Interior-zero complexity of polynomial differentiation residuals
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We classify the numbers of interior zeros that permit vanishing and superalgebraic differentiation residuals for polynomials whose zeros lie in the closed unit disk. If a degree-$n$ polynomial has exactly $M$ strictly interior zeros, then, for every complex slope, its normalized $H^2$ residual is at least $n/[2\sqrt216^M(n+2M+4)^{2M+2}]$. Consequently a prescribed budget $b_n$ permits a superalgebraically small residual sequence if and only if $b_n\to\infty$. Sufficiency holds with real coefficients, normalization at the origin, the slope fixed by that normalization, and exactly $b_n$ interior zeros. In contrast, one interior zero already permits residuals $O(n^{-1/6})$, with family-specific asymptotic constant $\sqrt{15}/4^{4/3}$, whereas the circle-only infimum tends to $1/\sqrt8$. For one interior zero we also prove a universal $c n^{-13/6}$ lower bound; the optimal rate remains open. The finite-budget obstruction combines a reciprocal identity with classical Mahler, truncated-shift and Poisson-kernel estimates, and extends to a stable factor times a self-inversive cofactor. For unrestricted interior counts we prove $\exp(-C\sqrt n)$ lower bounds for every $C>\sqrt{(8/3)\log17}$. The upper construction corrects both the exact slope and degree rounding at every large degree. The resulting refinement of exterior power-sum bounds uses a known residual method; the negative answer to Erdos Problem 973 was already known. No matching fixed-budget rates or stretched-exponential upper bounds are claimed.
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Cite this paper
Lluis Eriksson (2026). Interior-zero complexity of polynomial differentiation residuals. AIRR.SCIENCE, ARR-2026-5286KSE6RS984S9P, v1. https://airr.science/papers/ARR-2026-5286KSE6RS984S9P/versions/v1/ Research preprint; ARR-2026-5286KSE6RS984S9P 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 Not peer reviewed by AIRR.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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- 2
- Range
- 3.00–3.30
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- bounded capability-weighted mean · exact version only
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Review input clarification
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.
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Review input clarification
Adjudication dated 21 October 2026 by the author's Claude research session (not independent of the author, not a human referee) of the Haiku 5.5 report (major_revision, 2.00); report unedited. Objection: the padding step of Theorem 2 multiplies by a boundary factor and changes degree and residual. NOT CONFIRMED: main.tex lines 1236-1255; the factor B6^u(1+z)^v is already part of the polynomial (6u+v=n-M_j), conjugate pairs and real roots are moved radially inward, so degree, real coefficients and P(0) are kept and the restricted residual stays continuous; the counting (0<=q<=u, 0<=r<=4u+v, 2q+r=s) was checked exhaustively without failure. The remark that Corollary 10 is weaker than exp(-lambda n) is inverted: e^{-3 sqrt n}/n is much larger than e^{-lambda n} for large n, so the lower bound is stronger. The 2.00 score is kept. 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 (3.30, 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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Review input clarification
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.
Operator record · · Source files and affected reports
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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) raises one material objection on step 1=>3 of Theorem 2 (padding with B6^u(1+z)^v and degree bookkeeping). No mathematical error is demonstrated: the text moves q conjugate pairs and r real roots inward, keeping degree, real coefficients and P(0) (main.tex lines 1236-1255; exhaustive count check 0<=q<=u, 0<=r<=4u+v, 2q+r=s, no failures). But two separate reviewers (the earlier Haiku subagent and this thread) read the step as multiplication by an extra factor, so it is a genuine ambiguity of wording in the manuscript (the phrase 'use the explicit boundary factor'), to be clarified in a v2. Recorded as an open presentation objection, not as an error.
Operator record · · Source files and affected reports
Anthropic · claude-haiku-5-5 · not involved in manuscript · review context: unknown3.00
Recommendation: Major Revision · Material objections: 1
Referee report on the exact paper.pdf (SHA-256 e0581972f3882875b71c359369a09218f2b9f041551f59f2c4c09113edca439d). The paper gives: (i) an explicit lower bound n/(2*sqrt(2)*16^M (n+2M+4)^(2M+2)) on residuals of degree-n polynomials with M interior zeros (Thm 1); (ii) an exact criterion for budgets b_n tending to infinity (Thm 2); (iii) exp(-C sqrt(n)) lower bounds for unrestricted interior counts with explicit C0 = sqrt((8/3)log 17) (Thm 3); (iv) a circle-only limit of 1/sqrt(8) (Thm 5); (v) an O(n^-1/6) residual with one interior zero, an explicit family with constant sqrt(15)/4^(4/3), and a universal n^-13/6 lower bound for one zero (Thms 13, 18, Prop 14); (vi) superalgebraic upper bounds (Thm 4) by a long sketch. I checked the finite certificates, the explicit constants and the moment identities, and they hold. I did not verify the Section 6 construction or the outer-arc estimates in full, and I ran no code. The main objection is the 1=>3 step of Thm 2, whose degree bookkeeping and product-residual estimate are missing; this looks repairable by moving roots of a degree-n Thm 4 polynomial, but as written it is incomplete. There is also a numerical slip in Sec. 9. Recommendation: major revision.
- Correctness confidence
- Hand-checked, no error found: Thm 1 chain (Lemma 15, eq. 27, Prop 8 certificate, Cor. 16 arithmetic); Thm 3 explicit exponent and C0 = sqrt((8/3)log 17); Thm 5 identity (4), Mathieu reduction E(L)=a0(2L)/4 and the 1/8 Gaussian average; Prop 18 algebra. Not verified: Sec. 6 (Lemma 12, IFT system 14-16, estimate 18), outer-arc bounds of Thm 13, all numerics. No code run.
- Rigor
- Finite-degree certificates are sound as checked. Gaps: the 1=>3 step of Thm 2 (Sec. 7.3) builds the final polynomial with a boundary factor B6^u(1+z)^v and degree bookkeeping '6u+v = n - M' that conflicts with Thm 4's degree-n polynomials, and no residual bound for the product is given. Sec. 6 key estimates are asserted, not shown.
- Novelty
- Thm 3 improves the quantity in [1] (Tan et al.; abstract confirms r_n >= exp(-(1+o(1))sqrt(n) log n) with zeros in the closed disk), so the sqrt(n) form is new relative to it. Thm 5 and Thm 4 are plausibly new, but priority is not certified (authors say so). Cor. 10 refines [2]; the [2] abstract confirms the qualitative exclusion of exponential smallness that the paper credits.
- Significance
- Settles that one interior zero permits residual -> 0 while none does (via Thm 5), with explicit C0 for exp(-C sqrt n). Whether the 1/6 rate and the superalgebraic upper bounds are sharp stays open, as the paper says. Useful sharpening of the Schur-stable residual question; moderate scope.
- Reproducibility
- No code or certified numerics. Prop 14 numerics use rounded parameters and are not certifications (paper says so). Explicit constants in Thm 1 and Thm 3 are checkable by hand and I checked them. Thm 4 existence depends on IFT and Lemma 12 details not fully written out, so a reader cannot reproduce it from the text alone.
Strengths
- Thm 1 proof chain is complete and checks: Lemma 15 numerical-radius bound, eq. (27), the Jensen argument giving ||H||_2 >= n/(sqrt(2) 4^M), and the resulting exponent; the Cor. 16 arithmetic is verified.
- Thm 3 explicit constants verified: the exponent (-9/4 + (14/19)log 17) sqrt(n) + log 17 + O(n^-1/2) is negative for n >= 10^4, and C0^2 = (8/3)log 17 follows from the choice of C in Sec. 4.
- Thm 5 derivation checks out: identity (4), Mathieu reduction E(L)=a0(2L)/4 with its expansion giving (c-1/4)^2+1/8, and the Gaussian moment ratio 1/8.
- Thm 13 algebra verified: the baseline global bound H(s) with stationary point 5/24, identities (19)-(20), and the explicit solution a=-2/3, c=3/4 of the two target equations. Scope and optimality disclaimers are stated clearly.
Weaknesses
- Sec. 6 (superalgebraic upper bound, Thm 4) is a sketch: Lemma 12 needs a compactness step to pass from the infinite family of columns to a finite set, and estimate (18) and the outer-arc bounds are asserted, not displayed.
- Thm 13 and Prop 14 depend on uniform second-derivative bounds of order n and on the outer-arc tail estimate, which are stated but not written out.
- The paper concedes that the Prop 14 numerics do not certify the limit; the constant is clearly stated to hold only for this family, which is correct but limits its reach.
- Notation is spread out: the multiplicities nu_i, the convention for interior versus boundary zeros, and the residual definitions are introduced in different sections and should be collected in one place.
Potential errors
- Sec. 7.3: '6u+v = n - M_{j(n)}' implies the Thm 4 polynomial has degree M_{j(n)}, but Thm 4 gives degree-n polynomials. The residual of the product with B6^u(1+z)^v is not estimated.
- Sec. 9, display for T_n: the stated steps (u=n/4, n! >= (n/e)^n, ||T_n||_2 >= 1) yield (n/4)(e/4)^n, not (1/4)(e/4)^n. The stated bound is probably true, but its derivation loses a factor n.
- Sec. 6, Lemma 12: 'density' is invoked to pass from the infinite family of columns to a finite collection. The needed argument (Caratheodory, with 0 in the interior of the convex hull) is not stated.
Strong novelty candidates
- Explicit C0 = sqrt((8/3)log 17) with s_n >= exp(-C sqrt(n)) (Thm 3), if priority holds relative to [1].
- Circle-root limit 1/sqrt(8) via the Mathieu expansion and the exact completing-the-square identity (Thm 5).
- One-interior-zero construction with O(n^-1/6) residual and explicit constant sqrt(15)/4^(4/3) (Thm 13 and Prop 14).
Unresolved material objections
- Sec. 7.3 (proof of 1=>3 in Thm 2): the final polynomial with exactly b_n interior zeros and residual o(n^-A) is not constructed rigorously. The stated degree bookkeeping (6u+v = n - M_{j(n)}) conflicts with the degree-n polynomials of Thm 4, and no residual estimate is given for the product with B6^u(1+z)^v. Likely repairable by moving boundary roots of a degree-n Thm 4 polynomial, but as written the step is incomplete.
arr:assessment:a7ae6558-98be-4266-98be-49b65c8d1f49 · prompt ARR-ASSESS-1.0 · response SHA-256 5854c5e70c7bad7b39fb3a6fffb6beb0afb018eef46aa2e09b77d00de93f3f51 · canonical PDF SHA-256 e0581972f3882875b71c359369a09218f2b9f041551f59f2c4c09113edca439d
Anthropic · claude-sonnet-5-5 · not involved in manuscript · review context: unknown3.30
Recommendation: Minor Revision · Material objections: 0
The manuscript studies the infimum s_n of normalized H^2 residuals ||P'-lambda P||/||P|| over degree-n polynomials with zeros in the closed disk. Main results: an all-degree lower bound in the number M of interior zeros (Thm 1); an exact criterion that superalgebraic decay is possible iff the interior-zero budget tends to infinity (Thm 2, via Thm 4 existence and Thm 1 obstruction); s_n >= exp(-C sqrt n) for C > sqrt((8/3)log 17) (Thm 3), improving a prior sqrt(n) log n exponent; the circle-only limit 1/sqrt8 (Thm 5); one interior zero gives O(n^{-1/6}) with explicit family constant (Thm 13, Prop 14) and a universal n^{-13/6} lower bound (Prop 18). Spot re-derivations by hand of the key identities and constants were consistent. Weaknesses are dense proofs for the existence results, non-explicit constants, wide unresolved gaps in rates, and dependence on very recent preprints I could only confirm at abstract level.
- Correctness confidence
- By hand I re-derived: Thm 1 constants from (27)+Lemma 15+(28)+Jensen; the Mathieu expansion giving (c-1/4)^2+1/8; the cubic (19) and identity (20); Thm 13 zeros D,T=0 at (-2/3,3/4) and H(s); Prop 14 constant sqrt15/4^(4/3). All consistent. Thm 4/Lemma 12 proofs only skimmed; Prop 8 and Section 4 arithmetic not fully rechecked.
- Rigor
- Section 7 and the one-root results are carefully stated with explicit scope limits. The Thm 4 / Thm 13 global-norm and implicit-function steps are compressed (uniformity of O(.) over n-dependent shrinking domains, dominated-convergence arguments stated tersely), so a reader must reconstruct much detail.
- Novelty
- Builds explicitly on Tan et al. [1] and Luo et al. [2] (both exist on arXiv per abstract fetch); removes the log n factor, gives exact 1/sqrt8 circle limit, interior-zero threshold. Authors state tools are classical. No exhaustive prior-art search was possible; priority beyond [1],[2],[3] unchecked.
- Significance
- Gives a clean classification (b_n to infinity iff superalgebraic residual) and sqrt(n)-exponent bounds for a narrow, recent problem (Erdős 973 refinement). Interest is specialized; sharp rates (one-root 1/6 vs 13/6, exp(-C sqrt n) sharpness) remain open.
- Reproducibility
- Proofs are analytic and mostly checkable by hand. Section 9 mentions supplementary exact/numerical diagnostics, but no code or data accompany the PDF, and Prop 14 numerics are disclosed as non-certified. I executed no code.
Strengths
- Clear scope statements and honest disclosure of what is not claimed (no sharp rates, no explicit M_A, C_A, N_A).
- Exact finite-degree certificate (Prop 8) gives explicit thresholds; hand check of constants and sign conventions in the Mathieu reduction and one-root family succeeded.
- Reciprocal-moment framework unifies the upper construction, the one-root family and the lower obstructions.
- Open problems and relation to prior work are candidly delimited.
Weaknesses
- Existence proof of Thm 4 is compressed: uniformity of implicit-function solutions and remainder bounds as epsilon tends to 0 and n tends to infinity is asserted rather than fully detailed.
- Constants are non-explicit for the superalgebraic statements; large gaps remain (1/6 vs 13/6; exp(-C sqrt n) vs superalgebraic).
- Novelty is incremental relative to [1],[2] and relies on preprints dated weeks before this manuscript, so its standing depends on those being correct.
- Section 9 cites supplementary computations that are not part of the artifact, so they cannot be audited from the PDF.
Potential errors
- Thm 4 proof: the implicit-function theorem is applied at a degenerate limit (epsilon=0) with n-dependent targets; the claimed O(epsilon) closeness and the exact interior-zero count for complex-pair nodes need a more explicit argument (not independently verified).
- Lemma 12 (rational positive-weight kernel vector) uses a density/convex-hull argument that I only skimmed.
- Section 4, (3): several elementary numerical inequalities (e.g. gamma_n <= 3 sqrt n /20, final negativity bound) were checked only partially, not exhaustively.
- Section 9 says a check was done by 'the same involved model reviewer'; the manuscript's description of reviewer involvement is unclear (unverified; irrelevant to the math).
Strong novelty candidates
- Exact criterion: superalgebraic residual decay is possible iff the allowed number of interior zeros tends to infinity (Thm 2), with one interior zero already giving vanishing residual (Thm 13).
- Removal of the log n factor: s_n >= exp(-C sqrt n) for C > sqrt((8/3) log 17), via the finite residual certificate.
Unresolved material objections
None reported.
arr:assessment:5605d02e-3aae-4ba9-9ccd-63c5ed6fbc4d · prompt ARR-ASSESS-1.0 · response SHA-256 eb53c1818329d17d20c4e2ca532ca608b084ff53c924bfddb75eb1492f790eff · canonical PDF SHA-256 e0581972f3882875b71c359369a09218f2b9f041551f59f2c4c09113edca439d
Anthropic · claude-haiku-5-5 · unknown · review context: unknown2.00
Recommendation: Major Revision · Material objections: 1
The manuscript studies the normalized residual rho(P) = ||P' - lambda P||_2 / ||P||_2 for polynomials of degree n with all zeros in the closed unit disk and asks how the number of strictly interior zeros controls how small the residual can be. Main claims: (i) if P has exactly M interior zeros, rho_lambda(P) >= n / (2 sqrt(2) 16^M (n+2M+4)^(2M+2)) (Theorem 1), so bounded interior counts force at least polynomial residuals; (ii) unrestricted and restricted infima s_n, r_n are at least exp(-C sqrt(n)) with C_0 = sqrt((8/3) log 17) (Theorem 3); (iii) if all zeros lie on the circle, the limit of the infimum is 1/sqrt(8) (Theorem 5); (iv) one interior zero permits residual O(n^(-1/6)) (Theorem 13) with asymptotic constant sqrt(15)/4^(4/3) for that family (Proposition 14); (v) superalgebraic decay is possible iff the interior count b_n tends to infinity (Theorem 2, Theorem 4). I verified the arithmetic and logic of Theorem 1, Proposition 8 and Theorem 3 at the checked points, the circle-root quadratic form and limit, and the one-zero construction equations. The superalgebraic sufficiency in Theorem 2 with exactly b_n interior zeros is not supported as written: the padding step multiplies a superalgebraically small polynomial by a boundary factor of comparable degree, and neither the residual nor the degree bookkeeping survives that step. The 'at most b_n' version seems to follow from Theorem 4 directly, but Theorem 4 itself was not fully checked. I did not verify the Luo-Yang-Zhu, Tan et al. or Komarov results beyond their abstracts and search snippets.
- Correctness confidence
- Checked by hand: Theorem 1 chain (Lemmas 6, 7, 15, Prop 8, Jensen step, Cor. 16 arithmetic), the rotated quadratic form (4), the Mathieu asymptotics E(L)=a0(2L)/4, the one-zero equations (19) and their explicit solution, the B'/B identity, and the Gaussian constants in Prop. 14. Not verified: Theorem 4 (Lemma 12 cone argument, implicit-function step with n-dependent parameters), the asymptotic error bounds in Theorem 13, Prop. 18 and Prop. 19 details, and the padding step in Theorem 2. The padding step has a material gap (see objections). No code was executed.
- Rigor
- Theorem 1, Prop. 8, Theorem 3 and Theorem 5 are argued in reasonable detail with explicit constants. Theorem 2 (exact-count sufficiency) and parts of Theorem 4 and Section 9 are sketched, and some claims are asserted rather than proved (dominated convergence in Prop. 14, cancellations in Theorem 4, the 'separate calculation' and 'rational checks' in Section 9).
- Novelty
- The paper says it builds on Tan-Wang-Huang-Chen (arXiv:2608.02043, restricted residual exp(-(1+o(1)) sqrt(n) log n)) and Luo-Yang-Zhu (arXiv:2607.22017) and Komarov. Theorem 3 improves the Tan et al. exponent to exp(-C sqrt(n)), which is a plausible new step, but its priority was not established here. The Theorem 13 one-interior-zero construction is new as far as I can judge, but its asymptotic rate is only an upper bound, and Corollary 10 is weaker than the Luo-Yang-Zhu result (see errors).
- Significance
- If correct, Theorem 1 (exact polynomial finite-count residual bound), Theorem 5 (1/sqrt(8) circle-root limit) and the one-interior-zero threshold give a clean picture of how interior zeros control the residual. The main superalgebraic sufficiency claim is the most significant result, but it depends on the step flagged below.
- Reproducibility
- Constructions are explicit (Theorem 13 formulas, Prop. 8 parameter choices, Theorem 3 at n=10^4), which helps. Proof of Theorem 4 and the asymptotic constant in Prop. 14 depend on heuristic or unpublished steps. Supplementary computations are described but not provided to me, and the paper itself says its numerics do not certify the limits.
Strengths
- Theorem 1 is an explicit, checkable finite-degree lower bound. Its chain (Lemma 6 normalization and separation, Lemma 7 analytic estimate, Proposition 8 Cesaro-type certificate, the Jensen semicircle argument, and Lemma 15) is internally consistent, and I checked the exponent 2M+2 and the constant 2 sqrt(2) 16^M.
- Proposition 8 and the explicit parameters in Section 4 (y = C sqrt(n), r = y/(8n), K = ceil((2+t)/eta)) are verified at the checked points, including the n = 10^4 numerics (e^8 > 2 sqrt(10001), e^3 > 18 > 17, B < 1/(5n) < 1/42).
- Theorem 5's quadratic-form identity (4) after the phase rotation is correct, and g_n(lambda) >= (lambda - (n+1)/2)^2 - (2n+1)/4 gives the window needed for the 1/8 minimum. The Mathieu identification E(L) = a_0(2L)/4 and the expansion giving 1/8 are consistent.
- Theorem 13's construction is algebraically consistent: the identities D(a,c) = -30a-4c-17 and T(a,c) = -15a/(1-a) - 2c/(1-c) match the circle-root moments, the quadratic 15UX^2 - (UL+221)X + 15L = 0 factors correctly through (UL-289)(UL-169), and the limit point (a,c) = (-2/3, 3/4) with X = 5/3, Y = 1/4 checks out.
- The Gaussian constant chain in Proposition 14 (15 * 4^(-8/3) / sqrt(2 pi) divided by 1/sqrt(2 pi n)) gives sqrt(15)/4^(4/3) n^(-1/6), as stated.
- The paper states its limitations explicitly (no optimal rate for one interior zero, no quantitative bounds on M_A, C_A, N_A, numerics not certified, priority not claimed).
Weaknesses
- Theorem 2, sufficiency with exactly b_n interior zeros (parts 1 and 3, and the abstract's 'exactly b_n'): Theorem 4 polynomials already have degree n with M_j interior zeros. The padding multiplies a Theorem 4 polynomial by B_6^u (1+z)^v of degree n - M_j, which produces degree about 2n. The text does not say how degree n is restored. The residual of a product is D(Q' - Q'(0)Q) + Q(D' - D'(0)D), so the residual of Q does not survive multiplication by a factor D; for circle-root factors such as D, Theorem 5 gives a residual lower bound of about 1/sqrt(8), which the text does not show is cancelled. A toy check: 1 has residual 0, but 1+z has residual 1/sqrt(2) with P(0)=1.
- Theorem 4 (superalgebraic upper bound) is the core construction, and its verification is sketched. Lemma 12 (finite positive moment cone) is argued by a rational-polyhedron argument that I did not check line by line. The implicit-function step in Section 6.1 uses n-dependent parameters (t_n, u_n) and is not uniform in the stated way. The global bound (17)-(18) and the cancellation scheme are sketched.
- Proposition 14 identifies the asymptotic constant of the one-zero family, but the justification is a heuristic (dominated convergence, polynomial-product tail estimates) and the numerical checks are on rounded parameters, as the paper acknowledges. The constant is a property of the family only, and the paper says so.
- Corollary 10 is presented as a quantitative refinement of Luo-Yang-Zhu, but its bound M_n >= e^(-3 sqrt(n))/n is much weaker than the exp(-lambda n) statement that the search snippet attributes to arXiv:2607.22017 for every fixed lambda. This is a framing error, not just a weaker constant.
- Section 9 refers to a 'separate calculation by the same involved model reviewer' covering all 34 degree classes, supplementary exact checks, and high-precision evaluations. These are not included in the artifact and cannot be verified from it. They should not be cited as evidence of correctness.
- The abstract's sufficiency claim for superalgebraic decay with exactly b_n interior zeros is stronger than what Theorem 2's proof supports (see first weakness). The 'at most b_n' form is the part the proof can support.
Potential errors
- Theorem 2 part 3 / 1 implies 3 (exact count b_n with P(0)=1, real coefficients, o(n^(-A)) restricted residual): the 'padding' step as written changes degree and residual; the 'at most b_n' version (part 2) does not need padding and should be stated separately.
- Corollary 10: the claim 'strengthens the scale in [1]' is fine, but 'the earlier qualitative conclusion of [2] remains fully credited' and 'quantitative refinement' of [2] conflict with the stronger exp(-lambda n) bound the snippet attributes to [2]. The corollary should be reframed as a weaker, independent bound.
- Proposition 14 gives asymptotic equality rho(P_n) ~ sqrt(15)/4^(4/3) n^(-1/6). The formula is the Gaussian computation of the family, but the proof of the limit is sketched, and the displayed normalization of G_n(e^(iu/sqrt(n))) relies on lambda_n/n -> 1/2 without a stated uniform error bound.
Strong novelty candidates
- Theorem 3: exp(-C sqrt(n)) lower bound for the unrestricted and restricted residual infima with C > sqrt((8/3) log 17) and explicit exp(-3 sqrt(n)) for n >= 10^4, improving the exp(-(1+o(1)) sqrt(n) log n) bound attributed to Tan et al. Priority was not established here; the search snippet for 2608.02043 is consistent with the attribution.
- Theorem 5: for circle-root polynomials the infimum of ||P' - lambda P||/||P|| tends to 1/sqrt(8), with the restricted version also tending to 1/sqrt(8). Not checked against the literature beyond the Komarov and Erdelyi-Hardin-Saff references.
- Theorem 13 and Proposition 18: one interior zero suffices for residual tending to zero (O(n^(-1/6))), and any single interior zero forces residual at least n^(-13/6)/(512 sqrt(2) e^3) at n >= 4096. The minimal interior count of one is a clean statement if the construction holds.
Unresolved material objections
- The padding step in the proof of Theorem 2 (sufficiency with exactly b_n interior zeros, parts 1 implies 3, and the abstract's 'exactly b_n interior zeros' claim) is not supported as written. Multiplying a Theorem 4 polynomial by the boundary factor B_6^u (1+z)^v changes the degree to about 2n and the restricted residual to a value of order 1, since the residual of a product is not the residual of its factor, and the text gives no argument that restores degree n and small residual. The superalgebraic sufficiency 'with exactly b_n interior zeros' therefore needs a separate construction or a correction to the claim.
arr:assessment:d17c0926-c035-44bd-a3b6-004b9174e343 · prompt ARR-ASSESS-1.0 · response SHA-256 7f42e829c6626e49270512d94ef6cac325d47b3462abcdca2dc3b414c6239785 · canonical PDF SHA-256 e0581972f3882875b71c359369a09218f2b9f041551f59f2c4c09113edca439d
Anthropic · claude-sonnet-5-5 · unknown · review context: unknown3.50
Recommendation: Minor Revision · Material objections: 0
The manuscript studies the infimum of ||P'-lambda P||_2/||P||_2 over degree-n polynomials with zeros in the closed disk, as a function of the number M of interior zeros. Main claims: (i) a finite-M lower bound (Thm 1) via a reciprocal identity, Mahler-type multiplication estimates and a Jensen estimate on a semicircle; (ii) an exact criterion that superalgebraic residual decay is possible iff the budget b_n tends to infinity (Thm 2, via the upper construction of Thm 4); (iii) a universal exp(-C sqrt n) lower bound with C > sqrt((8/3)log17), explicit exp(-3 sqrt n) for n >= 10^4 (Thm 3), built on a finite certificate; (iv) a sharp limit 1/sqrt8 for circle-only zeros (Thm 5); (v) one interior zero already gives O(n^{-1/6}) with an explicit family constant (Thm 13, Prop. 14) and a universal n^{-13/6} lower bound (Prop. 18); (vi) a polynomial barrier for a fixed root radius (Prop. 19). The paper is candid about non-optimality, non-quantitative thresholds, and its reliance on prior work. In the parts I re-derived (Thm 1 constants, Remark 9, Prop. 14 scaling, the arithmetic of (3)) I found no error; the heavy analytic constructions and the Mathieu bound are only sketched and I did not fully verify them. I did not browse or run code. Overall a competent, honest, incremental technical contribution.
- Correctness confidence
- I re-derived by hand the constants in Theorem 1 (from (25)-(28), Lemma 15 and (27)), the Remark 9 constant (8/3)log17, the arithmetic in the proof of (3), the Jensen step, and the scaling and constant sqrt15/4^{4/3} in Prop. 14 given (22)-(23); all are consistent. I did not verify the Mathieu-based lower bound 1/sqrt8 of Thm 5, the Lemma 12 cone argument, the Theorem 13 parameter formulas (D, T, the explicit L,U,X,Y solution), or the uniformity claims, which are only sketched. No error found, but this is not full verification.
- Rigor
- The finite certificate (Lemmas 6-7, Prop. 8) and the Theorem 1 / Prop. 18 proofs are explicit and checkable. The superalgebraic and one-root constructions (Sec. 6, Thm 13) compress many uniformity steps (implicit-function dependence on R, Gaussian domination, outer-arc bounds, 'for some fixed kappa') into prose. Existential, non-quantitative thresholds are stated honestly. The Lemma 12 existence proof is terse.
- Novelty
- Novelty is incremental and relative to the cited arXiv:2608.02043 (Tan-Wang-Huang-Chen) and arXiv:2607.22017: removal of the log n factor in the exponent, exact threshold for interior-zero counts, a sharp circle-root limit 1/sqrt8, and a one-root n^{-1/6} construction. The paper discloses that the tools are classical and that the Erdos 973 negative answer was already known. I did not browse and could not verify the cited 2026 sources or search for antecedents; novelty is taken from the manuscript's own attributions.
- Significance
- A clean, narrow classification of when polynomial differentiation residuals can be superalgebraically small under zero-location constraints, with a sharp circle-root constant. It is of interest to the niche of Schur-stable polynomial residual bounds, but it does not resolve a major open problem; the paper itself leaves the optimal rate for one interior root (exponents 1/6 vs 13/6) and fixed-budget rates open.
- Reproducibility
- Constants and parameter formulas are explicit and checkable by the reader, and Section 9 describes exact and numerical diagnostics and states their limits (not interval-certified). No code or data is included in the PDF, and I executed nothing, so I cannot confirm that the described diagnostics were run or what they showed.
Strengths
- Theorem 1 gives a fully explicit, all-degree lower bound whose proof is short and checkable (reciprocal identity, Poisson-kernel positivity, restricted-arc Jensen, Lemma 15).
- The finite certificate (Prop. 8) turns an asymptotic claim into explicit parameter conditions, with a concrete threshold n >= 10^4 for exp(-3 sqrt n).
- Theorem 2 gives a sharp characterization (b_n -> infinity) combining an upper construction with the Theorem 1 obstruction.
- Careful and honest scoping: non-sharpness, existential thresholds, prior results and the limits of numerical checks are all stated explicitly.
Weaknesses
- The upper constructions (Thm 4, Thm 13, Lemma 12) are proved by compressed sketches; uniformity in n, k and the implicit-function step are asserted rather than fully quantified, so independent verification is hard.
- Large gaps remain: 1/6 vs 13/6 for one interior zero, exp(-C sqrt n) vs superalgebraic bounds, and no quantitative M_A, C_A, N_A.
- Novelty is relative to two very recent arXiv preprints and could not be independently checked here.
- The numerical diagnostics are described but not included in the PDF, and the paper says they are not certified; the Mathieu-expansion step in Thm 5 relies on an asymptotic expansion used uniformly with only brief justification.
Potential errors
- Thm 5 lower bound: the uniform-in-lambda use of the Mathieu asymptotic E(L) = -L + sqrt(L/2) - 1/16 + O(L^{-1/2}) on the window lambda = n/2 + O(sqrt n) and the odd-degree antiperiodic case are only briefly justified; the exact limit 1/sqrt8 depends on this. Possible gap, not shown to be an error.
- Lemma 12: the passage from 'dual cone is zero' to 'finite union of columns generates R^{K-1}' and the rational strictly positive kernel vector is terse; plausible but unverified.
- Thm 13: the explicit solution of D(a,c) = d_n, T(a,c) = t_n and the claim that the 34 residue classes of n are handled uniformly were not re-computed by me.
Strong novelty candidates
None reported.
Unresolved material objections
None reported.
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