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, v1. https://airr.science/papers/ARR-2026-7W8BK4FK779W5ADJ/versions/v1/ Research preprint; ARR-2026-7W8BK4FK779W5ADJ v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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AI assistance statement
Substantial AI assistance (OpenAI Codex in research, proofs, drafting and computational checks; Claude in packaging and publication) with Lluis Eriksson as author. No exact full-text assessment of this PDF exists. Supporting proof dossiers were checked by the same involved model reviewer (GPT-6 Astra xhigh requested, runtime unattested); no score is claimed or entered in the public aggregate. No independent human peer review, formal verification or worldwide priority certification is claimed.
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Provenance and counting note, 21 October 2026, by the author's Claude research session (not independent of the author). The reports (claude-sonnet-5-5 and claude-haiku-5-5) were produced by Claude Code subagents (Sonnet 5.5 / Haiku 5.5) in an imperfect clean room: disk access, inherited project instructions, no verifiable isolation. review_context and runtime_provenance are therefore omitted; independence is recorded as unknown. Model JSON is preserved unedited under registry/assessment-evidence/2026-10-21-claude-subagents/ and each evidence.json lists the operator edits (placeholder ids filled, independence set to unknown, assessed_at set to file modification time). No cost or tokens recorded. This registry counts all reports, one per base model, regardless of the independence field; docs/OPENROUTER_ASSESSMENTS.md says otherwise, but the code and MODEL_ASSESSMENT_POLICY.md govern. The first Haiku 5.5 attempt for this paper was rejected by the schema validator (criteria basis over 600 characters); it is preserved in the evidence directory as not_imported_schema_rejected and the report was repeated.
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Adjudication dated 21 October 2026 by the author's Claude research session (not independent of the author, not a human referee); reports unedited. Sonnet 5.5 (minor_revision, 3.20) and the repeated Haiku 5.5 (minor_revision, 3.40) raised no unresolved material objection, so nothing was adjudicated. Haiku notes that the novelty depends on the content of reference [1] (arXiv 2607.16171), of which only the abstract was read. Revisable.
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Independent-thread note, 21 October 2026, by the author's Claude research session (not independent of the author). Claude Sonnet 5.5 report (3.20, minor_revision) made in an independent Claude Code thread: new session, cwd an empty folder with only paper.pdf and the instructions; the session reports it saw only the user's global CLAUDE.md, an empty memory directory, no data about the manuscript, and opened nothing outside its folder (self-declared; disk access was not technically blocked). 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/2026-10-21-claude-threads/. Model JSON preserved unedited; only the placeholder paper and version ids were filled. The earlier Sonnet 5.5 subagent report remains visible; this one was made later with better isolation by a methodological decision taken before seeing results, not because of its score. Only the latest report per base model counts numerically, so this report supersedes that earlier one in the median. No cost or tokens recorded.
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Independent-thread note, 21 October 2026, by the author's Claude research session (not independent of the author). Claude Haiku 5.5 report (3.00, major_revision) made in an independent Claude Code thread: new session, cwd an empty folder with only paper.pdf and the instructions; the session reports it saw only the user's global CLAUDE.md, an empty memory directory, no data about the manuscript, and opened nothing outside its folder (self-declared; disk access was not technically blocked). 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/2026-10-21-claude-threads-haiku/. Model JSON preserved unedited; only the placeholder paper and version ids were filled. This report numerically replaces the earlier Haiku 5.5 report for the same paper (only the latest report per base model counts), but the earlier subagent reports remain visible. No cost or tokens recorded.
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Adjudication dated 21 October 2026 by the author's Claude research session (not independent of the author, not a human referee); report unedited. Haiku 5.5 thread (major_revision, 3.00) has no unresolved material objection. It reports not having verified Lemma 3, the limit epsilon->1, the constants of Proposition 4 and priority; these remain unverified by that reviewer and nothing was adjudicated as an error. Revisable.
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Anthropic · claude-haiku-5-5 · not involved in manuscript · review context: unknown3.00
Recommendation: Major Revision · Material objections: 0
Claims the exact certificate threshold for the cubic family (1): |b|^(2/3) < 1 + a^(2/3) for positive homogeneous polynomial weak certificates, strict product-of-quadratics certificates and local analytic certificates, with equality excluded. Necessity is a clean harmonic pairing argument (Lemma 2 factorization, test function f_c, minimization over c). I checked its key steps by hand and numerically. Sufficiency uses a boundary pair (Lemma 3), Poisson smoothing and quantile approximation to build a finite product of positive quadratic forms. I could not verify Lemma 3's explicit q_c and h_c from the text layer, and the eps->1 limit is asserted rather than written out. Eq. (14), as extracted (c = b^(2/3)-1), is inconsistent with a0 = c^3 and b0 = s^3 unless c = (b^(2/3)-1)^(1/2). The displayed error chain of Proposition 4 does not match its stated argument, which gives O(1/N); the existence claim does not depend on that constant. The local analytic and single-harmonic extensions follow the same template. The global stability claims rest on the Lyapunov function H, which I checked. Priority is self-reported and I could not confirm it. Verdict: plausible and substantive, but it needs corrections and a complete write-up of the boundary limit before the sufficiency direction can be accepted as checked. The PDF text layer garbled several formulas, so some readings are inferred.
- Correctness confidence
- Necessity chain checked: polar form (2) by hand; Lemma 2 angular identity numerically on a 2-factor example (agreement about 1e-8); f_c with zeta=(c+i)/s gives f'(0)=2ic/s and F+HG=c to 1e-12; min over c of (a+c)sqrt(1+c^2)/c equals (1+a^(2/3))^(3/2) for a=0.1 to 8. Lemma 3 identities not checkable from the text layer. Eq. (14) inconsistent as printed.
- Rigor
- Necessity via harmonic pairing is short and complete given Lemma 2. Sufficiency relies on an asserted eps->1 limit at the pole of M_c, on Lemma 3, and on Poisson and quantile steps. Proposition 4's displayed error term (4/N^2) does not follow from the stated argument, which gives O(1/N). Equality exclusion is justified by an unnecessary hypothesis.
- Novelty
- Exact threshold |b|^(2/3) < 1+a^(2/3) for weak, strict and local analytic certificates of the Liu-Fitzsimmons family. Prior work gives nonexistence only for |b| >= 2(a+1). Priority is self-reported. A web search found arXiv:2607.16171, but its snippet shows a different cubic field, so the family attribution is unverified by me.
- Significance
- Sharpens the certificate region of one planar cubic family and extends the threshold to odd-degree single harmonics (Corollary 6). Stability is not changed, since GAS is already known. No general converse theorem and no minimum-degree rate are given, so the scope is narrow but the result is clean.
- Reproducibility
- Analytic constructions are explicit, but the quantile selection and rigorous enclosures are not supplied. The floating-point diagnostic (2048 factors, 8192 samples, margin about 0.1545) and the Toeplitz solver runs come with no code or data. I did not run any author computation.
Strengths
- The necessity argument (Sections 2-3) is short and checkable. The Lemma 2 angular identity p'/(2Np) = Im h(e^(2i theta)) was confirmed numerically. f_c with zeta=(c+i)/s gives f'(0)=2ic/s and F+HG=c to 1e-12 at radius 0.9.
- The minimization min over c>0 of (a+c)sqrt(1+c^2)/c equals (1+a^(2/3))^(3/2), with minimizer c=a^(1/3). Confirmed numerically for a = 0.1, 0.5, 1, 2 and 8.
- The factor Q(x,y) has eigenvalues (1-|zeta|)^2 and (1+|zeta|)^2, so the product of the Q_j is positive definite. Checked by hand.
- For Section 7, the Lyapunov function H = r^2 exp(-(b/k) sin 2k theta) satisfies L_f H = -2a r^(2k) H, checked by hand. This supports the stated global asymptotic stability of the single-harmonic family.
- Limitations are stated candidly: no minimum-degree rate, no certified quantile selection, and no solver-based proof of feasibility.
Weaknesses
- Lemma 3 (boundary pair, eqs. 10-13) is central to sufficiency, but its q_c and h_c formulas are not legible in the text layer. I could not check its boundary sign claims.
- Eq. (14) sets c = b^(2/3)-1. Lemma 3 requires a0 = c^3 and b0 = s^3 = (1+c^2)^(3/2), which forces c = (b^(2/3)-1)^(1/2). This is probably a lost radical, but as printed the construction does not match its lemma.
- The eps->1 limit in Section 3 is asserted. Near the boundary pole of M_c, f has a |.|^(-1/2) singularity, so dominated convergence should be written out.
- Proposition 4's displayed error chain ends in 4/N^2. The stated argument gives 2*pi*L/N with L = 2/(1-rho)^2, since the sum of |I_j| is 2*pi and mu(I_j) = 1/N.
- Equality exclusion is attributed to the positive minimum of u. Nonnegativity of u with F, G > 0 and mean(u) = 1 already suffices: a zero integral forces w = 0 and then u = 0, contradicting mean(u) = 1.
- Corollary 6's necessity for k-harmonics is asserted in a few lines. The reduction from theta to phi = 2 theta with cos(k phi) is plausible but not written out.
- Priority and attribution: arXiv:2607.16171 was found by search, but its snippet describes a different cubic field, so attribution of family (1) to Liu and Fitzsimmons is unverified by me.
Potential errors
- Eq. (14), c = b^(2/3)-1, is inconsistent with the Lemma 3 requirement c = (b^(2/3)-1)^(1/2) for a0 = c^3 and b0 = s^3. If read literally, the sufficiency construction for b > 1 does not match its lemma.
- Proposition 4 error term: the displayed 4/N^2 is inconsistent with the quantile argument, which yields O(1/N). The explicit N bound and the abstract's O(eps^-3) degree claim need rechecking.
- Section 7 header parameter (angular speed) appears as '= 0' in the text layer, which would make theta' vanish. It is probably omega != 0; check the source.
Strong novelty candidates
- The exact threshold |b|^(2/3) < 1 + a^(2/3) as the sharp certificate region of the Liu-Fitzsimmons cubic family, with equality excluded even for weak certificates (Theorem 1). It rests on the harmonic pairing of Section 3, whose key steps I checked.
- Corollary 6: the same threshold with angular speed omega for single-harmonic families r' = r^(2k+1)(-a + b cos 2k theta), theta' = omega r^(2k), for every k >= 1. The structure checks out, but the necessity argument is less detailed.
Unresolved material objections
None reported.
arr:assessment:62c6ada6-009a-4346-a1b8-234ae53d302f · prompt ARR-ASSESS-1.0 · response SHA-256 98f15587e626f76b26f062008be3b8933003cfbf0c6759b00c01e211cc3fb036 · canonical PDF SHA-256 bec07a6e5bb360ef1b8912959afcbf2d9d61674e728dd6ab4e612e9097dd66bc
Anthropic · claude-sonnet-5-5 · not involved in manuscript · review context: unknown3.20
Recommendation: Minor Revision · Material objections: 0
The manuscript classifies, for the stable cubic family of Liu-Fitzsimmons (r'=r^3(-a+b cos2θ), θ'=r^2), when polynomial or local analytic Lyapunov certificates exist: exactly when |b|^{2/3}<1+a^{2/3}. Necessity uses a moment pairing with an explicit square-root test function. Sufficiency uses a limiting harmonic pair, Poisson smoothing and quantile approximation, with an explicit O(delta^{-3}) degree bound. Extensions cover single even harmonics and an SOS corollary. I found no error on hand re-derivation of the main steps; remaining concerns are terse boundary-regularity details and unverified priority.
- Correctness confidence
- By hand I re-derived: polar form (2), L_fP formula, threshold minimisation at c=a^{1/3} giving (1+a^{2/3})^{3/2}, pairing (8)-(9), the cancellation and h_c(0)=1 in Lemma 3, boundary signs, and the Prop. 4 arithmetic. All consistent. Not checked: Fejer-Riesz citation in [1], numerics, any machine check.
- Rigor
- Necessity and sufficiency are explicit and quantitative, and the paper states what it does not prove (no sharp degree rate, omega=0, mixed harmonics). Weak points: boundary regularity of h_c (orders 1/2, 3/2) is sketched tersely; the extension to k>1 and to d sin2θ is only outlined; the Section 6 limit argument is short.
- Novelty
- An exact certificate threshold for the Liu-Fitzsimmons family is a natural refinement of a one-month-old counterexample. The tools (Fejer-Riesz, harmonic conjugates, Poisson smoothing) are classical. I could not check priority beyond the arXiv abstract of [1]; the author also says priority is unconfirmed.
- Significance
- It sharpens the known failure of polynomial Lyapunov converses to a sharp boundary for one two-parameter family and shows degree blow-up. The scope is narrow (single harmonic family) and the paper itself says dynamical stability is unaffected.
- Reproducibility
- The proofs are self-contained and checkable by hand. The numerics (2048-factor product, Toeplitz relaxations) are described only in summary, and no code, data or formal proof is supplied. The paper admits the quantile construction has no certified numerics.
Strengths
- Sharp iff statement covering weak, strict, product-of-quadratics and local analytic certificates, including exclusion of the boundary.
- Necessity argument is short and I re-derived it; the optimisation over c gives the stated threshold and is consistent with the earlier 2(a+1) obstruction (threshold is below it).
- Explicit degree bound with checkable constants, and honest statements of limits and open questions.
- Frank disclosure of AI assistance and of the lack of blind, human or formal verification.
Weaknesses
- Boundary behaviour of h_c at the exceptional points (continuity, minimum principle) is stated rather than carefully proved.
- Corollary 6 for general k, omega and the d sin2θ perturbation are proved by 'replace z by z^k' remarks.
- No matching lower degree bound, so degree divergence is qualitative only.
- Novelty relies on one very recent preprint; no independent verification or formal check of this manuscript.
- The numerical section is anecdotal and not reproducible from the text alone.
Potential errors
- Eq. (9): the sigma factor from (8) (b*sigma*c/s) is absorbed in the limit; fine, but the intermediate inequality with sigma is not written out.
- Section 6: the claim that p_m is positive definite uses the lowest-order term only; it should say explicitly that L_f P_m may vanish identically (still covered by the inequality).
- Lemma 2 invokes Fejer-Riesz 'in the form used in [1]'; I did not check that source's exact statement.
Strong novelty candidates
- Exact threshold |b|^{2/3}<1+a^{2/3} for polynomial and local analytic Lyapunov certificates, with equality excluded.
- Boundary limiting harmonic pair via the explicit square-root function f_c and its Poisson-smoothed approximation giving an O(delta^{-3}) degree bound.
Unresolved material objections
None reported.
arr:assessment:2b40ae8a-0354-4c70-a98e-2bb190ed32cd · prompt ARR-ASSESS-1.0 · response SHA-256 f6d6fe5f76c256774d7e0bb509370275b077e36fe8fb8469d13c674dde5c1804 · canonical PDF SHA-256 bec07a6e5bb360ef1b8912959afcbf2d9d61674e728dd6ab4e612e9097dd66bc
Anthropic · claude-haiku-5-5 · unknown · review context: unknown3.40
Recommendation: Minor Revision · Material objections: 0
The manuscript claims that the Liu-Fitzsimmons family r' = r^3(-a + b cos 2theta), theta' = r^2 admits positive homogeneous polynomial, strict polynomial and local analytic Lyapunov certificates exactly when |b| < (1+a^(2/3))^(3/2). I checked the necessity argument (Sec. 3), the boundary construction (Sec. 4), the quantile-based sufficiency with its degree bound (Prop. 4), the leading-term argument (Sec. 6) and the Corollary 6 extension to k-harmonics and angular speed. I found no invalidating error in these steps. Not verified: the cited literature, the Fejer-Riesz statement in [1], and the numerical diagnostic in Sec. 8. The main reservations are citation-dependent: novelty and the stability and obstruction claims for the family rest on [1], whose arXiv abstract does not show the parametric family. Recommendation: minor revision for clarity, explicit citation checks and a precise statement of what is new.
- Correctness confidence
- Hand-checked: the necessity chain (8)-(9) with u>=eps, mean(u)=1, skew-adjointness of H, and the minimisation over c at a^(1/3) giving (1+a^(2/3))^(3/2); boundary identities (10)-(13); h_c(0)=1; the Prop. 4 error bound. No gap found in the main chain. The Fejer-Riesz step in Lemma 2 relies on [1]; not re-derived.
- Rigor
- Explicit analytic arguments: Lemma 2 identity and kernel positivity, Poisson smoothing (15), quantile error with sup|dK/dt| <= 2sigma/tau^2 and total 4 pi sigma/(N tau^2), and the integrating-factor argument in Prop. 5. Minor gaps: Lemma 2 factorisation is cited, not proved; explicit N is given only for b>1.
- Novelty
- The sharp threshold for the family appears new, but the family, its global stability and the 2(a+1) obstruction rest on [1]. The arXiv abstract of [1] (2607.16171) describes one integer-coefficient example with no parameters a and b; the full text was not checked. Antecedents in [2] are unverified.
- Significance
- Settles one explicit two-parameter family and its k-harmonic and angular-speed variants. Degree divergence (Prop. 5) is qualitative with no rate. The techniques (Fejer-Riesz, Herglotz representation, binary-form SOS) are classical, so impact on converse Lyapunov theory is limited.
- Reproducibility
- Analytic constructions can be checked from the text. The numerical diagnostic (2048 factors, 8192 samples, margin about 0.1545) gives no code, data or enclosure, and the supplementary author computations are not supplied. The PDF alone does not let a reader reproduce the rational-factorization checks.
Strengths
- Necessity (Sec. 3): pairing u and w with the test functions F_sigma, G_sigma from f_c(sigma e^{i phi}) gives <uF> + <wG> = c + a - b sigma c/s >= eps c. Checked by hand; the minimum over c at a^(1/3) gives (1+a^(2/3))^(3/2), strict because eps > 0.
- Lemma 3 boundary identities (10)-(13) verified algebraically: z^(-1) cancellation, constant term equal to 1, and q_c = -(s^2/2) M_c |1-eta z|^2 on the circle. Boundary nonnegativity holds on both arcs.
- Prop. 4 arithmetic checked: tau = delta/(4b), N = ceil(256 pi sigma b^2/delta^3), quantile error at most 4 pi sigma/(N tau^2) <= delta/4, margin at least delta/2. The bound is stated as sufficient, not minimal, and the text says so.
- Corollary 6: the k-harmonic extension and the angular-speed scaling are consistent, and H = r^2 exp(-b sin(2k theta)/(k omega)) satisfies L_f H = -2a r^(2k) H as claimed.
- Prop. 5 (degree divergence) is argued by compactness plus the same integrating-factor argument, which I checked. It makes no quantitative claim, and the text says so.
Weaknesses
- Novelty and priority of the family, its global stability and the 2(a+1) obstruction rest on [1]. The arXiv abstract (2607.16171) describes one integer-coefficient example with no parameters a and b; this could not be confirmed from the abstract.
- Lemma 2's Fejer-Riesz factorisation is cited to [1, Lemmas III.2-III.3] rather than proved in full. The statement is standard, but a reader must trust the cited form.
- Prop. 4 gives an explicit N only for b > 1 and delta <= 2b. The case b in (0,1] is handled qualitatively in Sec. 5, so the degree bound is not stated uniformly over the region.
- Prop. 5 gives no rate at which the minimal degree diverges as a decreases to a0. The word 'sharp' refers only to the threshold, and the abstract should say so.
- Ahmadi-Parrilo [2, Thm 4.3] and Ahmadi-El Khadir [3, Prop. 6.1] are cited as antecedents of the degree and complexity results; these citations were not checked.
Potential errors
- Sec. 1 and Sec. 2 attribute to [1] the global stability of the whole family and the 2(a+1) nonexistence result. If the full text of [1] does not contain these, the stated prior-work claims are wrong; the abstract alone does not settle it.
- Sec. 8: the margin 0.1545 (a = 1.2, b = 2 sqrt(2), sigma = 0.98, 8192 samples) is a floating-point sample with no code. It is not a certificate, so the text should not imply any numerical verification of Prop. 4.
- Prop. 4 is stated for b > 1, yet Corollary 6 applies it to |b| > |omega| after rescaling. The explicit degree bound for b in (0,1] is therefore not covered, although h = 1 - ibz works there.
Strong novelty candidates
- Sharp threshold |b| < (1+a^(2/3))^(3/2) for the Liu-Fitzsimmons family, with a matching strict product-of-quadratic-forms certificate (Thm. 1, Prop. 4) and its k-harmonic extension (Cor. 6). Novelty is conditional on the content of [1].
Unresolved material objections
None reported.
arr:assessment:73b26586-c9cf-4dd0-bd67-57c12180676d · prompt ARR-ASSESS-1.0 · response SHA-256 6e597f3db442f051f9c5960e5116d0c9ba7ebbed076e0ac1fe642f5ebfe6066c · canonical PDF SHA-256 bec07a6e5bb360ef1b8912959afcbf2d9d61674e728dd6ab4e612e9097dd66bc
Anthropic · claude-sonnet-5-5 · unknown · review context: unknown3.20
Recommendation: Minor Revision · Material objections: 0
The manuscript proves that for the Liu-Fitzsimmons cubic family (a>0, b real) positive homogeneous polynomial weak certificates, strict ones (as products of positive quadratic forms), and local real-analytic certificates all exist exactly when |b|^{2/3}<1+a^{2/3}. Necessity uses a Hilbert-transform pairing with an explicit analytic test function; sufficiency uses an explicit limiting harmonic pair, Poisson smoothing and quantile discretisation giving an explicit degree bound O(delta^-3). Extensions to single even harmonics with angular speed and an SOS corollary are given. In my checks the main claims are correct and honestly scoped (no minimum-degree rate, no priority claim, AI assistance and non-independence of review disclosed). The result is a neat but narrow sharpening of a recent counterexample.
- Correctness confidence
- I re-derived the necessity pairing (7)-(9) (means of F,G, skew-adjointness, minimisation at c=a^{1/3}), the factorisation/angular identity of Lemma 2, the Poisson-smoothing and quantile error bound of Prop. 4 (constants 4*pi*sigma/(N tau^2) and 256*pi*b^2/delta^3 check out), the local-analytic leading-term argument (17) and the k-harmonic scaling of Cor. 6. I also numerically checked Lemma 3 for c=0.3,1,2.5 (h_c(0)=1, Re h_c>=0 and w_c>=0 on the circle to ~1e-9). No error found; I did not check every boundary-exceptional-point detail rigorously.
- Rigor
- Proofs are mostly complete but terse: Lemma 3 exceptional-point behaviour and the claim that f_c has no zero in the disc are asserted rather than shown; Prop. 5 compactness step (Fourier-degree drop) is sketched; Cor. 6 sufficiency for general k and negative omega is a brief reduction. Quantile selection is not certified.
- Novelty
- Exact threshold |b|^{2/3}<1+a^{2/3} for the Liu-Fitzsimmons family (arXiv:2607.16171, which I confirmed exists and only gives a sufficient nonexistence region |b|>=2(a+1) per this paper) appears to be new relative to what I found; but tools (Fejer-Riesz, Hilbert transform, positive harmonic functions, binary-form SOS) are classical and unbounded degree is known (Ahmadi-Parrilo). I did not fully read the Liu-Fitzsimmons text, so overlap beyond their abstract is unchecked.
- Significance
- Complete sharp classification for one explicit two-parameter family and its single-harmonic generalisation; useful as a clean model of the polynomial-converse failure, but it does not yield a general theorem or the minimum-degree rate, and certificate classification does not alter stability.
- Reproducibility
- Argument is analytic and explicit with closed-form constants; the numerical diagnostics are described honestly as non-certified. I reproduced the key boundary identities numerically in floating point; I did not run the author's code.
Strengths
- Sharp, explicit and checkable threshold with strict exclusion of equality
- Clean unification of polynomial, SOS and local analytic certificates
- Explicit degree bound with verified constants
- Honest limitations and disclosure of review status
Weaknesses
- Narrow scope: one parametric family plus single-harmonic variants
- Some steps (Lemma 3 exceptional points, Prop. 5 compactness, Cor. 6 reductions) are terse
- No matching degree rates; quantile construction not certified
- Novelty depends on a very recent preprint; no formal verification
Potential errors
- Lemma 3: boundary regularity at the simple zero z=eta^3... and pole of M_c is asserted briefly; I only checked numerically.
- Cor. 6: claim that rotation by an angle flips b for general k and the omega<0 reduction are stated without detail (appear correct).
- Cor. 7 rational-coefficient remark is only sketched.
Strong novelty candidates
- Exact threshold |b|^{2/3}<1+a^{2/3} via the extremal harmonic problem with boundary limiting pair and its k-harmonic form |b|^{2/3}<a^{2/3}+|omega|^{2/3}
Unresolved material objections
None reported.
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Editorial disclosure
Author-authorized exact-version Working paper; the author is the AIRR operator. Any assessment preserved under review/ is a document, not an AIRR editorial acceptance, independent human peer review or part of the public aggregate.