A sharp circle-root residual barrier and its collapse with one interior zero
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
For degree-$n$ polynomials with all zeros on the unit circle, the least normalized residual $\|P'-\lambda P\|_{H^2}/\|P\|_{H^2}$ tends to $1/\sqrt8$, both for arbitrary complex $\lambda$ and for the restricted slope $\lambda=P'(0)$, $P(0)=1$. We prove the sharp lower limit through the classical Mathieu ground state and give an explicit circle-root family attaining it in every large degree. In contrast, exactly one zero strictly inside the disk, with all remaining zeros on the circle, permits restricted residual $O(n^{-1/6})$, giving the exact zero-count threshold for collapse. More generally, for each fixed $A>0$, a fixed number $M_A$ of interior zeros permits $O(n^{-A})$. Their distance and radial gap from 1 have order $n^{-1/3}$; explicit algebraic parameters in the one-root case, and an implicit-function construction in general, correct both degree rounding and the exact slope. For all disk-root polynomials and arbitrary slope, we also prove a uniform lower bound $\exp(-C\sqrt n)$ for every $C>\sqrt{(8/3)\log17}$, explicitly $\exp(-3\sqrt n)$ for $n\ge10^4$. This refines the known residual method for exterior power sums; the negative answer to Erdos Problem 973 was already known. The optimal disk and one-root decay rates, and small exterior power-sum constructions, remain unresolved.
Preview PDF on this page v1
Uses your browser's PDF viewer. Open the PDF directly if a preview is unavailable.
Cite this paper
Lluis Eriksson (2026). A sharp circle-root residual barrier and its collapse with one interior zero. AIRR.SCIENCE, ARR-2026-6TQRG7PY6C8JZR2D, v1. https://airr.science/papers/ARR-2026-6TQRG7PY6C8JZR2D/versions/v1/ Research preprint; ARR-2026-6TQRG7PY6C8JZR2D v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
Use this version link to cite the manuscript you read. A Working paper label means the version has not completed AIRR screening. AIRR screening is not journal peer review. Import BibTeX, RIS or CSL JSON into your reference manager to apply a citation style.
Version history
The paper identifier remains stable. Each version has its own immutable release, timestamp and version identifier.
- v1 · source snapshot available · viewing
Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Not assessed
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
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.
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
- Distinct models
- 2
- Range
- 3.40–3.40
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/2 benchmark matches. This is an evidence description, not a probability of correctness.
One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete; Adverse reports or unresolved objections require editorial disposition
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. 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.
Operator record · · Source files and affected reports
Review input clarification
Adjudication dated 21 October 2026 by the author's Claude research session (not independent of the author, not a human referee); reports unedited. Haiku 5.5 (repeated, major_revision, 3.00) raises one material objection: Theorems 2 and 11 depend on uniform estimates ((16), (17), outer-arc bounds) and on the Lemma 10 cone argument, asserted rather than proved. NOT resolved and NOT refuted: it remains an open objection (a presentation weakness or sketched proof also noted by Sonnet 5.5, 3.40, minor_revision, without a formal objection), not a demonstrated error. Its minor remark on Lemma 10 ('a_l real') is NOT a flaw: the functional lives in R^(K-1) because the columns V(zeta) are real (main.tex lines 596-621), so the a_j are real by construction. An earlier slip claim (B'/B-3 without z) from the rejected first attempt was NOT confirmed: main.tex line 245 carries z. Revisable.
Operator record · · Source files and affected reports
Review input clarification
Independent-thread note, 21 October 2026, by the author's Claude research session (not independent of the author). Claude Sonnet 5.5 report (3.40, 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.
Operator record · · Source files and affected reports
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 (3.40, 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
Review input clarification
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, 3.40) raises one material objection: Theorem 2 (superalgebraic upper bound with fixed M_A) asserts uniform estimates (outer arc, estimate (16), Lemma 10 weights) without writing them. NOT resolved and NOT refuted: it remains an open objection, a weakness of a sketched proof to be completed in a v2 if it can be proved, not a demonstrated error. Revisable.
Operator record · · Source files and affected reports
Anthropic · claude-haiku-5-5 · not involved in manuscript · review context: unknown3.40
Recommendation: Major Revision · Material objections: 1
Polynomials of degree n with all zeros in the closed unit disk; residual rho(P) = ||P' - P'(0)P||_2 / ||P||_2 (unsquared). Claims: (i) for circle-root polynomials the least residual tends to 1/sqrt(8), via a Mathieu ground-state lower bound and an explicit family in every large degree; (ii) for disk-root polynomials and arbitrary slope, s_n >= exp(-C sqrt(n)) for C > sqrt((8/3) log 17), and s_n >= exp(-3 sqrt(n)) for n >= 10^4, with corollaries for exterior power sums; (iii) one interior zero permits rho = O(n^(-1/6)) by an explicit construction; (iv) for fixed A, a fixed number M_A of interior zeros permits O(n^(-A)), without effective constants. Checks: SHA-256 matched; full text read via pdftotext and page images; hand checks of Lemmas 4-5, Proposition 6, parameters for (1) and (2), Corollaries 8-9, the Theorem 3 lower-bound arithmetic (Mathieu expansion as recalled, not fetched) and the Theorem 11 algebra. Float64 checks: the circle-root family gives a squared restricted residual of 0.12548 at n=1200 (k=0); the one-root construction gives P'(0)=P'(1)/P(1) to six digits for n=35 to 1120, with squared residual falling from 0.673 (n=70) to 0.0596 (n=1120). Not verified: Section 6 proof of Theorem 2, Lemma 10, Section 8 supplementary diagnostics (not supplied), and literature beyond search snippets. Recommendation: major revision, mainly because the Theorem 2 proof cannot be checked from the text.
- Correctness confidence
- Checked by hand: Lemma 4, Lemma 5, Proposition 6 chain, parameter choice and arithmetic for (1) and (2), Corollaries 8-9, the Theorem 3 lower-bound algebra and Theorem 11 algebra. Section 6 (Theorem 2) is only sketched and was not verified.
- Rigor
- Theorems 1, 3 and 11 are argued in enough detail to follow and check. Theorem 2 depends on uniform derivative and global estimates that are asserted rather than shown, and Lemma 10 is a sketch. Explicit constants are given for (1) and (2).
- Novelty
- Theorem 1 sharpens Tan et al. (r_n >= exp(-(1+o(1)) sqrt(n) log n)) by removing the log factor, reusing their normalization and Newton identity with attribution. The 1/sqrt(8) circle-root limit and the one-interior-zero threshold are new relative to the cited work. Literature checked only via search snippets.
- Significance
- Settles the circle-root limit and the zero-count threshold for a narrow extremal residual problem. The negative answer to Erdos Problem 973 is credited to Luo-Yang-Zhu and Tan et al.; the exterior power-sum corollaries are quantitative refinements.
- Reproducibility
- Explicit circle-root family and one-root formulas are complete. Float64 recomputation matched the Theorem 3 family (squared restricted ratio 0.1255 at n=1200, k=0) and the Theorem 11 slope identity to six digits. Supplementary diagnostics and audit cited in Section 8 are not included.
Strengths
- Theorem 1 proof chain checks: Lemma 4 normalization and separation, Lemma 5 Cauchy bound, the Proposition 6 Cesaro test, the choice giving C0 = sqrt((8/3) log 17), and the explicit arithmetic for n >= 10^4.
- Theorem 3 upper bound is confirmed numerically: P'_{m,k}(0) = n/2 + 1/4 exactly, and the squared restricted residual tends to 1/8 (0.12548 at n=1200, k=0). The lower-bound expansion gives (c-1/4)^2 + 1/8 exactly, consistent with a_0(q) = -2q + 2 sqrt(q) - 1/4.
- Theorem 11 explicit one-root family: identities (18)-(19), the slope and second-moment formulas, and the quadratic branch limit X -> 5/3, Y -> 1/4 check by hand. Numerically P'(0) = P'(1)/P(1) to six digits for n=35 to 1120.
- Claims are scoped honestly: non-effective constants for Theorem 2, unresolved optimal rates, and explicit credit to Tan-Wang-Huang-Chen and Luo-Yang-Zhu.
Weaknesses
- Section 6 (Theorem 2) is a sketch. The outer-arc derivative bounds, the global estimate (16) and the Lemma 10 positivity argument are asserted in a line or two each, so the superalgebraic claim cannot be checked from the text.
- Novelty over Tan et al. [1] is incremental: Theorem 1 keeps their maximum-point normalization and Newton identity and removes one log factor. The negative answer to Erdos Problem 973 is credited to prior work.
- The residual convention changes between sections. The abstract and Theorem 3 use the unsquared ratio (limit 1/sqrt(8)), but the proofs of Theorem 3 and Section 6 work with squared quantities. The square-root step is not written out.
- Supplementary diagnostics and the independent audit of all 34 degree classes (Section 8) are cited but not included, so they could not be checked. Numerically, the one-root squared residual decays faster than n^(-1/3) over n=560 to 1120, so the stated bound is not tight, which the paper does not claim.
Potential errors
- Theorem 11 is stated as rho = O(n^(-1/6)), while its proof concludes rho^2 = O(n^(-1/3)). These are consistent, but the paper should state which exponent is the claim, and the abstract should match.
- Section 2 gives the squared expansion (3) for the squared residual, and Theorem 3 is stated for the unsquared ratio. The passage from g_n to the limit 1/sqrt(8) needs an explicit square-root step.
Strong novelty candidates
- Exact limit 1/sqrt(8) for the least normalized residual over circle-root polynomials, with an explicit family attaining it in every large degree (Theorem 3).
- One interior zero suffices for a polynomial-rate residual, via an explicit algebraic construction, so one interior root is the minimal count for vanishing residual (Theorem 11).
Unresolved material objections
- Theorem 2 (superalgebraic upper bound with fixed M_A): Section 6 asserts uniform bounds without full estimates. These include the outer-arc derivative terms being polynomial in n times a Gaussian bound, the global estimate (16) with a uniform kappa, and the positivity of the weights from Lemma 10. Unless these are verified to be uniform in n for fixed K, the claim O(n^(-A)) with fixed M_A is unsupported. The reviewer could not check this from the text.
arr:assessment:6a614f0f-5b14-4ecf-a0bd-0541fd585c45 · prompt ARR-ASSESS-1.0 · response SHA-256 3839009ed23d8849c223c9e9a5ee718cf7ceb09ab89753bd293f82f85ddc3652 · canonical PDF SHA-256 28ed664ca672ed1cdf776969c2872dc65b4e491c3d81eec32e615efeaafe097e
Anthropic · claude-sonnet-5-5 · not involved in manuscript · review context: unknown3.40
Recommendation: Minor Revision · Material objections: 0
The manuscript studies the minimal normalized residual ||P'-lambda P||_2/||P||_2 over degree-n polynomials with zeros in the closed disk. It proves (i) a sharp limit 1/sqrt8 when all zeros are on the circle, via a Mathieu ground-state bound plus an explicit family; (ii) a lower bound exp(-C sqrt n) for every C>sqrt((8/3)log17), with explicit exp(-3 sqrt n) for n>=10^4, from a finite certificate refining Tan-Wang-Huang-Chen [1]; (iii) superalgebraic upper bounds with a fixed number of interior zeros at distance ~n^(-1/3), and collapse with exactly one interior zero (rate O(n^(-1/6))); (iv) a polynomial lower barrier if the root radius is fixed below 1. The finite-certificate chain and constants I recomputed hold. The asymptotic constructions (Sections 2, 6, 11) are argued compactly and I only checked them for plausibility. Claims of priority are properly hedged. External citations (arXiv 2026 items, Erdos 973) were not looked up by me.
- Correctness confidence
- I re-derived by hand: the Gaussian-moment limit 1/8 in Thm 3, Lemma 4/5 and Prop 6 (Cesaro test, remainder bound), and the explicit n>=10^4 arithmetic in (12) (e.g. -9/4+42/19=-3/76; 1/2-10/21=1/42). All consistent. Thm 2, Lemma 10, Thm 11 and the Mathieu uniformity step are terse and I only checked them for plausibility, not line by line.
- Rigor
- Finite certificate (Lemmas 4-5, Prop 6) and the explicit constant are clean and checkable. Sections 2 and 6 and the proof of Thm 11 compress key steps into 'uniformly bounded', 'by compactness', 'a polynomial times a Gaussian dominates'; the Mathieu expansion uniformity over the window n/2+c rests on cited asymptotics. Quantifiers are mostly explicit; non-effectivity of M_A, C_A, N_A is honestly stated.
- Novelty
- Removes the log n factor from the exponent of the cited Tan-Wang-Huang-Chen bound (via a local analytic estimate), gives the sharp 1/sqrt8 circle-root barrier, and a one-interior-zero threshold. Techniques build heavily on [1],[2]. The author discloses that a targeted search did not exhibit these bounds, and that priority is not certified; I did not browse and did not check these references.
- Significance
- Sharpens a quantitative refinement of a known negative answer (Erdos 973, per [2]) and gives a clean dichotomy: circle roots give a constant barrier, one interior root collapses it. Niche subject; gaps between the disk lower bound exp(-C sqrt n) and the superalgebraic upper bound remain, and the optimal rates are explicitly left open.
- Reproducibility
- The explicit constants (n>=10^4) and the one-root parameter formulas can be verified by hand, and I checked part of them. The paper cites supplementary symbolic/numerical audits (Gaussian-rational Newton identities, 34 degree classes), but they are not in the PDF, so I could not inspect or run them. Theorem 2 relies on a non-explicit implicit-function construction.
Strengths
- Clear scope statements: what is not claimed (optimal rates, finite extremizers, constants optimality) is listed in Sections 1 and 8.
- Finite certificate (Prop 6) is transparent and the explicit n>=10^4 constants check out in my hand recomputation.
- Sharp dichotomy between all-circle zeros (constant barrier 1/sqrt8) and one interior zero (vanishing residual), with an exact zero-count threshold.
- Explicit parameter formulas for the one-interior-zero family (cubic (18), circle-parameter solution) make that part of the construction checkable.
- Candid disclosure of AI assistance and of the limits of the verification and priority search.
Weaknesses
- Proofs of Theorem 3 (upper bound), Lemma 10 and Theorem 2 are summary-level; several uniformity claims (secondary maximum bound for H_c(s), Jacobian/IFT neighbourhoods, outer-arc estimates) are asserted without displayed computations.
- The supplementary computations cited in Section 8 are not part of the artifact and cannot be assessed from the PDF.
- The superalgebraic upper bound is non-effective (no bounds on M_A, C_A, N_A) and leaves a large gap to the exp(-C sqrt n) lower bound; the optimal rates remain open.
- Novelty relative to [1] and [2] depends on arXiv items I did not consult; the improvement is a log-factor removal plus an arbitrary slope.
- Theorem 3's lower-bound step relies on uniform Mathieu asymptotics over a growing window, cited from DLMF and not re-proved.
Potential errors
- Mathieu uniformity: the expansion E(L) = -L + sqrt(L/2) - 1/16 + O(L^(-1/2)) must hold with a uniform remainder for L=(n+1)lambda over the whole minimizing window; the paper asserts this but I did not verify it.
- Lemma 10 proof (rational conical combination, kernel vector strictly positive) is sketched; a non-degeneracy or full-rank detail could fail for particular node choices.
- Theorem 11: uniformity over the 34 residue classes k, and the claim that the discriminant and denominators stay positive with a_n, c_n in (-1,1), are stated rather than shown.
- Remark 7 gives a general sufficient condition C^2>2log(1+2/s)/(1-2s) without proof beyond 'same proof'; I did not rederive it (s=1/8 gives C0^2=(8/3)log17 and is consistent).
- Section 6: the 'unique global maximum / modulus gap' claim for the perturbed degree-6 factor is argued by local curvature plus compactness, not by explicit computation.
Strong novelty candidates
- The sharp circle-root constant 1/sqrt8 for the global residual, through the Mathieu ground state.
- The exact zero-count threshold: no collapse with zero interior zeros, collapse with exactly one, residual O(n^(-1/6)).
- Removing the log factor in the sqrt(n) exponent of the Tan-Wang-Huang-Chen lower bound, for an arbitrary scalar slope.
Unresolved material objections
None reported.
arr:assessment:cb10157a-3110-441f-a50e-9b47a1425eef · prompt ARR-ASSESS-1.0 · response SHA-256 5540e528284a9e4f16e350b9abbf98617589f8eb43e022bd4237ec5130508aec · canonical PDF SHA-256 28ed664ca672ed1cdf776969c2872dc65b4e491c3d81eec32e615efeaafe097e
Anthropic · claude-haiku-5-5 · unknown · review context: unknown3.00
Recommendation: Major Revision · Material objections: 1
Exact manuscript (SHA-256 verified); reviewer read all 14 pages. Main results: (Thm 1) s_n >= exp(-C sqrt n) for C > sqrt((8/3) log 17), explicitly exp(-3 sqrt n) for n >= 10^4; (Thm 2) superalgebraic decay of the restricted residual with a fixed number M_A of interior zeros; (Thm 3) the circle-root residual tends to 1/sqrt 8 via a Mathieu ground-state lower bound; (Thm 11) one interior zero gives residual O(n^{-1/6}). Checked in detail: Lemma 4 normalization, the Cesaro chain of Prop 6, the parameter choice and limit in Theorem 1, the Mathieu reduction and the value 1/8, Prop 12, Corollaries 8 and 9, and the Theorem 11 baseline identities (b_2 = 0, b_3 = -3/2, unique interior stationary point 5/24 of H(s), moment identity M_3 = (3/2)M_2 - N/4). No error found in these. Reference claims about Tan et al. and Luo et al. were checked only through search snippets, not the full papers. The reviewer did not execute code and did not verify the claimed audit. The main gap is Section 6, where uniform estimates behind Theorem 2 and Theorem 11 are asserted rather than written out; this is unresolved, hence major revision.
- Correctness confidence
- Checked in detail: Lemma 4 (Re v >= 1/2, |v|^2 <= Re v/eta), the Cesaro chain of Prop 6, the Theorem 1 parameter choice and limit (explicit n >= 10^4 case), the Mathieu reduction giving inf g_n -> 1/8, Prop 12 and Cor 8/9. No error found. Section 6 estimates only sketched and not verified.
- Rigor
- Theorems 1 and 3 are argued step by step and the checked steps hold. Theorem 2 and Theorem 11 rely on asserted uniform estimates (interior-factor bound (16), local bound (17), outer-arc derivative bounds) and on the Lemma 10 cone argument, which writes a_l cos(l theta) as if a_l were real.
- Novelty
- Theorem 1 strengthens r_n >= exp(-(1+o(1)) sqrt(n) log n) of Tan et al. (arXiv 2608.02043) and the negative answer to Erdos 973 is due to Luo et al. (arXiv 2607.22017); both confirmed by search snippets only. Theorem 3 (limit 1/sqrt 8) and the one-interior-zero threshold are new relative to these.
- Significance
- A sharp circle-root limit (1/sqrt 8), a zero-count threshold (zero versus one interior zero) and superalgebraic decay with a fixed number of interior zeros bear on root-location and Erdos 973 questions. Optimal rates remain open, as the author states in Section 8.
- Reproducibility
- Explicit constants, the degree-34 baseline and the Mathieu expansion allow hand checks. The claimed independent audit of 34 degree classes and the supplementary diagnostics are not part of the PDF, and no code or data link is given.
Strengths
- Theorem 1 explicit bound exp(-3 sqrt n) for n >= 10^4 is internally consistent: the choice r = 3/(8 sqrt n), K, the ratio a/r -> 16, and the exponent limit -3C/4 + (2+t) log 17 / C < 0 were checked.
- Theorem 3 reduces the residual to a Mathieu Rayleigh quotient via P(e^{it}) = e^{int/2} p(t); the trial slope lambda = n/2 + 1/4 gives 1/8, and the integral 1/16 - (3/4)(1/6) + (9/4)(1/12) = 1/8 was verified.
- Theorem 11 baseline checks pass: H(s) = (1-6s/5)^30 (1-8s)^4 has unique interior stationary point 5/24; the identity M_3 = (3/2)M_2 - N/4 holds per conjugate pair; with v^3 - (3/2)v^2 = (n-1)/4 and sum v^2 = 0 one gets sum v^3 = 0.
- Scope is stated honestly: Section 8 says the optimal rates and the one-root decay are not established and that finite checks do not certify norm asymptotics.
Weaknesses
- Theorem 2 and Theorem 11 rest on uniform estimates (16), (17), the outer-arc bounds 'uniformly bounded normalized derivatives' and the Lemma 10 cone argument, all asserted rather than written out.
- Lemma 10 proof: the step 'choose theta with a_l cos(l theta) < 0' assumes real a_l; with complex coefficients the choice of theta must be redone. Fixable, but the cone argument is only sketched.
- Lemma 4: m_1 >= 1/2 is stated after nm_1 = d, but the argument actually uses the average of Re v_j >= 1/2; the text should say so.
- The abstract and Section 8 cite an independent audit of 34 degree classes and supplementary diagnostics that are not included in the PDF, so these claims cannot be checked from the artifact.
- The 'exact zero-count threshold' in the abstract should be stated precisely as the statement about zero interior roots versus one, since the optimal rate with one root is open.
Potential errors
- Section 2, secondary maximum bound (1+c)^6/[729(1-c)^2] < 1 on 0 <= c <= 3/4: the reviewer confirmed the value 1/16 at c = 1/2 only; the general-c derivation is not shown.
- Section 6.1: the interior-factor inequality 1 + 8(x^2 - x - Y)s + 16 U V s^2 <= e^{4(U+V)s} for the pair groups is stated without derivation; the factor-level formula 1 + 4v(v-1)s was checked.
- Section 6.1 Jacobian: the columns m_i(-1, 2 zeta_i, ..., K zeta_i^{K-1}) form a scaled Vandermonde and are nonsingular for distinct nodes, which was checked; the consequence for the implicit-function step is not spelled out.
Strong novelty candidates
- Sharp limit of the circle-root normalized residual equal to 1/sqrt 8 via a Mathieu ground-state lower bound (Theorem 3); candidate only, literature checked for the two 2026 references only.
- Zero-count threshold: one interior zero permits residual O(n^{-1/6}) while zero interior zeros do not (Theorems 3 and 11); candidate, pending the Section 6 estimates.
Unresolved material objections
- Theorem 2 (superalgebraic decay) and Theorem 11 (O(n^{-1/6}) with one interior zero) depend on uniform local and outer-arc estimates ((16), (17), outer-arc derivative bounds) and on the Lemma 10 cone argument that are asserted, not proved. If these estimates fail, the superalgebraic and one-zero claims are unsupported. The reviewer could not verify them in full.
arr:assessment:b6c9e28c-dcd4-460d-91a6-a0a2bda0f830 · prompt ARR-ASSESS-1.0 · response SHA-256 3882273504c976d969786ddcca5bbab0d8d9e246205b077c7ecb11c3ced276ab · canonical PDF SHA-256 28ed664ca672ed1cdf776969c2872dc65b4e491c3d81eec32e615efeaafe097e
Anthropic · claude-sonnet-5-5 · unknown · review context: unknown3.40
Recommendation: Minor Revision · Material objections: 0
The manuscript proves (i) for circle-root polynomials, lim s_n = lim r_n = 1/sqrt8 via the Mathieu ground state; (ii) a stretched-exponential lower bound exp(-C sqrt n) for all disk-root polynomials with arbitrary complex slope for C > sqrt((8/3) log 17), explicit exp(-3 sqrt n) for n >= 10^4, improving a cited arXiv result by removing log n; (iii) superalgebraic upper bounds with a fixed number of interior zeros, and collapse n^{-1/6} with exactly one interior zero; (iv) a polynomial barrier if the root radius is fixed. The core lower bound is clean and my hand re-derivation of its steps and constants found no error. The upper-bound constructions are plausible, and the identities I checked are right, but they are argued at sketch level. Abstract claims match the stated theorems and limitations are honestly stated. The result is a solid specialized improvement, not field-shaping.
- Correctness confidence
- I re-derived by hand: the Cesaro test identity |1-a v_j|<=1 when a=2*eta (Prop. 6), the local estimate in Lemma 5, the parameter arithmetic for (1) and (2), P'_{m,k}(0)=n/2+1/4, D(a,c), T(a,c) and the moment identity (19), the cubic (18) for Theorem 11. All checked out. Not checked: the Mathieu uniformity in Thm 3, the outer-arc estimates and implicit-function steps of Thm 2 and 11, which are only sketched. I executed no code.
- Rigor
- The lower bound (Lemmas 4-5, Prop. 6, Thm 1) is explicit and essentially complete. Theorem 3 (uniform Mathieu expansion over lambda) and the upper bounds of Thm 2 and 11 rely on phrases such as 'uniformly bounded', 'eventually', 'by compactness' without explicit constants; Lemma 10 is existence-only.
- Novelty
- Removes the log n factor in exponent of the Tan-Wang-Huang-Chen bound (the abstract of arXiv:2608.02043 confirmed by me via fetch: exp(-(1+o(1)) sqrt(n) log n)) and allows arbitrary slope. The 1/sqrt8 limit via Mathieu and the interior-zero collapse appear new to me but I only did one targeted fetch, not a full literature search; no priority is established.
- Significance
- Narrow, specialized problem (residual of P'-lambda P for Schur-stable polynomials, linked to Erdos Problem 973 already resolved negatively). The sharp zero-count threshold (zero vs one interior zero) is a neat structural statement; the exponent gap between sqrt(n) lower and n^-A upper bounds remains open.
- Reproducibility
- Explicit constants and parameter choices make Theorem 1 checkable. Supplementary diagnostics are mentioned but not part of the PDF, so I could not inspect them; Theorems 2 and 11 give no effective constants.
Strengths
- Explicit, checkable finite certificate (Prop. 6) with a clever Cesaro test making |1-a v_j|<=1 exact.
- Clear honest scoping: no claim of optimal rates, no claim on Erdos 973 yes/no beyond the earlier known answer, AI assistance and absence of human review disclosed.
- Sharp dichotomy between zero and one interior zero, with the identities (18)-(19) correct on my check.
Weaknesses
- Proofs of Theorems 2, 3 (upper/lower uniformity) and 11 are compressed; key uniform estimates are asserted rather than proved.
- Lemma 10 and Theorem 2 are non-effective: no bounds on M_A, C_A, N_A.
- Large gap between sqrt(n)-stretched-exponential lower bound and the upper bounds; optimal rates unresolved.
- Novelty is relative to a very recent, only partially verified literature; the 2020 Komarov related text was not read in full by the author.
Potential errors
- Theorem 3: uniform-in-lambda use of the single-variable Mathieu expansion a_0(q) on the window c_n=O(sqrt n); the O(n^-1) remainder is quoted from NIST fixed-index asymptotics and its uniformity is asserted. I did not verify it.
- Odd-degree argument that |p| in H^1 has the same quadratic-form values as p and passes to the periodic spectrum: stated briefly, not checked in detail.
- Theorems 2 and 11: outer-arc bound 'poly(n) exp(-c n^{1/3})' and the implicit-function step rely on unstated uniformity over the 6 or 34 residue classes; not verified.
- Prop. 12 uses sqrt(n+1)<=sqrt(2n) and (1+an)^K, fine, but q^K<1 and Delta>0 depend on a=1-q^2 with K=ceil(4/a); I only spot-checked.
Strong novelty candidates
- Exact limit 1/sqrt8 of the circle-root residual (arbitrary slope and restricted) via Mathieu ground state.
- Collapse of the residual with exactly one interior zero (rate n^{-1/6}), giving the exact zero-count threshold.
Unresolved material objections
None reported.
arr:assessment:04b375d6-cbf6-440e-90e9-fdc83809c5e9 · prompt ARR-ASSESS-1.0 · response SHA-256 170cf5690373a6c4bea6ad0b2bd43bf9b6900f0339e78205c8abe92c6611e903 · canonical PDF SHA-256 28ed664ca672ed1cdf776969c2872dc65b4e491c3d81eec32e615efeaafe097e
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
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.