problem except that each net may be routed several times.
  • Listing each net as often as its capacity allows gives the same optimum, and the O(r³) ratio does not depend on the number of pairs.
  • The running time is then polynomial in the capacities.

Answered in the source itself

• AMR-029-0080 → solved (NP-complete). The EGRES page carries the "Solved" badge: Hörsch–Szigeti showed that the existence question is NP-complete (J. Comb. Optim. 45 (2023)). Only the EGRES index still lists the problem as open.
• AMR-029-0089 → solved (NP-complete). The page carries the "Solved" badge and is listed under solved problems. Hörsch–Szigeti, Inf. Process. Lett. 183 (2024) show that it is NP-complete to decide whether there is an orientation in which the vertices of S are pairwise mutually reachable. (The page notes that the case of constant |S| is open.)
• AMR-054-0074 → partially solved. The TOPP entry itself (2012 update) notes that Kaiser's Thm 1.4 (DCG 18 (1997)) gives f(m) ≤ 2m, and says: "This largely solves the problem." Since m − 1 pairwise independent rectangles need m − 1 lines, f(m) = Θ(m); only the exact value is open. The summary's claim that the asymptotics are open is out of date.

Literal statements that fail (the intended problems stay open)

• AMR-029-0058: the problem links to the EGRES definition page, which allows b = +∞ and speaks of a base polytope only when b is finite. Take b({1}) = 1, b({2}) = +∞, b(V) = 0. Then B = {(t, −t) : t ≤ 1} contains 0, and its only vertex (1, −1) has no opposite vertex. Please say "base polytope"; that version is open.
• AMR-030-0016: as worded (straight-line drawings), two edges cross at most once, so the two numbers coincide trivially. The intended Pach–Tóth question is about arbitrary drawings. That is OPG-37117, which the dataset itself treats as open. Restate for arbitrary drawings, or mark as a duplicate of OPG-37117. The summary's claim that the two numbers are known to differ is unsupported.
• AMR-030-0077: as worded it is impossible for every k, because a pattern that is avoidable over k − 1 letters is avoidable over k letters. The intended question is "avoidable over k letters but not over k − 1", i.e. avoidability index exactly k. Indices 4 and 5 are due to Baker–McNulty–Taylor (1989) and Clark (2006); index 6 is open.
• AMR-029-0079: "quasi-kernel" is not defined. The sister record AMR-029-0067 requires every vertex to be reachable from K in at most two steps.
  • Under that convention the statement as written fails: take three sources pointing to t, with t ⇄ u. All 5 vertices have positive out-degree, but every quasi-kernel contains the three sources.
  • With that convention the hypothesis would have to be positive in-degree; Clow, arXiv:2601.11847, states the conjecture for sourceless digraphs.
  • Please add the definition (every vertex reaches K in at most two steps); then it is the open Erdős–Székely conjecture.
  • The summary also needs fixing: the n/2 bound is the conjecture itself, not a Chvátal–Lovász theorem, and Clow notes that no (1 − ε)n bound is known.
• OPG-125: Royle's comment on the source page settles the literal version. Cay(Z₁₅, {±1, ±4}) contains a 5-cycle and maps onto C₅ by x ↦ x mod 5. So its core is C₅, which is not a Cayley graph on any power of Z₁₅. Restrict to M = Z₂ (cube-like graphs) or Z_p. That version is open; by Mančinska–Pivotto–Roberson–Royle, a counterexample for Z₂ needs at least 128 vertices.
• OPG-333: false for uncountable graphs (Oporowski, JGT 14 (1990), cited in the source's comments). Restrict to countable graphs, which the comments call the real problem.
• OPG-567: DeVos's comment on the source page shows that no fixed k works for any of the four questions. Give weight 1 to the edges of a K_{2k,4k} inside K_{6k} and weight 2 to all others. The procedure may then choose all k trees (or 1-trees) inside the unbalanced bipartite K_{2k,4k}, so their union has no Hamiltonian path or cycle. Mark as answered (no uniform k) or unpublish.
• OPG-46456: the source page itself has a typo. The hypothesis should be minimum out-degree f(d), not d; the page's own f(1) = 3 (Thomassen) shows this, and see also Christoph–Petrova–Steiner, CPC 34 (2025). As printed, f(d) is unused, and the bidirected K_{d+1} is a counterexample.
• Already flagged in the records' own reviews, but statement is unchanged:
  • OPG-2095: add finite, connected, at least 3 vertices; K₂ = Cay(Z₂, {1}) is not Hamiltonian.
  • OPG-37325: add n large relative to k; for n ≤ 2k + 1, C_n^k = K_n and eq = 1.
  • OPG-60001: add k ≥ 3, as in the conjecture of Brewster–McGuinness–Moore–Noel; K₂ and K₃ fail for k = 1, 2. The case k = 3 is proved by Kim–Picollelli.
  • OPG-37079: this is a finite game, so optimal strategies exist by backward induction, as a 2009 comment on the source says. Restate as an efficiency question or retire.

Corrections to notes and research summaries (statuses unchanged)

• AMR-029-0068: the summary says Kotlar–Ziv (2021) and Frankl–Kupavskii (2023) proved the Aharoni–Berger conjecture. They did not. Kotlar–Ziv (EJC 38 (2014)) proved the bound ⌊5k/3⌋. The best bound is k + o(k) (Pokrovskiy), as the EGRES page says.
• AMR-030-0015: the summary says Erdős–Hajnal was disproved, citing arXiv:2310.15628, which is an unrelated paper on arithmetic functions. EH is open. The source's remark that even C₅ is open is outdated: Chudnovsky–Scott–Seymour–Spirkl, Proc. LMS 126 (2023) proved that case.
• AMR-030-0043: the summary says the limit 2 was proven. In fact Tompkins, arXiv:2607.09497 (July 2026 preprint) disproves the diamond conjecture: La(n, Q₂) ≥ (2.1479… + o(1))·C(n, ⌊n/2⌋).
• AMR-030-0003: the summary's "solved" argument does not work. This is Graham's question, TOPP 58, which TOPP lists as open. It is also AMR-054-0058, which allows equilateral triangles as well. Consider marking the two records as duplicates.
• AMR-030-0011: the summary treats illumination as direct visibility (star-shapedness). In Straus's problem the walls are mirrors, and illumination from some point is still open for polygonal rooms (MathWorld, which the source links).
• AMR-029-0081: the summary's "solved" is unsupported. The problem asks for combinatorial algorithms that find unweighted subgraphs. Existence is already known (the EGRES page derives it from de Carli Silva–Harvey–Sato), and the cited results do not provide such algorithms.
• AMR-028-0001: Stanley's depth conjecture was disproved by Duval–Goeckner–Klivans–Martin, Adv. Math. 299 (2016), not by "2022 counterexamples" of Ichim–Katthän–Moyano-Fernández.
• AMR-030-0089 (pointer; the status stays open): the intended statement is the conjecture of Bixby–Flint–Miklós (arXiv:1303.6799, Involve 9 (2016)). Two successful pressing sequences are adjacent when their longest common subsequence has length at least L − 4.
  • H. W. Whitlatch's thesis (Univ. of South Carolina, 2019, §6.3; supervised by J. Cooper) defines the edit distance in exactly this way.
  • The constant 4 cannot be lowered (ibid., Example 6.6).
  • An exhaustive search over all black-and-white graphs on at most 7 vertices finds no counterexample.
• AMR-029-0034 (flag only): Du, arXiv:2604.01571 (April 2026, unrefereed) claims a deterministic O(n⁶) algorithm for bipartite exact matching. Its Lean formalization reduces the main theorem to eight explicit hypotheses. We have not checked it.

— Alper Ferudun

Two scope-specific preprints: AIM-LOGIC-0084 and AIM-LINEAR_ALGEBRA-0012

I am reporting two preprints with explicit scope limitations. Neither should
be recorded as an unqualified solution of every possible reading of the
corresponding source record.

AIM-LOGIC-0084: computability of nonforking sections

Sharp Computability Bounds for Nonforking Sections

• DOI: https://doi.org/10.5281/zenodo.23009494
• Public Zenodo record: https://zenodo.org/records/23009494
• Paper and files: https://eulersolve.org/papers/aim-logic-0084/
• Source: AIM Computable stability theory, Section 1.3, Question 6,
  https://aimath.org/pastworkshops/computestabproblems.pdf (printed p. 2).

For characteristic-function names of types and separately decidable elementary
diagrams of countable stable models, the nonforking section is computable from
the input type together with the fixed oracle 0'. The bound is sharp for a
decidable, omega-stable, omega-categorical equivalence-relation theory. For a
decidable-range inclusion, an oracle computes a section if and only if it
computes which classes of the larger model meet the smaller model; explicit
inclusions realize every c.e. degree as the least auxiliary degree.

The counterexample concerns a uniform section, not a noncomputable individual
output type. All individual types in the example are computable nonuniformly.
Importantly, a decidable full predicate-pair diagram instead yields a
computable section for every stable theory. The printed AIM question omits
effective-presentation conventions. Please retain that qualification rather
than marking the source as a presentation-independent negative result.

AIM-LINEAR_ALGEBRA-0012: the derivative-plus-constant branch

Weighted Root Deletions and Coefficientwise Toeplitz Positivity

• DOI: https://doi.org/10.5281/zenodo.23012273
• Public Zenodo record: https://zenodo.org/records/23012273
• Paper and files: https://eulersolve.org/papers/aim-linear-algebra-0012/
• Source: AIM Theory and applications of total positivity, Problem 1.4,
  https://aimath.org/pastworkshops/totalposrep.pdf (printed p. 2).

For independent indeterminates a, X=(x_1,...,x_N), and Y=(y_1,...,y_N), every
finite minor of the upper Toeplitz matrix of

b_k = a e_k(X) + sum_i y_i e_k(X without x_i)

has nonnegative integer coefficients. This includes the coefficientwise total
nonnegativity of the coefficients of (u D_z + v) product_i(1+x_i z). The proof
uses bordered Gram minors, Schur-complement characters, and a letter-content
grading to produce squared-norm coefficient certificates.

This closes the D+a branch, strengthened to independent root weights. It does
not close the other operator conjectures grouped in the same source
record. The anonymous affine result reported in Sokal's 2025 talk is credited;
no absolute priority claim is made.

Both releases include English PDFs, LaTeX sources, proof/audit notes and exact
finite regression scripts. They are AI-assisted, self-audited, unrefereed
preprints. Publication, finite tests, and self-audit are not independent peer
review or formal verification. I remain responsible for the manuscripts and
welcome specific mathematical corrections.

Release-time access note (28 September 2026): both Zenodo records and all
files are public and verified, and Zenodo has assigned the DOIs above. At our
release-time check, doi.org still returned "DOI Not Found" for those identifiers;
please use the direct Zenodo or EulerSolve links if DOI resolution is not yet
available. This note does not imply a registration-completion time guarantee.

Alper Ferudun — Mercury Software GmbH

AIM-COMBINATORICS-0177: a counterexample for fixed-order, vertex-averaged reachability

I have released A Six-Vertex Counterexample to Vertex-Averaged Ordered Reachability.

• Paper and files: https://eulersolve.org/papers/aim-combinatorics-0177/
• Public Zenodo record: https://zenodo.org/records/23016087
• Assigned DOI: https://doi.org/10.5281/zenodo.23016087
• Primary source: Sullivan's 2006 survey, Conjecture 6.10 (attributed to DeVos), printed p. 7: https://aimath.org/WWN/caccetta/caccetta.pdf

There are two loopless digraph layers on six vertices, each of outdegree two, with no rainbow directed cycle. In the fixed label order 1<2, the increasing-path reachability sizes (including the starting vertex) are 5,5,5,5,5,4. Thus their sum is 29<30, contradicting the proposed bound under the uniform-start-vertex interpretation with fixed label order. Uniform blow-ups give examples on 6m vertices with average 1+23m/6<1+4m for every positive integer m. The proof is a direct neighborhood calculation; the search is not needed to verify it.

The source does not specify the averaging variable. The reverse label order gives total 35, so this is not a counterexample to an order-averaged alternative. It does not disprove the Caccetta–Haggkvist conjecture or the unordered rainbow-path conjecture, and I do not claim minimality of six vertices. Please preserve these distinctions rather than marking every interpretation of the source as solved. In the frozen v1.6.0 corpus, 0177 is the split heading and 0178 contains the body of the same conjecture; these are not two separate results.

The release includes the English PDF, LaTeX source and exact standard-library verifier. This is an AI-assisted, self-audited, unrefereed preprint, not an independently refereed or formally verified result; no absolute priority certificate is claimed. I remain responsible for the text and welcome specific corrections.

Access note (28 September 2026): the Zenodo record and files are public and verified. Zenodo has assigned the DOI, but doi.org returned "DOI Not Found" at the release-time check. Please use the direct Zenodo or EulerSolve link if necessary; no activation-time guarantee is implied.

Alper Ferudun — Mercury Software GmbH

AIM-REPRESENTATION_THEORY-0102: a degree-24 obstruction for three explicit supercharacter families

I have released A Degree-24 Obstruction to Hopf Structures on Symmetric-Group Supercharacter Spaces.

• Paper and files: https://eulersolve.org/papers/aim-representation-theory-0102/
• Public Zenodo record: https://zenodo.org/records/23018019
• Assigned DOI: https://doi.org/10.5281/zenodo.23018019
• Primary source: AIM, Supercharacters and Combinatorial Hopf Algebras (2010), Problem 1.18, p. 3: https://aimath.org/WWN/supercharacters/supercharacters.pdf

For the supercharacter family whose superclasses are the individual S_n-conjugacy classes inside A_n together with the single odd-permutation block, the superclass-function spaces admit no connected graded Hopf algebra structure in characteristic zero, with the ordinary degree-n grading and one-dimensional components in degrees zero and one. This excludes arbitrary graded Hopf operations, not only quotients of symmetric functions.

The proof applies the known nonnegative Euler-product condition for Hopf Hilbert series (credited to Zhou–Shen–Lu, Proposition 3.5, and its antecedents). The first negative Euler exponent is a_24=-1: lower degrees force 795 basis monomials in degree 24, but the required dimension is 794. The coarse alternating-group partition and the maximal two-block partition both fail the same test in degree 4. The ordinary conjugacy-class family is the classical positive case.

Scope: the original question does not name its four families. The manuscript defines the three obstructed families explicitly and does not claim to classify all symmetric-group supercharacter theories or resolve the entire unnamed-family source question. Signed, positive-characteristic, ungraded and regraded variants are not covered. Please record this as a scoped obstruction, retaining the unresolved source-level qualification.

The release includes an English PDF, LaTeX source, exact coefficients and reproducible checks. It is AI-assisted, self-audited and unrefereed; finite checks are not formal verification or independent peer review, and no absolute priority certificate is claimed. I remain responsible for the manuscript and welcome specific corrections.

Access note (28 September 2026): the Zenodo record and all files are public and verified. Zenodo has assigned the DOI, but doi.org returned "DOI Not Found" at the release-time check. Please use the direct Zenodo or EulerSolve link if necessary; no activation-time guarantee is implied.

Alper Ferudun — Mercury Software GmbH

AIM-COMPUTATION-0073: strict likelihood modes and septic equations for the closed Birkhoff mixture model

I have released Strict Likelihood Multimodality and Septic Equations for Two-Component Birkhoff Models.

• Paper and files: https://eulersolve.org/papers/aim-computation-0073/
• Public Zenodo record: https://zenodo.org/records/23019398
• Assigned DOI: https://doi.org/10.5281/zenodo.23019398
• Primary source: AIM, Computational algebraic statistics, Problem 58, printed p. 11 (model on p. 3): https://aimath.org/WWN/compalgstat/compalgstat.pdf

An explicit positive dataset of 2013 observations on S_4 has at least three distinct strict local likelihood maxima in the closed two-component Birkhoff statistical model, as probability distributions, not component-label copies. All probability coordinates are positive. An exact nonnegative-decomposition certificate, continuous parameter recovery and compactness establish image-space strictness.

The paper also proves at least three global maximizers for unequal positive parity counts for every n >= 4, and determines the first nonzero homogeneous equation degree of the complex S_4 secant variety to be seven. A 96-term septic and its 24 symmetry images supply local equations at the displayed smooth point.

Scope: one component at each exhibited strict mode is a boundary component; strict modes with both component matrices positive are not established. Isolation of the separate parity global maxima and the global defining ideal/radical are not claimed. The entire source record should therefore not be relabeled fully solved. Classical methods and the upstream dimension-19 computation are credited.

The release includes an English PDF, LaTeX source, exact certificates and reproducible verification scripts. It is an AI-assisted, self-audited, unrefereed preprint, not independent peer review or formal certification; no absolute priority claim is made. I remain responsible for the manuscript and welcome specific corrections.

Access note (28 September 2026): the Zenodo record and files are public and verified. Zenodo has assigned the DOI, but doi.org returned "DOI Not Found" at the release-time check. Please use the direct Zenodo or EulerSolve link if necessary; no activation-time guarantee is implied.

Alper Ferudun — Mercury Software GmbH

AIM-PROBABILITY-0108: scoped polynomial rigidity and a sharp free-covariance bound

I have published Polynomial Rigidity and a Sharp Degree Bound for Free Quadratic Variation (Alper Ferudun, Mercury Software GmbH).

• Zenodo: PDF, LaTeX source and exact checker
• EulerSolve manuscript page
• Assigned DOI: 10.5281/zenodo.23021763. The direct Zenodo link works; DOI.org had not resolved it at the release-time check.

For a nonconstant free polynomial of degree d and any unital moment functional, its free quadratic-variation polynomial has degree exactly 2d-2. The highest coefficients form a positive Gram matrix. This classifies self-adjoint polynomial unit-covariance maps at algebraically free tuples as affine coisometries. For an m-variable standard semicircular tuple, the L2 distance of the covariance from scalars is at least 1/sqrt(m) times the squared L2 norm of the polynomial's highest Wick component. The constant is sharp; in degree two this gives sharp stability relative to affine polynomials.

Scope: this is a completed restricted theorem about prescribed polynomial pathwise changes in the same filtration. It does not solve general terminal-law steering, nor the adjacent entropy and pressure questions fused into this dataset record. Please do not mark the whole source record solved on this basis.

The inherited one-variable obstruction and affine converse, standard free Itô/Fock tools, and adjacent rigidity literature are credited. The 4003 exact checks supplement the all-degree proof; they are not formal verification or independent review. This is an AI-assisted, self-audited, unrefereed preprint; the author remains responsible. A bounded search did not locate the precise degree/bound statements, but elementary rigidity may be folklore and no absolute-priority claim is made.

New preprint relevant to AIM-REPRESENTATION_THEORY-0007:

Alper Ferudun, “Compact-Unit Types and Fixed Vectors for Symplectic Similitudes.”

The paper proves that every smooth character of the compact subgroup H_r = {diag(I_r, a I_r): a in Z_p^times} occurs in every irreducible infinite-dimensional smooth complex representation of GSp_{2r}(Q_p), for every rank r >= 1 and every prime p, including p = 2. Consequently some congruence subgroup R_r(m) from the source problem fixes a nonzero vector.

This addresses AIM Problem 3.2 with fixed rank and varying level, as in Saha's cited subgroup definition; it does not claim m = r or a minimal level. The known rank-two result and compact averaging mechanism of Roberts–Schmidt are explicitly credited. The all-rank argument uses finite character localization and symplectic transvections.

• Zenodo record, PDF, LaTeX and reproducibility archive — DOI: 10.5281/zenodo.23023079
• EulerSolve paper page

Access note: Zenodo has assigned the DOI and the record and files are public. At the release-time check, doi.org returned "DOI Not Found"; the direct Zenodo link above works.

This is an unrefereed, self-audited, AI-assisted preprint, not an independently certified or formally verified result. The exact checker is diagnostic, and the bounded literature search does not exclude folklore or unlocated prior art. I am reporting the manuscript for consideration, not treating publication as dataset acceptance or claiming absolute priority.