Datasets:

---

ulamai

/

UnsolvedMath

Like 80

Follow

ulam.ai 20

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

AMR-022-5039 (Hayman–Lingham Problem 5.39, Duren's problem on integral means of the derivative of a subordinate function): r_p = 1/2 for 0 < p ≤ 2 and two-sided bounds for p > 2; a partial answer, the exact value for p > 2 stays open

I have released an unrefereed preprint, Duren's Problem on Integral Means of the Derivative of a Subordinate Function: the Radius 1/2 for p ≤ 2 and Bounds for p > 2 (Alper Ferudun, Mercury Software GmbH).

• Zenodo (PDF, LaTeX source, the exact programs with their outputs, verification report): https://zenodo.org/records/23252438
• DOI: https://doi.org/10.5281/zenodo.23252438

Question. Problem 5.39 of Hayman and Lingham, Research Problems in Function Theory (posed by P. L. Duren): let g be subordinate to f in the unit disc, g = f∘φ with φ analytic, |φ| < 1, φ(0) = 0. Find the largest number r_p such that M_p(r,g′) ≤ M_p(r,f′) for 0 < r < r_p, where M_p(r,·) is the p-th integral mean on |z| = r. Recorded with the problem: the inequality holds for p = 2 and r ≤ 1/2 (Goluzin), it can fail for r > 1/2 (f = z, g = z²), and r_p ≥ √2 − 1 for every p > 0.

Result (partial; the exact value of r_p for p > 2 is not determined).

• r_p = 1/2 for every 0 < p ≤ 2, in the quantitative form M_p(r,g′) ≤ (α² + 4r²(1 − α²))^{1/2} M_p(r,f′) for r ≤ 1/2, α = |φ′(0)|. For p = 1 this says that the length of the image of the circle |z| = r (counted with multiplicity) does not increase under subordination as long as r ≤ 1/2. The proof is short: Hölder's inequality between Littlewood's subordination theorem and Goluzin's theorem, applied to a power of the zero-free part of f′.
• p ↦ r_p is non-increasing and left-continuous, r_p → 1/2 as p ↓ 2, and r_p → √2 − 1 = r_∞ as p → ∞. Hence r_p = 1/2 exactly for p ≤ p_*, with some p_* ≥ 2.
• Lower bound for p > 2: r_p ≥ r_1(p) > √2 − 1, where r_1(p) is the root in (√2 − 1, 1/2) of 8r⁴ − 16r³ + (p+4)r² + 2pr − p = 0 (for example r_3 ≥ 0.45226 and r_12 ≥ 0.42108). So the bound √2 − 1 recorded with the problem is not best possible for any finite p.
• Upper bounds, computer-assisted: r_p < 1/2 for every p ≥ 12.0068, so 2 ≤ p_* < 12.0068; further r_p < 0.4779 for p ≥ 20, r_p < 0.4372 for p ≥ 100 and r_p < 0.4207 for p ≥ 1000. They rest on 19 explicit pairs (f, φ) that violate the inequality; each is a strict inequality between two rational numbers, certified in exact arithmetic by three independently written programs.
• Numerical evidence only, nothing proved: floating-point experiments suggest r_p = 1/2 up to p ≈ 12.0065 and, beyond that, the critical radius of the family φ(z) = z(z + a)/(1 + az). This is stated as a conjecture.

Credit. The problem is Duren's. Goluzin (1951) proved the case p = 2 with the radius 1/2 (his Theorem 5) and the radius √2 − 1 for all p > 0 (his Theorem 6, which is the note attached to the problem); the device of dividing by the Blaschke product of a smaller disc is from his proof of Littlewood's theorem. The other ingredients are Littlewood's subordination theorem, Rogosinski's coefficient inequality (used in a slightly refined form, without a claim of novelty), and Dieudonné's estimate (1931) of the derivative of a bounded function, which gives |φ′| ≤ 1 for |z| ≤ √2 − 1, the sharp bound beyond and the extremal function behind r_∞ = √2 − 1.

What stays open. The exact value of r_p for every p > 2. For 2 < p < 12.0068 only r_1(p) ≤ r_p ≤ 1/2 is known, and for larger p there is a gap between r_1(p) and the certified upper bounds. The number p_*, the conjecture, and continuity from the right of p ↦ r_p are open as well.

Caveat. The result for 0 < p ≤ 2 has a short proof, built from tools that were available in 1951, and it may be known to experts. The two parts of Goluzin's 1951 paper, which were read in full, do not contain it; Goluzin's book, the books of Duren, Pommerenke and Pavlović, and MathSciNet were not checked.

Suggested status: partially_solved (the dataset currently labels the record open). The number r_p is determined for 0 < p ≤ 2 and bounded from both sides for p > 2; the problem as posed, for all p > 0, is not settled.

Three independent AI-assisted verification runs re-checked the proofs line by line; two of them certified the 19 inequalities again with separately written exact programs. Our literature search (arXiv, Crossref, zbMATH, OpenAlex, web) found no earlier statement of these results; the books named above were not checked. This is AI-assisted, self-audited and unrefereed work, not independent human peer review or formal certification. No absolute-priority claim is made. Corrections and prior-work pointers are welcome. — Alper Ferudun, Mercury Software GmbH

OWR-15427-019: log-free dense-regime bounds for collinear triples

Preprint: Log-Free Supersaturation of Collinear Triples in Three-Dimensional Grids, Alper Ferudun (Mercury Software GmbH).

• DOI: https://doi.org/10.5281/zenodo.23250910
• Paper, source and verification report: https://eulersolve.org/papers/owr-15427-019/

Let F(n,m) be the minimum number of unordered collinear triples in an m-point subset of [n]³. The paper proves, for all integers 12n² ≤ m ≤ n³,

$$\frac{m^4}{62208n^6}\le F(n,m)\le 134784\frac{m^4}{n^6}\mathrm{.}\backslash frac\{m^4\}\{62208n^6\}\backslash le\; F(n,m)\backslash le134784\backslash frac\{m^4\}\{n^6\}.$$

Consequently F(n,⌈n^(3−s)⌉) = Θ(n^(6−4s)) for every fixed 0 ≤ s < 1. The primitive-chain lower-bound argument removes the logarithmic loss in Balogh–Solymosi's supersaturation estimate (https://arxiv.org/abs/1704.05089, Lemma 4.2). The matching upper method is explicitly credited to Roche-Newton's periodic finite-field cap construction (https://arxiv.org/abs/2607.25742v1, Theorem 4 and Section 6); the paper gives a uniform adaptation to arbitrary side lengths and exact cardinalities.

Scope: this determines the order in the dense regime of Question 11.1 in the 2017 Oberwolfach Discrete Geometry problem session (report p. 1200, https://doi.org/10.4171/owr/2017/19). It does not determine exact minima, optimal leading constants or the threshold near m = n². Indeed, F(p,p²) = 0 for odd primes by the known quadratic-cap construction. The separate rigid bipartite unit-distance question in the same dataset record is already answered in prior literature, including Solymosi–White (https://arxiv.org/abs/1808.04005, Theorem 2); it is not a second new result.

This is an AI-assisted, self-audited, unrefereed preprint with a complete written proof of the stated dense-regime theorem. The accompanying exact regression checker passes 68,705 checks in normal and optimized Python with identical output; those finite checks are supplementary, not a proof of all parameters. No independent review, formal proof-assistant verification or absolute priority is claimed. Please retain the distinction between this complete scoped theorem and the broader source record, which remains only partially addressed.

OWR-11127-007: the qualitative certificate is already answered in the cited source

Record 30001839 is still partially_solved at commit 372682f. For the question as currently worded, I suggest solved in the literature / answered in the source, not a new result of ours.

In Kowalski's joint-work report with Zywina, OWR 35/2011, printed pp. 1992-1995, p. 1993 explicitly gives the certificate: the observed G-conjugacy classes must exclude every proper subgroup of G. Equivalently, for each maximal proper subgroup M, some observed class must be disjoint from M. Jordan's derangement theorem ensures that suitable classes exist; Chebotarev ensures eventual occurrence when the Galois injection is onto.

The full paper, Kowalski-Zywina, The Chebotarev Invariant of a Finite Group, Experimental Mathematics 21 (2012), 38-56, gives the criterion in Definition 2.1 and Lemma 2.2 and an expected waiting-time formula in Proposition 2.7. An author-hosted PDF is available.

Scope: this concerns full conjugacy classes and a finite positive certificate, not a uniform bound on the primes inspected, independent Frobenius samples, or arbitrary coarser labels such as cycle types. If a quantitative question was intended, please state that additional requirement explicitly. This is an AI-assisted source check, not a new preprint or priority claim. — Alper Ferudun, Mercury Software GmbH

AMR-044-0001 (Dragović–Radnović, periods of pseudo-integrable billiards for two concentric half-circles): a periodicity criterion and the periods in special cases; a partial answer

I have released an unrefereed preprint, On the Billiard Problem of Dragović and Radnović for Two Concentric Half-Circles: A Periodicity Criterion and the Periods in Special Cases (Alper Ferudun, Mercury Software GmbH).

• Zenodo (PDF, LaTeX source, programs with recorded outputs, verification report): https://zenodo.org/records/23256238
• DOI: https://doi.org/10.5281/zenodo.23256238

Question. V. Dragović and M. Radnović (Arnold Math. J. 1 (2015) 69–73) consider the billiard table bounded by two concentric half-circles on opposite sides of a diameter and by the two segments of this diameter between them, and the trajectories tangent to a fixed concentric circle (the caustic), with rotation numbers ρ1 > ρ2 with respect to the two half-circles. Are these trajectories periodic, how many periodic and non-periodic regions do they form on the boundary, and what are the periods? (The period counts the reflections off the arcs and off the segments.)

Result (a partial answer).

• Reduction: an unfolding turns the billiard into an explicit map of two circles, which is one of McMullen's coupled rotations, with parameters (L1, τ1, L2, τ2; t) = (1, 2ρ1, 1, 2ρ2; 2ρ2). Its first-return map is an exchange of three arcs of a circle, and the period of a trajectory is given by a formula in terms of the reduced orbit.
• Criterion: all trajectories with the given caustic are periodic if and only if ρ1 + ρ2 is rational. Through the reduction this is an application of published theorems, not new mathematics: Boshernitzan's theorem that minimal interval exchange transformations of rank two are uniquely ergodic (1988), or alternatively McMullen's theorems on measured foliations of genus two (2015; 2003 for quadratic irrationals).
• Regions: there are at most two, each symmetric under the mirror symmetry of the table. If ρ1 + ρ2 is rational: one or two periodic regions (two if ρ2 is irrational). If ρ1 + ρ2 is irrational: exactly one non-periodic region, in which every trajectory is dense, and at most one periodic region. The count is again an application of the published decomposition of interval exchanges into periodic and minimal components.
• Periods in special cases. If ρ1 + ρ2 = 1/2: periods 3s − 2 (s even) or 6s − 4 (s odd), where s is N = ⌊1/(2ρ2)⌋ or N + 1. If ρ1 + ρ2 is irrational, the periodic region is decided, with its period, in these cases: none if 1, ρ1, ρ2 are rationally independent or ρ2 is rational; for ρ1 = s/t it exists iff ρ2 < 1/(2t) (t odd) or ρ2 < 1/t (t even), with period t + 2s; for ρ1 + qρ2 ∈ ½Z (q ≥ 2) it exists iff q divides N or N + 1; for pρ1 + ρ2 ∈ ½Z (p ≥ 2) it exists iff pρ1 + ρ2 = p/2 and {1/(2ρ2)} > 1 − 2/p, with period (2p + 1)N − 2 or twice this number. For rational pairs the regions and periods are the cycle data of an explicit permutation (a table for denominators up to 12 is in the package).
• The examples of the source and of the companion paper are recovered: (1/3, 1/4): 12 and 7; (1/4, 1/6): 5 and 6; (1/3, 1/5): 13 and 21; (1/4, 1/√30): 6 and a non-periodic region; ((5−√5)/10, √5/10): 14 and 4; the example of Section 8.2 of the companion paper: no periodic trajectory.

Credit. The problem, its conventions and the examples are due to Dragović and Radnović (the problem article, and "Pseudo-integrable billiards and arithmetic dynamics", J. Mod. Dyn. 8 (2014)). The criterion and the count of regions rest on M. Boshernitzan, "Rank two interval exchange transformations" (1988), and on C. T. McMullen, "Cascades in the dynamics of measured foliations" (2015) and "Teichmüller geodesics of infinite complexity" (2003); Boshernitzan derives his theorem from a criterion of Veech and quotes Keane's minimality theorem. What the note adds, to our knowledge, is the reduction with the dictionary to coupled rotations, the period formula, the mirror symmetry of the regions and the explicit cases.

What stays open. A closed formula for the periods when ρ1 + ρ2 is rational and different from 1/2. The existence and the period of the periodic region when ρ1 + ρ2 is irrational and the generating relation pρ1 + qρ2 = n/2 has p, q ≥ 2 (both answers occur on one and the same relation). Question 3 of the source, on tables with more arcs, is not treated.

Note on the record: its research summary gives the pages of the source as 69–85; the journal and Crossref give 69–73.

Suggested status: partially_solved (the dataset currently labels the record open). The note is a partial answer and supports no more than that.

Three independent AI-assisted verification runs re-checked the proofs line by line, compared the cited theorems with their sources and tested the statements with separately written programs (a direct simulation of the billiard against the exact reduced map). Our literature search (arXiv, Crossref, zbMATH, OpenAlex, web) found no earlier answer to these questions for this table. This is AI-assisted, self-audited and unrefereed work, not independent human peer review or formal certification. No absolute-priority claim is made. Corrections and prior-work pointers are welcome. — Alper Ferudun, Mercury Software GmbH

Follow-up 46 (2026-10-09): one status correction and one literature note, both for AMR records (J. M. Sullivan, F. Morgan (eds.), Open problems in soap bubble geometry, Internat. J. Math. 7 (1996), 833–842, doi; R. Fenn, D. P. Ilyutko, L. H. Kauffman, V. O. Manturov, Unsolved problems in virtual knot theory and combinatorial knot theory, Banach Center Publ. 103 (2014), doi).

• Both items were checked against the current version of the dataset (commit 372682f) and against the source, and the cited statements were read in full in the papers named. Suggested changes are in bold.

Partly settled in the literature (label → partially_solved)

• AMR-058-0019 → partially_solved (the second question is answered: yes; the first stays open). Problem 19 (Heppes) asks (a) whether an optimal planar partition crossed with a short interval is optimal in the slab, and (b) whether the least-area way to divide a very long cylinder or prism into two halves of equal volume is the horizontal slice at mid-height.
  • (b), round cylinder B²(r) × [0, L]: among hypersurfaces that divide a product of Euclidean balls into two equal volumes, the least area is attained by a central section of one factor times the other factors (A. Ros, The isoperimetric problem, Clay Math. Proc. 2 (2005), 175–209, Theorem 23(d) and Figure 17; credited there to F. Barthe and B. Maurey, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), and F. Barthe, J. Funct. Anal. 182 (2001)). The two candidates are the horizontal disc (area πr²) and the axial rectangle (area 2rL), so the slice is optimal exactly when L ≥ πr/2.
  • (b), rectangular prisms: in a box with sides a₁ ≤ … ≤ aₙ the hyperplane that halves the longest side has least area among the hypersurfaces dividing the box into two equal volumes (Ros, Theorem 8; the cube is a theorem of Hadwiger). So the slice is optimal exactly when the height is at least the longer side of the base.
  • (b), general cross-section K: Ros's Theorem 23(a), with the interval as one factor, gives the slice as soon as L ≥ √(2π)/c(K), where c(K) is the Gaussian isoperimetric constant of the normalised measure on K (it is positive for compact manifolds with boundary; see the proof of Theorem 23(b)). This two-line deduction is ours. Compare E. Milman, arXiv:2403.06602 (v3), Corollary 2.15: if the profile of the base, normalised at 1/2, dominates the Gaussian profile normalised at 1/2, the least area of a bisecting hypersurface of the slab is the smaller of the best vertical wall and the slice.
  • None of these sources mentions Problem 19; the link is ours. We found no answer to (a).

Literature note (label unchanged: partially_solved)

• AMR-105-0047 (free knots: connected sums, recognition, commutation). For the third sub-question, an existence statement for long free knots is asserted in the literature without proof: M. W. Chrisman, arXiv:1311.5748, Section 1.1, says that distinct irreducibly odd free long knots do not commute, by a combinatorial argument that is not given there. The natural tool is the bracket for oriented long free knots of V. O. Manturov, arXiv:0909.2230, Section 2. As worded the sentence needs a qualifier: A and A # A are distinct, both have irreducibly odd diagrams when A has one, and they commute. We found no written proof and no statement about general pairs of long free knots; the problem list (2014) cites the preprint and still poses the question. The first two sub-questions are untouched by this note.

These are AI-assisted, self-audited checks, not independent human peer review. Corrections are welcome. — Alper Ferudun, Mercury Software GmbH

OPG-46575 update: version 1.1 (relation to the preprint of Yang and Zeng)

This updates my notice of 29 September on this record (Melnikov's valency-variety problem; preprint A Negative Answer to Melnikov's Valency-Variety Problem).

• Version 1.1 on Zenodo: https://zenodo.org/records/23261539
• DOI of this version: https://doi.org/10.5281/zenodo.23261539 (all versions: https://doi.org/10.5281/zenodo.23034083)

What changed. On 7 October 2026 Z. Yang and Q. Zeng posted the preprint arXiv:2610.09649 on the same problem (it is mentioned in follow-up 45 above). It does not refer to my note, and I regard the two works as independent. Version 1.1 adds one paragraph on how the two are related, one sentence in the abstract and one in the concluding remarks, the reference, and one program:

• They prove the non-strict inequality χ(G) ≥ ⌈⌊w(G)/2⌋/(n − w(G))⌉ for all graphs. My note has it only for graphs with at most 400 vertices and, for each fixed chromatic number, for all large n.
• Their main bound is an explicit form of Theorem 1.3(b) of the note, with the same constant μ_k.
• Their counterexamples G_q have the parameters of the graphs of Theorem 1.1(b) of the note (14q − 5 vertices, chromatic number q). For q = 3 the two 37-vertex graphs have 263 edges and the same degree sequence, but they are not isomorphic: they contain 615 and 599 triangles.
• Not in their preprint: the minimality statements (no counterexample with fewer than 37 vertices; smallest orders 14k − 5 and 22k − 9 for 3 ≤ k ≤ 60), the connected counterexamples, and the violation that grows linearly in n.

The theorems and proofs of the note are unchanged.

Status. Unchanged: solved (no — counterexample).

This is AI-assisted, self-audited and unrefereed work, not independent human peer review or formal certification. Corrections and prior-work pointers are welcome. — Alper Ferudun, Mercury Software GmbH

AIM-ANALYTIC_NUMBER_THEORY-0071: literature correction for the endpoint smoothing input

In the frozen v1.6.0 research report, the estimate for the average of the endpoint remainder is treated as an ICM announcement without a located full proof. A full paper is available: A. Ivić, The mean value of the zeta-function on sigma=1, Ramanujan Journal 26 (2011), 209–227, DOI, Theorem 2 and Section 3.

For the endpoint remainder

$$R(X)={\int }_1^X\mathrm{\mid }\zeta (1+it){\mathrm{\mid }}^{2}dt-\zeta (2)X+\pi \log X,R(X)=\backslash int\_1^X|\backslash zeta(1+it)|^2dt-\backslash zeta(2)X+\backslash pi\backslash log\; X,$$

that theorem proves

$${\int }_1^{X}R(t)dt=BX+O(X\mathrm{/}\log X)\mathrm{.}\backslash int\_1^X\; R(t)dt=BX+O(X/\backslash log\; X).$$

Hence the report's formula (7.13) is unconditional: for a fixed twice continuously differentiable weight with compact support in the positive half-line, two integrations by parts give

$$\int \psi (t\mathrm{/}T)\mathrm{\mid }\zeta (1+it){\mathrm{\mid }}^{2}dt=\zeta (2)T\int \psi (u)du-\pi \int \psi (u)\mathrm{/}udu+O_{\psi }(1\mathrm{/}\log T)\mathrm{.}\backslash int\; \backslash psi(t/T)|\backslash zeta(1+it)|^2dt=\backslash zeta(2)T\backslash int\backslash psi(u)du-\backslash pi\backslash int\backslash psi(u)/u\backslash ,du+O\_\backslash psi(1/\backslash log\; T).$$

The weight vanishes near zero, avoiding the pole. This is a consequence of Ivić's theorem, not a new result of ours. It does not establish the report's simultaneous-limit estimate at σ = 1 − λ/log T, or settle higher moments/the full AIM agenda. Please retain the partial scope; no whole-record solved label is suggested. This is an AI-assisted, self-audited literature check, not a new preprint or priority claim.