Research log — hilbert16-limit-cycles

Papers, OEIS, failed lookups. Cite every URL you opened, including the ones that gave nothing. Forum numbers (MSE, Reddit, MathOverflow, AlphaXiv) are leads, not citations.

2026-08-27

Fetched with python3 scripts/arxiv_fetch.py <id> --text none --research … then opened the abs page (and, for 2604.12883 and 2407.13465, the HTML). Stubs from the fetcher are expanded below.

• Eshkobilov–Kadyrov–Mamayusupov, Limit-Cycle Replication via Chebyshev Pullbacks and a Quadratic Ceiling for Separable Schemes, arXiv:2604.12883v1 (14 Apr 2026). Opened abs and the HTML. Theorem 1: H(nm+m−1) ≥ m² H(n) for m≥2, via Φ(u,v)=(T_m(u),T_m(v)) and the pullback field (u̇,v̇)=(T_m'(v) P∘Φ, T_m'(u) Q∘Φ). Theorem 2: pure separable replication is at most quadratic in the final degree. Table 1 / Appendix A lift published seeds to H(14)≥252, H(29)≥1080, H(31)≥1380, H(39)≥2012. Section 6: explicit cubic with one hyperbolic circle, T_3-pullback of degree 11 with nine lifts. Remark 4 leaves non-separable maps open. Does not claim a new H(2) or H(3). Does not replay Prohens–Torregrosa or Han–Li centers.

• Gasull–Santana, A note on Hilbert 16th Problem, arXiv:2407.13465v2 (1 Oct 2024). Opened abs and the HTML. If H(n)<∞ then it is realized by a structurally stable field with only hyperbolic cycles, and H is strictly increasing. Small-n table in the introduction: H(2)≥4, H(3)≥13, H(4)≥28. Cites Shi; Li–Liu–Yang JDE 246 (2009); Prohens–Torregrosa; Christopher–Lloyd.

• Buzzi–Novaes, A note on a recent attempt to solve the second part of Hilbert's 16th Problem, arXiv:2411.09594v1 (14 Nov 2024). Opened abs. The Entropy 26 (2024) formula H(n)=2(n−1)(4(n−1)−2) is quadratic, hence incompatible with Christopher–Lloyd n² log n. Not a record.

• Gasull–Santana, On a variant of Hilbert's 16th problem, arXiv:2405.04281v3 (26 Sep 2024). Opened abs. Hilbert number as a function of the number of monomials, not of degree. Introduction restates H(2)≥4, H(3)≥13, H(4)≥28.

• Novaes–Pereira, A version of Hilbert's 16th Problem for 3D polynomial vector fields: Counting isolated invariant tori, arXiv:2212.12006v2 (25 Sep 2024). Opened abs. Quotes the Prohens–Torregrosa table: H(2)≥4, H(3)≥13, H(4)≥28, H(5)≥37, H(6)≥53, H(7)≥74, H(8)≥96, H(9)≥120, H(10)≥142, and the 4-fold lifts H(13)≥212, H(17)≥384, H(21)≥568, H(31)≥1184, H(35)≥1536, H(39)≥1920, H(43)≥2272. 3D tori, not a planar H(n) improvement.

• Carvalho–Cruz–Gouveia, New lower bound for the Hilbert number in low degree Kolmogorov systems, arXiv:2304.05111v1 (11 Apr 2023). Opened abs. Kolmogorov local cyclicity M_K(3)≥6, M_K(4)≥13, M_K(5)≥22. Restates H(2)≥4, H(3)≥13, H(4)≥28 and M(2)=3, M(3)≥12, M(4)≥21, M(5)≥33. Different function from H(n).

• Gasull–Santana, Limit cycles and invariant algebraic curves, arXiv:2510.11705v2 (22 Dec 2025). Opened abs. Lower bounds given a prescribed invariant algebraic curve; a recurrence for H(n); Kolmogorov and game-theory families. Does not replace Li–Liu–Yang 13 or Prohens–Torregrosa 28.

• Yang, The cyclicity of period annulus of cubic isochronous Hamiltonian systems, arXiv:2512.19046v1 (22 Dec 2025). Opened abs. Weak Hilbert 16th for a cubic isochronous Hamiltonian: cyclicity of the period annulus is n−1 and is attained. Restates Li–Liu–Yang 13. Not a new H(3).

Journal / PDF pages opened (not arXiv ids):

• Prohens–Torregrosa, Nonlinearity 32 (2019), UAB copy. Theorem 1: H(4)≥28, H(5)≥37, H(6)≥53, H(7)≥74, H(8)≥96, H(9)≥120, H(10)≥142 with configurations ⟨8,12,8⟩, …, ⟨45,52,45⟩. Corollary 2: the 4-fold lifts listed above.

• Shi Songling, Sci. Sinica 23 (1980) landing page. Concrete quadratic with four limit cycles; English version pp. 153–158. Coefficients taken from later expositions, not from a scanned PDF of Shi (no open PDF on that DOI page).

• Yu–Zhang, visualization preprint. Writes Shi as ẋ=λx−y−10x²+(5+δ)xy+y², ẏ=x+x²+(−25+8ε−9δ)xy with Shi's parameters λ=−10^{−250}, ε=−10^{−52}, δ=−10^{−13}. Chen–Wang is a different quadratic with a order-2 fine focus.

• Galias–Tucker–Wilczak, AMC 2022 preprint. Interval-arithmetic proof that this one Songling system has exactly four cycles (parameters δ=−10^{−13}, ε=−10^{−52}, λ=−10^{−200}). Not a bound on H(2).

• Han–Li, JDE 252 (2012) ScienceDirect landing. Abstract: H(m) grows at least as (m+2)² log(m+2)/(2 log 2); improves existing lower bounds for all m≥7 from seeds in degrees 3–6. Full table not extracted from the paywall HTML; numbers used later are those quoted by 2604.12883 Appendix A and 2212.12006.

• Li–Liu–Yang, JDE 246 (2009) ScienceDirect landing. Abstract: a planar cubic with at least 13 cycles, via zeros of Abelian integrals. No open PDF.

• Giné–Gouveia–Torregrosa slides. M(2)=3; M(3)≥12; M(4)≥21; M(5)≥33; M(6)≥48. Local, not H(3).

• PlanetMath, Hilbert 16th for quadratic vector fields. Lead only: writes the Shi system and 0<−λ≪−ε≪−δ≪1. Not a citation.

• Eshkobilov–Kadyrov–Mamayusupov HTML. Theorem 1, Theorem 2, §6 cubic, Remark 4, Appendix A seed table. Same claims as the abs fetch; this is the page the Chebyshev replay was read from.

• Llibre–Valls, Global asymptotic stability in quadratic systems, EJDE 2025 no. 36 abstract. Opened as the source line E used for the w1 = Aα − Bβ cross-check of L1. The paper characterises globally asymptotically stable quadratics; it does not move H(n).

Failed / not used as record (q1):

• Llibre–Schlomiuk and Coppel were not opened in q1 (opened in the q2 section below).
• No OEIS lookup (H(n) is not an OEIS sequence we needed).
• Wikipedia and MathOverflow numbers were not opened as citations.
• Piecewise paper arXiv:1809.03433 (H_p(2)≥16) is a different function; abs not used as an H(n) bound.
• Entropy 26 (2024) quadratic formula: not fetched; the refutation is 2411.09594.
• Li–Liu–Yang JDE 246 (2009) full text: paywalled in q1. Their 13 is cited from later papers’ introductions, not replayed.

2026-08-27 (q2)

Papers q1 left closed, plus later 2025–2026 arXiv. Every URL below was opened this session.

Previously closed, now opened

• Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966) 293–304. Opened the PDF (Columbia copy). Lemma: three finite critical points of a quadratic are never collinear; a line meets the field in at most two contacts-or-equilibria. Theorem 1: the interior of a closed path is convex. Theorem 2: a unique critical point in the interior of each closed path. Theorems 3–4: two closed paths are oppositely (resp. similarly) oriented according as their interiors are disjoint (resp. nested). Theorem 5: two foci/centers are oppositely oriented, hence at most two. Theorem 6: the interior point is a focus or a center. Theorem 7: four critical points, convex quadrilateral, opposite saddles. Does not prove H(2) finite. Does not give an explicit 4-cycle field.

• Llibre–Schlomiuk, The geometry of quadratic differential systems with a weak focus of third order, Canad. J. Math. 56 (2004) 310–343. Opened the Cambridge PDF (34 pages). Class QW3: 18 topological portraits. A neighborhood of those QW3 systems that have graphics but no polycycle and no limit cycle has at most four limit cycles. Quotes: a quadratic cycle surrounds a unique singularity; if two foci carry cycles, one nest has at most one; no cycle surrounds the unperturbed order-3 weak focus. Does not move H(2)≥4. Does not produce a fifth cycle.

• Prohens–Torregrosa, Nonlinearity 32 (2019), UAB PDF. Re-opened the full preprint. Theorem 1 and Corollary 2 as in q1. Proposition 6: the explicit rational first integral H = (2x⁴ − x² + y² − 2x − 2)⁵ / (8x⁵ − 5x³ + 5xy² − 10x² − 5x − 4)⁴ is a degree-4 Darboux center at (0,0) and (1, ±2); degree-4 perturbations give configurations ⟨6,10,6⟩, ⟨8,11,8⟩, ⟨8,12,8⟩ at Lyapunov orders 1, 3, 5. Cubic examples (14), (17)–(20) with polynomial first integrals; they did not beat H(3)≥13. The 28-cycle claim is a Lyapunov-budget argument, not an explicit perturbed field written term-by-term.

• Li–Liu–Yang, JDE 246 (2009). Re-tried the landing page, Unpaywall (is_oa bronze, PDF URL is the Elsevier /pdf endpoint), and four Elsevier PDF URLs. All 403. Unpaywall API reports no repository copy. Full text still not fetched. Their 13 is still not replayed.

• Scholarpedia, Limit cycles of planar polynomial vector fields (Han–Li–Li, 2010). Opened the article. Lead, not a citation. Catalogue: Ye class I has at most one cycle; a quadratic with a straight-line solution has at most one; Zhang: (2,2) is impossible; Liénard numbers H(2,1)=H(1,2)=H(2,2)=H(1,3)=1, H(3,1) conjectured 1, H(2,3)≥5; weak Hilbert Z(3,2)=2 generic / 3 degenerate; Z(2,n)=⌊(n−1)/2⌋.

Later 2025–2026 arXiv (and two older ones q1 skipped)

• Chen–Dai–Kaloshin–Li, Cyclicity of lips and center-focus problem near infinity, arXiv:2608.17773v1. Opened abs and HTML. Theorem 3: Liénard Hilbert number H(2n+1, 5) ≥ B(n) = 2n + ⌊n/3⌋ + ⌊(n+1)/3⌋ − 2 for n≥2, from infinity cyclicity plus a two-cuspidal lips. Beats Han–Romanovski for n≥7 and a later +2 bound for n≥13; weaker than some fixed-degree tables at small n. C(3)=+∞ for a typical three-parameter lips family (no uniform cyclicity). Necessary and sufficient condition for a polynomial Liénard center at infinity. Different function from H(n). Not a new planar H(n).

• Lu, Local Uniform Finite Cyclicity of the H₁₄³ Semihyperbolic Hemicycle, arXiv:2607.13785v3 (26 Aug 2026). Opened abs. Completes one labelled open case in the quadratic finite-cyclicity program: a uniform bound in a two-sided annular neighborhood of that graphic. Bound existential, not claimed sharp. Not an H(2) number and not an explicit field.

• Gavrilov–He–Xiao, On the cyclicity of the period annulus of quasi-homogeneous polynomial vector fields, arXiv:2606.22137v1. Opened abs and HTML. Upper bound on zeros of the k-th Melnikov function for a one-parameter polynomial perturbation of a quasi-homogeneous center, in terms of k, max(s₁,s₂), and perturbation degree n. Completely solves one named application. Unperturbed quasi-homogeneous centers have a period annulus, hence no isolated cycles. Not a new H(n).

• Gasull–Santana, Limit cycles and invariant algebraic curves, arXiv:2510.11705v2. Re-opened HTML. Theorem A: H_C(n+c) ≥ H(n) + O(C) for a prescribed non-degenerate invariant curve C of degree c. Corollary 2: H(n+m) ≥ H(n) + Har(m) with Harnack number Har(m) = (m−1)(m−2)/2 + [1+(−1)^m]/2. Weaker than H(n+1)≥H(n)+1 when m=1 (Har(1)=0). Corollary 3: Kolmogorov H_K(n) ≥ H(n−1), and they print H_K(5)≥28 (uses the unreplayed H(4)≥28 seed; beats Carvalho–Cruz–Gouveia 22 as a Kolmogorov number, not as H(5)). Corollary 4: game-theory square H_□(n) ≥ H(n−2). Does not beat Li–Liu–Yang 13 or Prohens–Torregrosa 28 as planar H(n).

• Ascoli–Novaes, arXiv:2601.21865v1. Opened abs. Crossing Hilbert number H_c(n) for piecewise polynomial fields; H_c(n) grows at least as n²/4. Different function.

• Cruz–Oliveira–Torregrosa, arXiv:2509.06198v1. Opened abs. Piecewise quadratic Kolmogorov: at least 6 crossing cycles. Smooth quadratic Kolmogorov have none (they say so). Different function.

• Marín–Villadelprat, The cyclicity of hyperbolic hemicycles, arXiv:2501.16924v1. Opened abs. Cyclicity of an unbounded hemicycle graphic; for quadratic class Q₃^R, cyclicity of two hemicycles and an alien-cycle statement. Not a new H(2).

• Yeung, arXiv:2402.12506v1. Opened abs. Claims a gap in Ilyashenko 1991 Dulac asymptotics. Context for per-system finiteness, not a table number.

• Ilyashenko–Llibre, arXiv:0910.3443v1. Opened abs and PDF. Restricted upper bound for quadratic fields σ-distant from centers and κ-distant from singular fields, counting only cycles δ-distant from equilibria and infinity. Bound depends on (σ,κ,δ); not a uniform H(2). Quotes Zhang: only one quadratic nest can have more than one cycle.

• Binyamini–Novikov–Yakovenko, arXiv:0808.2952v3. Opened abs. Tangential Hilbert 16th: Abelian-integral zeros bounded by a double exponential in the degree. That is Z(m,n), not H(n).

Failed / not used as record (q2):

• Li–Liu–Yang full text: still 403 after Unpaywall and four Elsevier URLs. No author-page PDF found.
• Chen–Wang 1979 Acta Math. Sinica 22: Chinese journal; Chen Lansun’s index lists the title; no open PDF opened.
• Yu–Han “four limit cycles in quadratic near-integrable systems” was cited as arXiv:1002.1005; that id is a 2010 software-engineering paper (Waignier et al.). Wrong id; not a Hilbert paper.
• ddd.uab.cat and Scholarpedia live fetches timed out once each; the Prohens–Torregrosa PDF succeeded on retry, Scholarpedia was already read from the first search-tool fetch.
• Llibre’s mat.uab.cat/~jllibre/ homepage returned 164 bytes, not a publication list. The 2004 PDF was the Cambridge file above.
• Piecewise 2025–2026 papers are a different Hilbert number; not used as H(n).
• No OEIS lookup.

2026-08-27 (q3)

No new published H(n). Li–Liu–Yang full text still 403. Papers opened or re-opened while scoring the ten ideas:

• Gavrilov–He–Xiao, arXiv:2606.22137v1. Re-opened abs and HTML for line HH. Their §5 application is the period annulus of a constant multiple of H = x⁴ + 4y² (after rescaling y). They bound every Melnikov order for one-parameter polynomial perturbations of a quasi-homogeneous center and get cyclicity 1 for a linear unfolding. Not a new H(3). Not beaten here.

• Eshkobilov–Kadyrov–Mamayusupov, arXiv:2604.12883v1. Re-opened abs and HTML for line II (Remark 4, T3 degree budget N = 3n+2). Same claims as q1.

• Scholarpedia, Han–Li–Li 2010. Re-opened for line JJ. Z(2,n) = floor((n−1)/2) because M(h) is a polynomial. Lead for the formula; the attaining family is replayed in compute/q3/jj-weak-hilbert/.

• Villanueva–Tucker, Center conditions and cyclicity for generic planar polynomial vector fields, arXiv:2602.22558v1. Opened abs and HTML. Generic (residual) upper bounds on small-amplitude cyclicity: M_h^(n) ≤ n+2 for homogeneous nonlinearities, and a quadratic-in-n bound for generic non-homogeneous fields. They note 3 = M(2) < M^(2) = 4, so the generic bound is not Bautin. Not a uniform H(n). Not a new lower bound.

• Gasull–Santana, A note on Hilbert 16th problem, PAMS 153 (2025) 669–677. Opened the UAB postprint. Same two theorems as arXiv:2407.13465v2 (already in q1): H(n+1) ≥ H(n)+1, and if H(n) is finite it is attained by a structurally stable field with only hyperbolic cycles. Table still H(2)≥4, H(3)≥13, H(4)≥28.

Failed / not used as record (q3):

• Li–Liu–Yang JDE 246 (2009) full text: still 403. Their 13 is still not replayed.
• Piecewise crossing numbers (arXiv:2601.21865) remain a different function.
• No OEIS lookup.

2026-08-27 (q3 extra)

No published planar H(n) moved. Li–Liu–Yang full text still 403; their 13 is still not replayed. No 2025–2026 paper claiming H(3)≥14 was found. Fetched with python3 scripts/arxiv_fetch.py <id> --text none --research … then opened abs/HTML (or the author PDF) and expanded the stubs. Every URL below was opened this session. Already-logged ids were not re-entered as new.

Li–Liu–Yang JDE 246 (2009) retry

• ScienceDirect landing, Li–Liu–Yang JDE 246. HTTP 403 (Cloudflare challenge). The search-tool snippet of that landing still only gives the abstract: a planar cubic with at least 13 cycles, via zeros of Abelian integrals. No article body.
• DOI 10.1016/j.jde.2009.01.038. 302 to linkinghub (200, 2.6 KB interstitials). Not the article.
• Elsevier PDF endpoints all 403: /pdf, /pdfft, and the md5=c114a9579f640438744a304af416cef8 /pdf URL from the landing snippet.
• Elsevier API PII S0022039609000643. 200 with empty body; WARNING - Unauthorized request results in minimized metadata response.
• Unpaywall API timed out (25 s, then 60 s, 0 bytes). unpaywall.org/10.1016/j.jde.2009.01.038 302 to oadoi.org; oadoi.org returned no usable PDF this session.
• Semantic Scholar API, DOI 10.1016/j.jde.2009.01.038. openAccessPdf.status is BRONZE; the PDF URL is the same Elsevier /pdf endpoint (403). S2 HTML page is a JS captcha, not a paper.
• ADS abs 2009JDE...246.3609L. AWS WAF captcha. Full-text source listed as Elsevier only.
• Changjian Liu, SYSU Zhuhai. Publication list; no PDF of the 2009 JDE paper (the visible items skip that year). No author copy.
• Chengzhi Li, Abelian Integrals and Limit Cycles, QTDS 11 (2012) 111–128 landing. Abstract and reference list only (no article PDF). Cites Li–Liu–Yang 2009 as the cubic with thirteen cycles. Survey of the weak problem; does not replay the 13 and does not move H(n).
• Pei Yu UWO homepage and publication list. Opened. The /pub/ directory itself is 403. Yu hosts later cubics (below); he does not host Li–Liu–Yang.

Status: full text still not fetched. Do not treat 13 as replayed.

Christopher–Lloyd 1995 (the 4-fold paper)

• DOI 10.1098/rspa.1995.0081. 302 to the Royal Society article page, then 403 (Cloudflare). RS landing and RS PDF both 403. Abstract on the DOI/search page: H_n grows at least as n² log n (it was already known to be ≳ kn²).
• Semantic Scholar API, same DOI. openAccessPdf.status CLOSED; empty PDF URL. No author-page or institutional PDF found (Plymouth/Aberystwyth searches returned no file).
• Li–Llibre–Pérez–Torregrosa 2001, UAB PDF. Opened the PDF. This is not Christopher–Lloyd 1995. It quotes their bound as H_m ≥ ½(m+1)²(log₂(m+1)−3)+3m. Weakened 16th (Abelian zeros), not a new planar H(n).
• The (x,y)↦(x²−A,y²−A) 4-fold recipe is stated in already-logged arXiv:2601.21865 HTML (re-opened by search this session; not re-logged as new): degree doubles, four copies of the cycles, plus new axis centers, recurrence S_{k+1}=4S_k+(2^k−1)² on degrees 2^k−1. That is a secondary account. The 1995 PDF itself was not read.

Han–Yang–Yu five Abelian / Hopf zeros

• Han–Yang–Yu, Hopf bifurcations for near-Hamiltonian systems, IJBC 19 (2009) 4117–4130, Slideblast extract. Opened. This is Scholarpedia’s [46]. Algorithm for the coefficients b_ℓ of M(h)=h ∑ b_ℓ h^ℓ near an elementary Hamiltonian center. Theorem 4: a quadratic Hamiltonian with cubic perturbation, after using the Hamiltonian coefficients as extra parameters, has Hopf cyclicity 5 at the origin (explicit δ₀, σ₀ with b^=(2/13)√26, c^=2b^*). Theorem 5: the a=0 slice has cyclicity 3. This is five small cycles from one quadratic center, not H(3)≥14 and not a global 13. Does not move the table.
• World Scientific abs 10.1142/S0218127409025250. Cloudflare interstitial; no PDF.
• Semantic Scholar API, same DOI. CLOSED.
• Yu-hosted HYY PDF. 404 (the publication list still points here). The Slideblast extract is the text that was actually read.
• Tian–Han, Hopf and homoclinic bifurcations for near-Hamiltonian systems, JDE 262 (2017) ScienceDirect landing. Cites Han–Yang–Yu as the Hopf-coefficient method. Homoclinic expansion at h=h_s; not a new H(n).
• Yang–Han–Li–Yu, Existence conditions of thirteen limit cycles in a cubic system, IJBC 20 (2010), Yu PDF. Opened the 9-page PDF. Theorem 1: a concrete (k,b,δ) condition (they take k=100, b=7/2) under which their cubic (2) has 13 cycles. They write Li–Liu–Yang (1) as ẋ=y(y²−k²), ẏ=x(x+1)(x−λ)+εy(α₁+α₂x+α₃x²+α₄y²) and say LLY need k sufficiently large; the 13-cycle distribution is the same as LLY Fig. 1 (five centers, double homoclinics, one surrounding nest). Still 13, not 14. This is a second 13, not a replay of the 2009 JDE text.

Ye class I / invariant-line cubics

• Artés–Chen–Ferrer–Jia, Quadratic vector fields in class I, UAB preprint. Opened the PDF (Dyn. Syst. 40 (2025) 191–222). Ye class I is ẋ=y, ẏ=−x+fy+ℓx²+2mxy+ny². Theorem 1: 49 topological portraits, 261 parameter regions; seven portraits have one limit cycle (V5, V6, V13, V31, 1S5, 5S5, 9S3). They quote Rychkov: class I has at most one cycle, and they use that to rule out a double-cycle bubble. Does not produce two class-I cycles. Does not move H(2).
• DOI 10.1080/14689367.2024.2436223. Same paper; landing/abstract only in the search fetch.
• Gasull–Giacomini, arXiv:2210.15803v1. Opened abs and the PDF. Quadratic with an invariant line: at most one cycle (they take this as known). Invariant parabola reduces to the line case (Theorem 1.1). Eight known algebraic-cycle quadratics have no extra cycle (Theorem 1.2). Two generalized invariant curves: at most 2M+2 cycles (Theorem 1.3). Section 7 treats some cubics with a quadratic or cubic invariant curve; those are uniqueness/non-existence statements, not H(3)≥14. Does not move the table.
• Kooij, Cubic systems with four real line invariants, Math. Proc. Cambridge Philos. Soc. 118 (1995). DOI/search landing only. Semantic Scholar CLOSED. Cambridge /content/view/…pdf redirected to a product page, no PDF. Christopher Uniqueness of limit cycles in polynomial systems with algebraic invariants was opened via search: a cubic with four real line invariants has at most one (hyperbolic) cycle. Uniqueness, not a lower bound.

New 2025–2026 papers (not already in this file)

• Yang–Zhang, Limit cycles appearing from the perturbation of a cubic isochronous center, arXiv:2503.09014v1. Opened abs and HTML. Pleshkan S₂^* under degree-n perturbation: if the Abelian integral is not ≡0 then at most 4⌊(n+1)/2⌋+1 cycles from that period annulus (Theorem 1.1). S₃^, S₄^ left open. Weak Hilbert for one isochronous cubic, not H(3). For n=3 the bound is 9, below Li–Liu–Yang 13. Table unchanged.

• Yang–Zhang, Bifurcation of limit cycles from a cubic reversible isochrone, arXiv:2503.09004v1. Opened abs and HTML. CR₅ under Liénard perturbation of degree n: at most 22n+6 from the annulus; they exhibit 1 cycle for n=1,2 and 2 for n=3. Journal version QTDS 2026 DOI landing states the same numbers. Not a planar H(n) seed.

• Fazoli–Santana, Planar vector fields without invariant algebraic curves, arXiv:2511.13596v1. Opened abs and HTML. Theorems A–C: generically (residual, full measure) a degree-n field has no invariant algebraic curve other than a prescribed nodal F; Kolmogorov fields generically have only the axes. Theorem D: if H(n)<∞ it is attained by a structurally stable simple field without algebraic limit cycles (they restate H(2)≥4, H(3)≥13, H(4)≥28). Refines Gasull–Santana 2407.13465; does not raise any H(n).

• Arakaki–Novaes, arXiv:2504.13832v1. Opened abs. 3D normally hyperbolic limit tori: N_h(m+1)≥N_h(m)+1 if that number is finite. Analog of Gasull–Santana increase, not planar H(n).

• Arakaki–Gouveia–Novaes, arXiv:2507.17932v1. Opened abs. N_h(2)≥3, N_h(3)≥5, N_h(4)≥7, N_h(5)≥13, then a Christopher–Lloyd-style lift in 3D. Introduction restates the planar table H(2)≥4, H(3)≥13, H(4)≥28, …. Different function.

• Sui–Zhang–Li, arXiv:2504.01661v1. Opened abs. Piecewise-cubic perturbations of the unique weight-degree-2 quasi-homogeneous center; switching line x=0 or y=0. Different Hilbert number (crossing / non-smooth).

H(3)≥14 search

• arXiv API queries all:"fourteen limit cycles", all:"H(3)" AND all:cubic AND all:"limit cycles", and all:"thirteen limit cycles" for 2025–2026 returned no new planar paper.
• Wang–Mao–Jing, Fourteen limit cycles in a cubic Hamiltonian system with nine-order perturbed term, Chaos Solitons Fractals landing. Opened via search. The perturbation is degree 8 (n=8 in εx(mx^n+ℓy^n−λ)); not a cubic field. Not H(3)≥14.
• 2025–2026 introductions that we opened (2511.13596, 2507.17932, 2503.09014) still print H(3)≥13.

Failed / not used as record (q3 extra):

• Li–Liu–Yang full text: still 403 after ScienceDirect, four Elsevier PDF URLs, Unpaywall/oadoi timeout, Semantic Scholar bronze→Elsevier, ADS captcha, Liu SYSU page, Li 2012 survey landing, Yu’s site. 13 not replayed.
• Christopher–Lloyd 1995 PDF: Royal Society 403, S2 CLOSED, no author PDF. The 4-fold map was not read from the 1995 paper.
• Han–Yang–Yu official PDF: World Scientific Cloudflare; Yu HYY_IJBC2009.pdf 404. Text read from the Slideblast extract and from YHLY 2010.
• Kooij 1995 Cambridge PDF: not obtained.
• Piecewise 2025–2026 (2504.01661 and the Pleshkan-piecewise hits 2605.05010 / 2604.19483 from the API sweep) remain a different function; 2605.05010 and 2604.19483 abs pages were not opened.
• No OEIS lookup.