A sharp circle-root residual barrier and its collapse with one interior zero
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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.
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Cite this paper
Lluis Eriksson (2026). A sharp circle-root residual barrier and its collapse with one interior zero. AIRR.SCIENCE, ARR-2026-6TQRG7PY6C8JZR2D, v2. https://airr.science/papers/ARR-2026-6TQRG7PY6C8JZR2D/versions/v2/ Research preprint; ARR-2026-6TQRG7PY6C8JZR2D v2. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Revision statement
Minor revision. Adds Appendix A with explicit-constant lemmas and rewrites the proofs of Theorems 2 and 11 on them; statements and numbering unchanged.
Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Not assessed
- Lean 4
- Not applicable
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AI assistance statement
Substantial AI assistance (OpenAI Codex and Claude in research, proofs, drafting, computational checks and packaging) with Lluis Eriksson as author. This version was revised by Claude Sonnet 5.5 acting as author-reviewer (not human, not independent of the author). Assessments or scores of the earlier version v1 do not carry over to this version and none exist for this exact PDF. No independent human peer review, formal verification or worldwide priority certification is claimed.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 2
- Range
- 3.40–3.40
- Method
- bounded capability-weighted mean · exact version only
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Review input clarification
Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Sonnet 5.5 report (3.40, minor_revision) made in an independent Claude Code thread: new session whose working folder held only paper.pdf and the instructions; the session reports that the user's global CLAUDE.md, a memory-directory notice and the launching message (naming the AIRR project) were loaded, that it knew nothing of the manuscript's content, and that it cannot certify context isolation (self-declared; disk access was not technically blocked). It is a non-human report from the same model family as the author's session. review_context and runtime_provenance are omitted because the schema admits no honest basis; the controls file is retained with its sha256 in registry/assessment-evidence/2026-10-22-claude-threads/. Model JSON preserved unedited; only the placeholder paper and version ids were filled (and the version label corrected from the template's v1 to v2). No other report exists for this version. No cost or tokens recorded.
Operator record · · Source files and affected reports
Review input clarification
Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Haiku 5.5 report (3.40, major_revision, independence 'not_involved_in_manuscript' as the model reported), non-human, from an independent Claude Code thread: new session with only paper.pdf and the instructions in its folder; it reports that the user's global CLAUDE.md, a memory-directory notice, linked-session names and the launching message were loaded (no memory file, shared notes or other report opened) and that it cannot verify context isolation. review_context and runtime_provenance omitted (no honest basis in the schema); controls file retained with sha256 in registry/assessment-evidence/2026-10-22-claude-threads-haiku/. Model JSON unedited; only placeholder ids filled (and version label v1 corrected to v2). A separate base model from the Sonnet 5.5 thread report on the same version, which it does not replace.
Operator record · · Source files and affected reports
anthropic · claude-haiku-5-5 · not involved in manuscript · review context: unknown3.40
Recommendation: Major Revision · Material objections: 0
Version-2 manuscript on residuals ||P' - lambda P|| / ||P|| for polynomials with all zeros in the closed unit disk. Main claims: (i) for all-circle-zero polynomials the unrestricted and restricted infima tend to 1/sqrt(8) (Thm 3); (ii) s_n >= exp(-C sqrt(n)) for every C > sqrt((8/3) log 17), and s_n >= exp(-3 sqrt(n)) for n >= 10^4 (Thm 1, eq. 2); (iii) one interior zero permits restricted residual O(n^{-1/6}) (Thm 11), while superalgebraic decay for every A > 0 needs finitely many interior zeros (Thm 2). I hand-checked the identities and constants of the main lower-bound and upper-bound arguments and found no error. The Thm 2 and Thm 11 constructions were checked only at the level of moment identities and exponent bookkeeping. Recommendation: major revision for verification, effective constants and prior-art documentation.
- Correctness confidence
- Hand-checked: the squared-residual identity (3), the Lemma 4 separation bound, the Lemma 5 Cauchy estimate, Prop. 6 arithmetic and the explicit n>=10^4 certificate for eq. (2), and the Thm 3 upper-bound limit 1/8 from the Gaussian ratio. No error found. Not checked line by line: the Thm 2 and Thm 11 parameter systems, the IFT and Lemma 10 existence step, and the numerical bound in Lemma A.1(e).
- Rigor
- Constants in Thm 1 and eq. (2) are traceable. Thm 2 is non-effective: the IFT neighbourhood is unquantified and M_A, C_A, N_A are not computed, as the paper admits. The Thm 3 tail step ('a polynomial times a Gaussian dominates') is asserted, not written out. Prop. A.5 has a compressed proof.
- Novelty
- Thm 1 is presented as a refinement of Tan et al. and of the qualitative Luo-Yang-Zhu result, and the Erdos 973 negative answer is conceded as prior. The sharp 1/sqrt(8) circle-root constant and the one-interior-zero collapse appear new in the text. The prior-art claims rest on arXiv items (2607.22017, 2608.02043, Komarov) that I could not check.
- Significance
- Relevant to exterior power sums and root-residual barriers. The zero-count statement (no interior zero: no collapse; one interior zero: restricted residual O(n^{-1/6})) is clean and citable. Superalgebraic decay (Thm 2) is non-effective, and the one-root rate is not optimal (Sec. 8), which limits impact.
- Reproducibility
- The PDF contains no code, no numerical tables and no supplementary artifact. It refers to supplementary diagnostics, a grid evaluation and quadrature checks that are not provided. Thm 2 and Thm 11 rely on solving systems by IFT or numerically, which the text does not reproduce.
Strengths
- The Mathieu-based lower bound in Sec. 2 is clean: identity (3) checks, and the expansion E(L) = a0(2L)/4 with E(L) = -L + sqrt(L/2) - 1/16 matches the standard a0 asymptotics.
- The finite certificate (Prop. 6) gives an explicit path to exp(-3 sqrt(n)) for n >= 10^4; the arithmetic I checked holds (K <= (14/19) sqrt(n) + 1, a/r <= 16, 1/(K eta) <= 10/21, B < 1/(5n)).
- The upper-bound family P_{m,k} in Thm 3 checks: P'(0) = n/2 + 1/4, P'(1)/P(1) = n/2, and the Gaussian ratio with variance 1/6 gives exactly 1/8.
- The paper scopes its claims honestly: Thm 2 is non-effective, the one-root rate is not claimed optimal, and the Erdos 973 negative answer is credited to prior work.
Weaknesses
- Thm 2 depends on an implicit-function neighbourhood that is never quantified and on Lemma 10, whose existence argument is sketched. M_A, C_A and N_A are not computed, so the superalgebraic claim is an existence statement with unverified steps.
- The Thm 11 rate O(n^{-1/6}) rests on Prop. A.5 with K = 3. Its proof is compressed, and the solving of (a_n, c_n) from D and T, with the uniformity over 34 residue classes, is described but not written out.
- The Thm 3 dominated-convergence step on the outer arc is asserted, not proved.
- Priority and novelty claims rely on preprints (Tan et al., Luo et al., Komarov) and on a 2020 journal version that the author states was not available. None of these were checked.
- The PDF header says version 2 (21 October 2026), while the review template field says v1. This is not a mathematical defect but should be reconciled before release.
- No code, numerical tables or supplementary checks are included, so the numerical claims in Lemma A.1(e) and Sec. 8 cannot be reproduced from the artifact.
Potential errors
- Unconfirmed: Sec. 6 states P'(0) = P'(1)/P(1) as an 'exact identity' from (13)-(15). It holds because each interior factor satisfies f'(0) - f'(1)/f(1) = -v^2/(v-1) = d(v), so the first-moment equation enforces it, but this step is not written out and should be stated explicitly.
Strong novelty candidates
- The sharp circle-root constant 1/sqrt(8) for both the unrestricted and the restricted residual (Thm 3), provided the prior literature does not already contain it.
- The minimal interior-root count: zero interior zeros cannot collapse the residual (Thm 3), while one interior zero does, at rate O(n^{-1/6}) (Thm 11).
Unresolved material objections
None reported.
arr:assessment:f43c8f1a-aa47-4066-9b9f-97ac66f689b6 · prompt ARR-ASSESS-1.0 · response SHA-256 d8a49714abbaa21a22fc7453580eb42582284ed85840afccd6b67eb3c3fb370d · canonical PDF SHA-256 75332dab56cc67dbb3df0f61ffa49f532e258ce8c22ac2b7439477dc35ebaf44
anthropic · claude-sonnet-5-5 · not involved in manuscript · review context: unknown3.40
Recommendation: Minor Revision · Material objections: 0
The paper proves lim s_n = lim r_n = 1/sqrt8 for polynomials with all zeros on the unit circle (Mathieu ground state lower bound, explicit family for the upper bound), shows one interior zero suffices for vanishing residual O(n^{-1/6}), a fixed number of interior zeros gives superalgebraic decay, and a universal disk lower bound exp(-C sqrt n), C>sqrt((8/3)log 17). I rechecked the key identities and the Mathieu and Gaussian computations and found no error, but did not verify every appendix estimate, and no code was run. Main caveats are non-effective constants and unverifiable external citations.
- Correctness confidence
- Hand-rechecked: identity (3) for the squared residual, Mathieu expansion E(L)=-L+sqrt(L/2)-1/16 from a0(2L)/4, Gaussian ratio 1/8, cubic (16) from (17), Lemma 4/5/Prop 6 chain, Cor 9. No error found. Asymptotic uniformity in Thm 3 and Appendix A (not re-derived in full) rest on dense, unchecked estimates.
- Rigor
- Core chains are explicit and mostly checkable, and limits are honestly stated. But Thm 2 uses a non-quantified implicit-function neighborhood and eps_0 is non-effective; Prop A.5 constants only bind for n>~1e21; the dense Lemma A.1/A.3/A.5 estimates and the Thm 3 lower-bound uniformity are compressed and hard to verify fully.
- Novelty
- Removes the log factor of Tan et al. (exp(-(1+o(1))sqrt(n) log n)) to exp(-C sqrt(n)), and gives the sharp 1/sqrt8 circle-root barrier plus a one-interior-zero collapse. Novelty relies on arXiv 2026 preprints I could not check; priority search is admittedly non-exhaustive. Methods are largely adaptations.
- Significance
- Solid, specialized contribution to residual bounds for Schur-stable polynomials and exterior power sums; the sharp circle barrier and the exact zero-count threshold are clean. Does not settle optimal rates (disk, one-root) and the Erdos 973 answer was already known, so impact is moderate and narrow.
- Reproducibility
- Proofs are written out, appendix added, explicit parameters given. Supplementary scripts and numerical checks are described but were not available to me and I ran nothing. Non-effective constants (M_A, C_A, N_A) and the references (arXiv 2026) limit independent reproduction.
Strengths
- Sharp, explicit circle-root limit 1/sqrt8 with matching upper and lower arguments
- Exact threshold (zero vs one interior zero) with explicit parameters in the one-root case
- Elementary finite certificate giving explicit bound exp(-3 sqrt n) for n>=10^4
- Candid statement of what is not proved (optimal rates, effectiveness, priority)
Weaknesses
- Theorem 2 relies on a non-quantified implicit-function neighborhood; M_A, C_A, N_A are non-effective
- Appendix constants in Prop A.5 give non-trivial bounds for the one-root family only for n of order 1e21 or more
- Lower and upper disk bounds differ greatly (exp(-C sqrt n) vs n^-A); optimal rates open
- Novelty and priority depend on 2026 arXiv preprints and are not exhaustively searched
- Several dense asymptotic arguments (Thm 3 uniformity, Lemma A.1(e)) are compressed
Potential errors
- Thm 3 lower bound: uniform control of the Mathieu remainder on the window lambda=n/2+O(sqrt n) is argued briefly; the O(q^{-1/2}) uniformity is imported from DLMF
- Lemma 4 requires sqrt(n+1)e^{-y}<=1 and eta>0; the parameter regime for n>=10^4 is checked only by sketched numerics (log 17<3, monotonicity of gamma_n/sqrt n)
- Lemma A.3(ii) bound f''>=-8 Sigma_2 and the tau_1 integral lower bound are stated tersely
Strong novelty candidates
- Removal of the log n factor in the exponent for restricted residual lower bound (exp(-C sqrt n))
- Sharp circle-root residual limit 1/sqrt8 and exact interior-zero-count threshold
Unresolved material objections
None reported.
arr:assessment:3a71d982-aae2-448d-ba62-c84b6c6a0419 · prompt ARR-ASSESS-1.0 · response SHA-256 fb77ce8d63bd6bf280909af3c4096abf254893250a2a250559f3ad761b6b96ea · canonical PDF SHA-256 75332dab56cc67dbb3df0f61ffa49f532e258ce8c22ac2b7439477dc35ebaf44
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Editorial disclosure
Working-paper version v2 supersedes v1. Adds Appendix A with explicit-constant lemmas and rewrites the proofs of Theorems 2 and 11 on them; statements and numbering unchanged. Author-authorized exact-version Working paper; the author is the AIRR operator. It remains outside the AIRR accepted collection.