{"id":"aa4d83c9-464f-44cd-95fb-d751c9f5a4ee","arxiv_id":"2607.16171","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A homogeneous cubic planar vector field, globally asymptotically stable, admits no homogeneous polynomial Lyapunov function and no local real-analytic one, disproving Ahmadi's conjecture.","lead":"This paper exhibits a simple cubic planar equation whose origin attracts every trajectory, yet no polynomial — indeed no real-analytic — energy function can certify that stability, refuting a conjecture open since 2011. It matters because it closes a clean route to deciding stability algorithmically and leaves a reusable Fourier inequality for proving when polynomial certificates cannot exist.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma III.2's printed proof is invalid for complex Fourier coefficients: B#(z)=z^L B(1/z) roots are reciprocals, not reciprocal conjugates, so A=κBB# fails; the lemma and bound survive a one-line correction.","rationale":"The reader's weakest assumption correctly identifies Lemma III.3's strict Fourier bound as the load-bearing technical estimate. My concern sits one level below: the printed proof of Lemma III.2, which supplies the factorization used in Lemma III.3, contains a false root-pairing statement for complex Fourier coefficients. The standard Fejér–Riesz theorem is cited and the intended proof is easily repaired, so the bound itself and the main contradiction remain valid. Hence my read does not change the ACCEPT verdict, but it does require a correction to the manuscript. I chose 'partial' agreement because the reader pointed to the lemma's conclusion while I point to a flaw in the proof of the lemma that the reader's verification did not catch.","tokens_in":8977,"tokens_out":24253,"duration_ms":171747,"concrete_test":"Take q(φ)=1+0.5 sinφ. Compute A(z)=z(1−(−i/4)z+(i/4)z^{−1}) = z+(i/4)−(i/4)z², whose outside root is β≈−3.732i. With the printed B#(z)=z B(1/z)=1−βz, check whether A(z)=κ(z−β)(1−βz); it fails because the second root would be 1/β≈0.268i rather than the true inside root 1/\\bar β≈−0.268i. Then replace B# by z \\overline{B(1/\\bar z)} and verify A(z)=κ(z−β)(1−\\bar β z), which establishes the claimed factorization. This also confirms Lemma III.3's |ĝ₂|<1 for this example, showing the defect is typographical and does not affect the main theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma III.2, for a real-valued trigonometric polynomial q, A(z)=z^L Σ c_j z^j satisfies A(z)=z^{2L} \\overline{A(1/\\bar z)} because c_{−j}=c̄_j. Hence roots pair as ζ ↔ 1/\\bar ζ, not ζ ↔ 1/ζ as printed. The proof then defines B#(z)=z^L B(1/z), whose roots are 1/β, whereas the actual paired inside roots are 1/\\bar β. Consequently A=κBB# is false for generic complex Fourier data. Example: q(φ)=1+0.5 sinφ (from P=x²+xy+y²) has A(z)=z+(i/4)−(i/4)z² with roots i(−2±√3); the printed BB# would pair β with 1/β, but the true inside root is 1/\\bar β. Thus the 'elementary proof' of Fejér–Riesz factorization is wrong as written. This matters because Lemma III.3's bound |ĝ₂|<1 is derived from the factorized form (15). However, Lemma III.2 is a standard cited theorem [15], and replacing B# by z^L \\overline{B(1/\\bar z)} repairs the proof; the factorization Q=C∏(1−α_ℓ z), |α_ℓ|<1, and hence the strict bound, are correct. The central contradiction in §III-C is not threatened, but the manuscript's self-contained proof has a genuine gap that must be corrected.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper disproves the homogeneous polynomial Lyapunov converse conjecture by constructing a planar cubic homogeneous polynomial vector field (2)–(3) with integer coefficients. The origin is globally asymptotically stable, and the explicit 2-homogeneous function H in (4) satisfies L_fH = −2(x²+y²)H < 0, with H ∈ C¹(R²) ∩ C∞(R²\\{0}). In polar form the system is ṁ = r³(−1+5cos2θ), θ̇ = r². The authors show that no positive definite homogeneous polynomial P satisfies L_fP ≤ 0: via Fejér–Riesz factorization and a logarithmic Fourier-coefficient bound |ĝ₂| < 1, the weak Lyapunov inequality forces |ŵ₂| ≤ 1 while ŵ₂ + ĝ₂ = −5/2, giving 5/2 ≤ |ŵ₂| + |ĝ₂| < 2. A leading-term argument extends the obstruction to local real-analytic Lyapunov functions. A two-parameter family (27)–(28) provides an open set of counterexamples, and a rational Lyapunov function is exhibited. A Lean 4 formalization is also claimed.","tokens_in":9165,"tokens_out":7601,"duration_ms":58954,"significance":"If the main result is correct, it settles a long-standing open question in a surprising direction: the homogeneous-polynomial converse Lyapunov theorem fails even for planar homogeneous cubics. The example is remarkably simple (integer coefficients), and the method is elegant and degree-independent. The paper also supplies a machine-checked Lean 4 formalization, which is a substantial contribution to trustworthiness, as is the explicit rational Lyapunov certificate. These features make the result significant to the systems and control community and to algebraic/geometric stability theory.","major_comments":[{"comment":"The displayed identity (13), A(z)=z^{2L}A(1/z), is false for general real-valued trigonometric polynomials because c_{−j}=\\bar c_j does not imply c_j=c_{−j}. The correct symmetry is A(z)=z^{2L}\\overline{A(1/\\bar z)}, so roots pair as ζ ↔ 1/\\barζ, not ζ ↔ 1/ζ. Consequently the defined B^#(z)=z^L B(1/z) has reciprocal (not reciprocal-conjugate) roots, and the equality A=κ BB^# is false in general; e.g. q(φ)=1+0.5 sinφ. This invalidates the printed proof of the factorization. Since Lemma III.3's bound (14) rests on the factorized form (15), and Lemma III.3 is the load-bearing estimate in the contradiction of §III-C, the proof must be corrected. The standard Fejér–Riesz theorem is correct, and replacing B^# with \\tilde B(z)=z^L\\overline{B(1/\\bar z)} gives a valid proof; but as printed the paper's self-contained argument is incomplete. This is a major, though locally repairable, gap.","section":"III-B, Lemma III.2 and Eq. (13)"}],"minor_comments":[{"comment":"“Degree-two homogeneous Lyapunov function” may be misread as a polynomial of degree two; H is 2-homogeneous but not polynomial. Suggest “2-homogeneous” or “non-polynomial 2-homogeneous.”","section":"Abstract"},{"comment":"In the displayed necessary condition |b| ≤ 2(a+1), it may be worth noting explicitly that this condition is necessary only and is not claimed sufficient; the surrounding text already implies this, but a short clarifying sentence would help.","section":"IV-D"},{"comment":"The proof that the lowest Taylor term P_m is positive definite relies on F vanishing on an interval; the argument is correct but somewhat terse. A brief justification that F′≤0 and F≥0 with F(θ₀)=0 forces F≡0 would improve readability.","section":"III-D"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is likely publishable after the proof of Lemma III.2 is corrected. The result and the Lean formalization make it valuable. I recommend major revision for the stated reason; with the correction, acceptance would be appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is that this paper gives a genuine counterexample to the homogeneous polynomial Lyapunov converse conjecture: a planar cubic homogeneous field with integer coefficients, globally asymptotically stable, with a C^1 exponential homogeneous Lyapunov function, yet no homogeneous polynomial—or even local real-analytic—Lyapunov function. If it holds up, it settles a question open since Ahmadi's 2011 thesis and the 2012 ACC paper. The polar reduction is clean, the H function is explicit and verifiable, and the Fourier-coefficient contradiction in Section III-C is genuinely clever. I checked the key expansions (6)-(7), the identity Hdot/H = -2r^2, and the leading-term argument in III-D; those all work. The rational Lyapunov function in IV-C and the parametric robustness in IV-D are nice extras. The Lean 4 formalization is claimed, but no commit hash or extraction details are in the manuscript, so it is not independently checkable from the text.\n\nThe main soft spot is real. The stress-test note is correct: in Lemma III.2 the proof defines B#(z)=z^L B(1/z) and claims roots pair as beta and 1/beta, but for real-valued q with complex Fourier coefficients the correct pairing is beta and 1/conjugate(beta). The printed factorization A = kappa B B# is therefore false for generic q. The lemma itself is a standard Fejer-Riesz result, and the proof is repairable by replacing B# with z^L conjugate(B(1/conjugate(z))) (or by citing a textbook), and the strict bound |g_2|<1 survives. But as written, the self-contained proof has a gap. A referee must insist on this correction. Everything downstream of the lemma—the contradiction in III-C and the analytic obstruction in III-D—is built on the corrected version, so the central theorem is not threatened, but the manuscript is not fully sound in its current form.\n\nThe other caveats are minor: the literature claim that the conjecture was open rests on the authors' reading of [6],[8],[9], which the reader checked; and the Lean formalization needs a stable URL or commit. The paper is worth serious review and, once the lemma proof is repaired, it should be accepted. I would bring it to a reading group and cite it once the corrected version appears.","headline":"A strong paper that likely kills the homogeneous polynomial Lyapunov converse conjecture, but the printed proof of Lemma III.2 has a real reciprocal-vs-conjugate gap that must be fixed.","tokens_in":9857,"tokens_out":1710,"would_cite":true,"duration_ms":14411,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34D20","37C75","93D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper gives a planar homogeneous cubic vector field that is globally asymptotically stable but has no polynomial — and indeed no local real-analytic — Lyapunov function, disproving a converse conjecture.","keywords":["Lyapunov functions","homogeneous polynomial vector fields","global asymptotic stability","converse Lyapunov conjecture","positive trigonometric polynomials","Fejér–Riesz factorization","Fourier coefficient bounds","planar cubic systems"],"falsifier":"A direct refutation would be to find any positive definite homogeneous polynomial P, of any even degree, with L_f P ≤ 0 for the field (2)–(3); a semidefinite-programming search over coefficient vectors at increasing degree would locate one if Theorem II.2(3) were false. For the real-analytic claim, a local real-analytic V with V(0)=0, V>0 near 0, and L_f V ≤ 0 near 0 would refute item (4). At the technical level, computing a positive definite homogeneous polynomial whose normalised logarithmic derivative has |ĝ₂| ≥ 1 would falsify Lemma III.3, the load-bearing bound.","tokens_in":8654,"feed_emoji":"🌀","tokens_out":9883,"duration_ms":81194,"temperature":0.7,"pith_summary":"Lyapunov functions certify stability by decreasing along trajectories, and for homogeneous polynomial vector fields it was conjectured that a polynomial certificate always exists. This paper disproves that conjecture with a planar homogeneous cubic vector field with integer coefficients whose origin is globally asymptotically stable. The field has no positive definite homogeneous polynomial with nonpositive derivative along trajectories, and a leading-term argument rules out even local real-analytic Lyapunov functions. Stability is proved instead by an explicit degree-two homogeneous function H = (x²+y²) exp(−10xy/(x²+y²)), which is positive definite, radially unbounded, C¹, and strictly decreasing along every trajectory. A machine-checked formalization of the main theorem accompanies the proof.","feed_headline":"No polynomial Lyapunov function exists for a stable cubic field","feed_subtitle":"A 2D cubic counterexample disproves the homogeneous polynomial converse conjecture.","key_machinery":"The key machinery is the reduction to polar coordinates, which turns the field into ṙ = r³(−1 + 5 cos 2θ), θ̇ = r², and the resulting Fourier argument on the unit circle. A homogeneous polynomial candidate P of degree 2N restricts to a positive trigonometric polynomial p; writing g = p′/(2Np), the weak Lyapunov inequality forces w = 1 − 5 cos 2θ − g ≥ 0, whose Fourier coefficients satisfy |ŵ₂| ≤ ŵ₀ = 1. The strict Fejér–Riesz factorization of p gives the degree-independent bound |ĝ₂| < 1, and since ŵ₂ = −5/2 − ĝ₂, the triangle inequality yields the contradiction 5/2 < 2. The factor 5 in the angular dynamics is the forcing term that no polynomial's logarithmic derivative can compensate.","core_discovery":"The central discovery is a single planar system, the homogeneous cubic vector field f₁ = 4x³ − x²y − 6xy² − y³, f₂ = x³ + 4x²y + xy² − 6y³. In polar coordinates it becomes ṙ = r³(−1 + 5 cos 2θ), θ̇ = r², so the angle increases monotonically while the radius is governed by a second-harmonic forcing. The paper proves that the origin is globally asymptotically stable, that H = (x²+y²) exp(−10xy/(x²+y²)) is a valid non-polynomial Lyapunov function with L_fH = −2(x²+y²)H < 0, and that no positive definite homogeneous polynomial P — of any degree — satisfies the weak Lyapunov inequality L_fP ≤ 0. Since the vector field is minimal in dimension and degree for this phenomenon, the result closes the h","pith_inferences":["The second-harmonic obstruction suggests a general threshold: for the family ṙ = r³(−a + b cos 2θ), θ̇ = r², polynomial Lyapunov functions may exist only when |b| is below roughly 2(a+1); establishing existence in that regime would show that absence is governed by a sharp parameter inequality.","Because the proof only uses the second Fourier coefficient of the angular forcing, analogous non-existence arguments should hold for higher-degree homogeneous fields whose polar form has a dominant 2θ component; constructing such higher-degree examples would test the generality of the mechanism.","The regularity–algebraicity coupling observed here — a C^m homogeneous function of degree m is a polynomial — suggests that smooth non-polynomial certificates occupy a distinct complexity class from algebraic ones; that distinction may matter for designing certificate-search algorithms."],"forward_implications":["The homogeneous polynomial Lyapunov converse conjecture is false: polynomial certificates are not guaranteed for homogeneous polynomial systems.","For this system, the explicit C¹ function H and the rational function R both prove global asymptotic stability, so non-polynomial certificates can be simpler than polynomial ones.","Any algorithm that searches only over polynomial Lyapunov functions of bounded degree cannot certify all stable homogeneous cubic systems.","The counterexample persists on an open set of the two-parameter family (27)–(28), so the failure of polynomial certificates is robust under parameter perturbation.","The real-analytic obstruction means the phenomenon is not an artifact of restricting to polynomials; no analytic certificate exists locally either."],"fun_headline_variants":["Stable cubic field has no polynomial Lyapunov function","Counterexample to Lyapunov conjecture: stable cubic system","No polynomial Lyapunov for globally stable cubic field","Cubic stability without polynomial Lyapunov: conjecture false","Lean-verified: stable cubic defies polynomial Lyapunov existence"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument's load-bearing premise is Lemma III.3: for every positive definite homogeneous polynomial of degree 2N, the normalised logarithmic derivative g(θ) = p′(θ)/(2N p(θ)) has second Fourier coefficient strictly below 1 in modulus; if that strict bound fails, a polynomial Lyapunov function could pass the Fourier test.","fun_headline_variants_meta":{"raw":{"variants":["Stable cubic field has no polynomial Lyapunov function","Counterexample to Lyapunov conjecture: stable cubic system","No polynomial Lyapunov for globally stable cubic field","Cubic stability without polynomial Lyapunov: conjecture false","Lean-verified: stable cubic defies polynomial Lyapunov existence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001137,"raw_usage":{"total_tokens":4523,"prompt_tokens":676,"completion_tokens":3847,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":420,"completion_tokens_details":{"reasoning_tokens":3765}},"tokens_in":420,"tokens_out":3847,"duration_ms":23776,"temperature":1.0,"reasoning_tokens":3765,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T21:12:09.674615+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct refutation would be to find any positive definite homogeneous polynomial P, of any even degree, with L_f P ≤ 0 for the field (2)–(3); a semidefinite-programming search over coefficient vectors at increasing degree would locate one if Theorem II.2(3) were false. For the real-analytic claim, a local real-analytic V with V(0)=0, V>0 near 0, and L_f V ≤ 0 near 0 would refute item (4). At the technical level, computing a positive definite homogeneous polynomial whose normalised logarithmic derivative has |ĝ₂| ≥ 1 would falsify Lemma III.3, the load-bearing bound.","supporting_citations":[],"review_version":1}