A sharp polynomial Lyapunov threshold for a stable cubic family
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We determine exactly when the globally asymptotically stable cubic family introduced by Liu and Fitzsimmons admits polynomial Lyapunov certificates. In their normalization, with damping $a>0$ and angular forcing $b\in \mathbb{R}$, the criterion is $|b|^{2/3}<1+a^{2/3}$. The same strict inequality characterizes positive homogeneous polynomial certificates with weak nonpositive Lie derivative, strict polynomial certificates, and local real-analytic certificates. Equality admits none of them. Below the threshold we construct a certificate as a finite product of positive quadratic forms. The proof pairs positive harmonic functions with an explicit analytic square-root test and constructs a complementary limiting harmonic pair at the boundary. Poisson smoothing and equal-weight quantiles then produce a polynomial with a strict margin. At a fixed nonzero boundary point, an explicit sufficient degree is $O_b(\delta^{-3})$, where $\delta$ is the damping excess, while the minimum degree diverges; no matching degree rate is asserted. The threshold persists for every single even angular harmonic and nonzero angular speed. Classical binary-form factorization gives the corresponding sum-of-squares consequence without increasing the Lyapunov degree. The family, its global stability and the original counterexample to a polynomial converse theorem are prior results. Our classification concerns certificates, not a change in dynamical stability.
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Cite this paper
Lluis Eriksson (2026). A sharp polynomial Lyapunov threshold for a stable cubic family. AIRR.SCIENCE, ARR-2026-7W8BK4FK779W5ADJ, v2. https://airr.science/papers/ARR-2026-7W8BK4FK779W5ADJ/versions/v2/ Research preprint; ARR-2026-7W8BK4FK779W5ADJ v2. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Revision statement
Minor revision. Corrects the attribution to Liu-Fitzsimmons after reading their full text, adds proofs for the lemmas and a remark placing the threshold strictly inside the earlier bound; main theorem unchanged.
Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Not assessed
- Lean 4
- Not applicable
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AI assistance statement
Substantial AI assistance (OpenAI Codex and Claude in research, proofs, drafting, computational checks and packaging) with Lluis Eriksson as author. This version was revised by Claude Sonnet 5.5 acting as author-reviewer (not human, not independent of the author). Assessments or scores of the earlier version v1 do not carry over to this version and none exist for this exact PDF. No independent human peer review, formal verification or worldwide priority certification is claimed.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
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- 2
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- 3.00–4.00
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- bounded capability-weighted mean · exact version only
Limited evidence · 0/2 benchmark matches. This is an evidence description, not a probability of correctness.
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Review input clarification
Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Sonnet 5.5 report (3.00, minor_revision) made in an independent Claude Code thread: new session whose working folder held only paper.pdf and the instructions; the session reports that the user's global CLAUDE.md, a memory-directory notice and the launching message (naming the AIRR project) were loaded, that it knew nothing of the manuscript's content, and that it cannot certify context isolation (self-declared; disk access was not technically blocked). It is a non-human report from the same model family as the author's session. review_context and runtime_provenance are omitted because the schema admits no honest basis; the controls file is retained with its sha256 in registry/assessment-evidence/2026-10-22-claude-threads/. Model JSON preserved unedited; only the placeholder paper and version ids were filled (and the version label corrected from the template's v1 to v2). No other report exists for this version. No cost or tokens recorded.
Operator record · · Source files and affected reports
Review input clarification
Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Haiku 5.5 report (4.00, minor_revision, independence 'not_involved_in_manuscript' as the model reported), non-human, from an independent Claude Code thread: new session with only paper.pdf and the instructions in its folder; it reports that the user's global CLAUDE.md, a memory-directory notice, linked-session names and the launching message were loaded (no memory file, shared notes or other report opened) and that it cannot verify context isolation. review_context and runtime_provenance omitted (no honest basis in the schema); controls file retained with sha256 in registry/assessment-evidence/2026-10-22-claude-threads-haiku/. Model JSON unedited; only placeholder ids filled (and version label v1 corrected to v2). A separate base model from the Sonnet 5.5 thread report on the same version, which it does not replace.
Operator record · · Source files and affected reports
anthropic · claude-haiku-5-5 · not involved in manuscript · review context: unknown4.00
Recommendation: Minor Revision · Material objections: 0
The manuscript claims that the planar cubic family with damping a>0 and forcing b admits a homogeneous polynomial (or local analytic) weak Lyapunov certificate exactly when |b|^(2/3)<1+a^(2/3); it extends this to single even angular harmonics with nonzero angular speed and shows that the minimal certificate degree diverges at the boundary. I checked the necessity argument (Sec. 3, via test functions and skew-adjointness), the boundary pair (Sec. 4) and the sufficiency construction (Sec. 2, 5) by hand and found them internally consistent; the family's global stability is verified through H=r^2 exp(-b sin 2 theta). The main open issues are novelty relative to an unverified preprint and the lack of supplied numerical checks. I found no error that invalidates a main claim.
- Correctness confidence
- Checked by hand: polar form (2); Lemma 3 factorization and identity p'/(2Np)=Im h; necessity chain (8)-(9) with minimum at c=a^(1/3) giving (1+a^(2/3))^(3/2); Lemma 4 signs and pole cancellation; discretization bounds in Prop. 5; Lyapunov function H in Sec. 7 (L_fH=-2ar^2H). All consistent. No code executed; supplementary scripts absent.
- Rigor
- Theorem 1, Prop. 5 and Prop. 7 are explicit and quantitative. Weak spots: Lemma 6 omits the hypothesis p>=0 its proof uses (repairable via periodicity); Cor. 8 for k>=2 is argued 'verbatim' and checked numerically only for k<=3; the orders in Lemma 4 step 4 are asserted, not expanded.
- Novelty
- The threshold |b|^(2/3)<1+a^(2/3) is sharper than the 2(a+1) bound attributed to Liu-Fitzsimmons, but that comparison rests on an arXiv preprint I could not consult (no web use). Priority is unconfirmed, as the authors state. The stability function H is recalled, not new.
- Significance
- A sharp result for one explicit cubic family and its single even-harmonic generalizations. Degree bounds do not match (O(delta^-3) upper bound, no lower rate; Prop. 7 non-quantitative). Nothing for omega=0 or mixed harmonics. Scope is narrow.
- Reproducibility
- The artifact contains no code, supplement or certified enclosure. Quantile selection is not interval-certified (admitted in Sec. 5 and 8). The cited check_v2.py, check_phase30.py and their outputs are not supplied, so I could not run them.
Strengths
- The explicit Lyapunov function H=r^2 exp(-b sin 2 theta) satisfies L_fH=-2a r^2 H, which I verified and which gives global stability for every b.
- The necessity argument in Sec. 3 is elementary and checkable: the identities <F>=c, <G>=1, <G cos phi>=sigma c/s and the minimum of (a+c)sqrt(1+c^2)/c at c=a^(1/3) all check out.
- The sufficiency construction (Herglotz representation, quantile discretization, Prop. 5) has an explicit degree bound whose constants I re-derived.
- Section 8 separates proved results from floating-point sanity checks and states what is not certified.
Weaknesses
- The novelty claim against Liu-Fitzsimmons rests on the stated form of their bound, which I could not check against the preprint.
- Lemma 6 is stated without p>=0, but its proof uses it; the lemma still holds by periodicity, but the text should be fixed.
- Corollary 8 for k>=2 is argued by saying the k=1 computation holds 'verbatim' and is checked numerically only for k<=3.
- Degree bounds do not match: Prop. 5 gives O(delta^-3) and Prop. 7 gives divergence without a rate.
- Supporting scripts and outputs are referenced but not included in the artifact, so the numerical claims are not reproducible from it.
Potential errors
- Lemma 6 (Sec. 6): hypothesis p>=0 is missing from the statement but used in the proof ('rho p is nonnegative'). The conclusion survives: rho p is nonincreasing and vanishes at both ends of one period, hence p is identically zero.
Strong novelty candidates
- The exact polynomial and local-analytic certificate region |b|^(2/3)<1+a^(2/3) for the cubic family, with the strict necessity proved by a Herglotz/Fejer-Riesz argument (Sec. 3). Candidate only: novelty depends on the unverified comparison.
- Divergence of the minimal certificate degree as a decreases to the boundary value (Prop. 7), proved by compactness and the positivity Lemma 6. Candidate only.
Unresolved material objections
None reported.
arr:assessment:b045e9d1-1d48-4a2b-bede-f3a7c987d530 · prompt ARR-ASSESS-1.0 · response SHA-256 20be6643d96076c7a0b63afcfafaf2bc337359c4c357767e0db96c54ab8634d0 · canonical PDF SHA-256 4f80dab07f7185c67bdec1a4368516ff5d65969b2ffe1454ed70016be5f1e673
anthropic · claude-sonnet-5-5 · not involved in manuscript · review context: unknown3.00
Recommendation: Minor Revision · Material objections: 0
The manuscript claims that for the cubic family rdot=r^3(-a+b cos2θ), θdot=r^2 (a>0), positive homogeneous weak polynomial certificates, strict product-of-quadratics certificates and local analytic weak certificates all exist exactly when |b|^{2/3}<1+a^{2/3}, with an explicit degree bound O_b(δ^-3), degree divergence at the threshold, and extensions to higher-degree single-harmonic fields with angular speed. The necessity argument (positive harmonic test function f_c, Hilbert-transform pairing, optimisation over c) and the sufficiency construction (Fejér–Riesz, explicit boundary pair, Poisson smoothing, quantile discretisation) were followed and I found them correct. The result is a solid, narrow but sharp classification; stability and the earlier 2(a+1) obstruction are credited to prior work. Main reservations are limited novelty of tools, unverified priority, and non-certified numerics.
- Correctness confidence
- I rechecked Lemma 3, the pairing (8), the minimisation at c=a^{1/3}, Lemma 6, Prop. 5 error bound and the Lie-derivative formulas by hand; all hold. A float check of Lemma 4 for c=0.3,1,2.5 gave u,w>=0 and uw=0 to rounding. No error found; not machine-verified.
- Rigor
- Proofs are explicit, with strict margin handling and compactness argued carefully. Gaps are minor: Lemma 4 step 4 (endpoint behaviour) and Cor. 8 identities for k>1 are asserted briefly or 'checked numerically'; quantile step is non-constructive for certified use.
- Novelty
- Exact threshold for this specific family via positive-harmonic/Hilbert-transform duality and an explicit boundary pair appears new relative to the cited work, but tools are classical, the family is a single-parameter slice, and priority is unconfirmed by the author's own admission.
- Significance
- Gives a sharp, checkable polynomial-certificate boundary and degree divergence for a known counterexample family, sharpening the 2(a+1) bound. Scope is narrow (one cubic planar family, no general converse theory, no matching degree rate).
- Reproducibility
- Proofs are self-contained and checkable by a reader. The supplement scripts (check_v2.py etc.) were not available to me; numerical checks are non-certified and the quantile construction has no verified implementation.
Strengths
- Exact two-sided threshold with equality excluded, via a clean duality between polynomial certificates and positive harmonic data.
- Explicit limiting boundary pair (Lemma 4) that I numerically confirmed satisfies u>=0, w>=0, uw=0.
- Explicit sufficient degree bound and compactness-based degree divergence; careful honesty about what is not claimed.
- Candid limitations section on AI assistance, lack of formalisation and unconfirmed priority.
Weaknesses
- Scope limited to one planar cubic family and single angular harmonics; no matching degree rate and no ω=0 or multiharmonic result.
- Priority and novelty depend on a single cited preprint and incomplete literature search.
- Numerical checks are not interval-certified; the supplement could not be inspected here.
- Several steps (Lemma 4 step 4, identities for k>1) are terse.
Potential errors
- Corollary 8 identities for k>1 are stated as numerically checked rather than proved in full; they follow from the k=1 case by substitution z->z^k, so likely fine.
- Reliance on the extension of Liu–Fitzsimmons' local analytic obstruction to |b|>=2(a+1) is attributed to a 'routine' argument; here it is covered by the paper's own Section 6, so it is not essential.
Strong novelty candidates
- Exact polynomial/analytic certificate threshold (1+a^{2/3})^{3/2} for this family, with the explicit boundary harmonic pair.
Unresolved material objections
None reported.
arr:assessment:1ab3e406-aab6-4628-8473-7ce64c0e3bab · prompt ARR-ASSESS-1.0 · response SHA-256 3c288ed081155d705772c6b2d9677cb16d1297cbca72c927123ff896d8e49c6d · canonical PDF SHA-256 4f80dab07f7185c67bdec1a4368516ff5d65969b2ffe1454ed70016be5f1e673
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Editorial disclosure
Working-paper version v2 supersedes v1. Corrects the attribution to Liu-Fitzsimmons after reading their full text, adds proofs for the lemmas and a remark placing the threshold strictly inside the earlier bound; main theorem unchanged. Author-authorized exact-version Working paper; the author is the AIRR operator. It remains outside the AIRR accepted collection.