Report

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Subject and independence

Mathematical verdict: refutation-failed, for the frozen cubic statement and its supplied proof under the external source premises identified below. This is a mathematical examination, not its own grade or a tier assertion. The distinct grader must adjudicate the disclosed import exposure and the weight of the failed optional calibration checks.

The subject is the complete extracted statement, all eight proof sections, the complete evidence description and checker, and the supplied Eremenko–Hayman text, headed On the length of lemniscates, through Lemma 6 with bibliography. The source mathematics was frozen on 2026-10-15 at 14:29:00 UTC; the supplied neutral extraction was prepared that day at 15:09:22 UTC. The examination date is 2026-10-15. No conclusion here concerns later changes.

The preserved subject paths, relative to this report, are:

The project-owned mathematical copies preserve the delivered bytes, including extraction cut markers. The copyrighted source excerpt is cited by its locator instead of being reproduced; its actual reading depth and scope remain below. The intended filing home is wiki/theory/polynomials/L144_cubic_lemniscate_length/evidence/verify/review/; the paths above travel with this record. The copied assets/subject/runtime/ is execution support, not an additional mathematical subject. The independent derivations, diagnostic, exact diagnostic inputs, and reproduction instructions are part of this examination. Original project mathematics is credited to Plasma AI. Source mathematics retains the authorship of Alexandre Eremenko and Walter Hayman.

The reviewer was assigned a fresh adversarial context, was neither an author nor a construction collaborator, and had not built on this result. The disclosed model is gpt-6-astra. The assignment sought a counterexample, unsupported essential deduction, or canonical failure mode. No earlier review, sibling verdict, author plan, live claim standing, or research narrative was supplied or read. There was no web search, outside-worktree source reading, native subagent, synchronization, or message-inbox/feed read. Subscription checks at the first boundary, after startup, and at the examination boundary all returned empty tables. That is evidence about wiring, not proof of what was read.

The full statement, proof, checker, evidence description, and supplied source excerpt were actually read. The complete verification standard and its campaign adapter, the required anatomy sections, evidence instructions, authoring and weaving instructions, and commissioned operating context were also read. Runtime reading included the complete Checker and parser implementation, package import surfaces, constants, CLI wrappers, and copied project specification. Other supplied runtime modules and the lockfile were identity checked and used as import support, not individually audited as repository tools. The supplied installed python-flint API capture was read; its links and examples were not followed or executed.

Exposure disclosure. An initial environment-only probe, before any mathematical execution, used Python on standard input from the checkout root. Despite the intended PYTHONPATH, Python's initial search entry selected the live checkout's tools package rather than the frozen copy. This implicitly imported excluded implementation bytes. Only environment identities were printed; no live module source or mathematical corpus text was displayed, and no corpus command or mathematical check was invoked. The exposure was disclosed immediately. A repeated probe with safe-path mode selected the frozen package. Both mathematical executions subsequently verified the correct package path in their own process. The first import is not erased by this remedy; its materiality remains for the distinct grader to rule on.

The supplied inputs remained unchanged under the recorded identity checks. No frozen proof, checker, input, or public corpus file was edited. The original source PDF and the underlying books were outside the read set. This is text-only inspection of the commissioned corrected-1999-preprint excerpt, not a claim of independently checked PDF transcription fidelity.

Restatement

For a monic complex polynomial $p$, let $\ell(p)$ be the Euclidean arclength of the geometric set $\{z\in\mathbb C:|p(z)|=1\}$. For every real $a$ with $0<a\le2^{-1/3}$, set $c=\sqrt{1-4a^6}$ using the nonnegative real square root. Then

$$\ell(z^3+3a^2z-c) <L_0:=\ell(z^3-1) =\frac{2\pi\Gamma(1/3)}{\Gamma(2/3)^2}.$$

For every monic polynomial of degree exactly three with complex coefficients, $\ell(p)\le L_0$. Equality holds if and only if there exist $z_0\in\mathbb C$ and $w\in\mathbb C$ with $|w|=1$ such that $p(z)=(z-z_0)^3-w$. The strict family statement excludes $a=0$; that auxiliary endpoint gives equality. The endpoint $a=2^{-1/3}$, where $c=0$, is included. The geometric length is not replaced by a different multiplicity convention at ramification points.

Findings on all proof components

For the normalization, write a translated cubic as $z^3-3u^2z+b$. A unit rotation of the variable together with the inverse cubic phase of the value makes $b\le0$ real. Unit-modulus critical values imply

$$\operatorname{Re}(b\overline{u^3})=0, \qquad b^2+4|u|^6=1.$$

When $b<0$, $u^3$ is imaginary, and one of the rotations fixing $b$ makes $u=\pm ia$, $a=|u|$. When $b=0$, an arbitrary phase rotation is available; when $u=0$, $b=-1$. Thus the stated closed parameter interval is exactly the needed normalized class. No scale change has been introduced.

For the constant, the three inverse derivatives of $z^3=s$ sum to $|s|^{-2/3}$. Therefore

$$\begin{aligned} L_0 &=\int_0^{2\pi}2^{-2/3}|\cos(\theta/2)|^{-2/3}\,d\theta\\ &=2^{1/3}B(1/2,1/6) =\frac{2\pi\Gamma(1/3)}{\Gamma(2/3)^2}. \end{aligned}$$

The last equality uses $\Gamma(1/6)\Gamma(2/3)=2^{2/3}\sqrt\pi\Gamma(1/3)$. For $c<1$, absolute convergence of the binomial series gives Parseval; for $c\ge1/2$, the identity $|c+e^{i\theta}|^2=(1-c)^2+c|1+e^{i\theta}|^2$ supplies an integrable majorant proportional to $|1+e^{i\theta}|^{-2/3}$. Thus the endpoint passage is justified. The nonzero coefficient at degree one proves strict increase, including comparison to the endpoint. Finally, $G_a(s)=a^{-2}G_1(s/a^3)$ and $|ds|=a^3|d\sigma|$ give $L(a)-\Phi(c)=aI(a)$ exactly.

A false sentence in the supplied source excerpt

The final paragraph of the supplied proof of Lemma 4 says that the length tends to zero as the root vector $Z$ tends to infinity. Taken literally, this sentence is false. Let $Z_R=(R,R,R)$ and $p_R(z)=(z-R)^3$ for real $R\to\infty$. Then $\|Z_R\|\to\infty$, but $E(p_R)$ is the circle $|z-R|=1$ and has constant length $2\pi$.

The preceding source argument concerns root diameter, which remains zero in this witness. A compactness argument for existence must first normalize translations, for example by putting one root at zero. Bounded root diameter then bounds the normalized root vector. The frozen native proof consumes Lemma 4's existence and continuity statements, not the false unnormalized asymptotic sentence; its own section 8 explicitly centers its cubics. This finding limits any claim to have verified every sentence of the source proof. It is not a counterexample to the commissioned cubic statement or to the stated source theorem. With no PDF inspection, this record does not decide whether the wording error is in the print or its supplied transcription.

Weakest steps and strongest attempted refutation

The three independent weakest-step derivations are retained in derivations.md:

1. The source deformation for any maximizer, its monic normalization and preservation of ramification, and the cubic compactness argument extending local moves to the circle without collisions.
2. The uniform small-parameter bound, including the scaled graph, both tail constants, the signed head subtraction, and the square-root window.
3. Branch-independent root enclosures, signed rectangle sums, exhausted-depth behavior, the large-parameter singular window, and the endpoint extensions and full parameter cover.

The strongest attack was to separate the existence of one normalized maximizer from classification of every maximizer. A second possible failure was that local deformations could fail to reach the circle because of coefficient escape or coalescing critical values. The supplied source construction and the explicit identities $b=(t+w)/2$, $|u|^3=|w-t|/4$ defeat both attacks when the chosen arc avoids the fixed value. The remaining $u=0$ family independently has equality only at $|b|=1$. This supplies the required entailment rather than assuming that an existential result classifies all extremizers.

The strongest computational attack used a prescribed exact root and quadratic division to challenge the Cardano enclosures across cuts and near critical values. It found no conflicting root-density value in its finite domain. The four failed calibration equalities, described below, are failures of the reviewer's optional diagnostic assertions; they do not supply an inequality contradicting an author's enclosing bound.

Computational findings and reproduction

The complete frozen checker was executed with no quick mode and no cached results, using one thread, python-flint 0.9.0, and the installed CPython 3.14.6 environment. The running process verified its environment prefix, frozen tools import, standard-library typing import, and python-flint version before executing the unchanged file. The only execution substitutions were the commissioned file relocation/runtime import and an operating wrapper that printed and checked those identities, then invoked the unchanged main file with its original argument list.

The run exited 0, with 22 checks passed, in about five minutes (observed elapsed time approximately 288 seconds). Its results were:

The distinct diagnostic route starts from an exact Gaussian-rational root $r$, sets $s=r^3+3a^2r$, and finds the other roots from $z^2+rz+r^2+3a^2$. It never chooses a Cardano resolvent or takes a cube root. The 128-bit inverse-derivative sums for 60 prescribed inputs lay inside the 64-bit Cardano density balls for twelve boxes. All generated inputs were checked to lie in their corresponding boxes. Cases cover both signs of real and imaginary cuts, near-critical values, general complex values, scaling, and zero linear coefficient. This attacks the root-generation implementation; it shares the ball library and does not independently validate that library or prove uniformity over the parameter ranges.

The diagnostic also confirmed the analytic instance $G_1(0)=2/3$, rejected the deliberately wrong value $4/3$, checked the finite numerator and zero denominator at $s=2i$, and observed nonfinite density there. Its nonfinite depth-cap, invalid-window, and unresolved-cover inputs produced the required failure behavior.

The diagnostic as a whole exited 1: four of its 48 checks failed. They are the finite-cap exact endpoint equality and the three signed-rectangle exact endpoint equalities. The driver required exact ideal endpoints even though a constructed arb radius may round outward. Such equality is stronger than the required enclosure inequality. The independent signed-rectangle derivation establishes the relevant inequalities without these tests.

Actual endpoint values for those four assertions were not printed, so no exact numerical cause or size of their discrepancy is asserted here. Their failed status is retained; the unchanged driver is not advertised as a passing certificate. No third mathematical execution or repaired diagnostic was run. A corrected instrumented diagnostic would be a separate record if its added coverage is wanted. The reproduction instructions distinguish the successful full run from this reproducible failed diagnostic.

Premises and source boundaries

The premise census was derived from the full extracted statement and proof, including their consequence sentences. No native L-claim is used as an established premise. The self-reference supplies definitions already present in the extraction. The research-sketch link identifies provenance for algebra reproduced in section 1; it supplies no additional theorem. Those definitions are meaningful independently of the truth of L144. No live ledger read was authorized or performed, and there is no pending native dependency standing to substitute for a current one.

The supplied Lemma 6 and its following connectedness remark were read but are not consumed. Neither the source's general linear-in-degree bound nor its quadratic extremizer theorem is a premise of the cubic argument. The review is neither a batch nor a commissioned second review cycle; no prior-cycle attack comparison was supplied or inferred.

Checklist

Canonical failure modes

Named patterns

Erdos-specific checklist

The neighboring discipline about correlated statistical samples is inapplicable because there is no statistical inference. Evidence coverage is checked under the supplied anatomy/evidence contract: the computation certifies only its integral bounds, and the written mathematics supplies their connection to the complete statement.

Verdict, limits, and remaining ownership

refutation-failed is the mathematical finding for the complete frozen cubic assertion and its supplied proof, relying on the identified external source theorems and the documented ball-arithmetic contract. The strict family inequality, exact gamma value, universal upper bound, and equality classification have each been examined. No counterexample to them or unsupported essential inference was found. The false unnormalized asymptotic sentence in the supporting excerpt is retained at its actual, nonconsumed scope.

The distinct grader owns acceptance of this examination. In particular, the initial live-package import requires an explicit materiality ruling; this report does not supply one. The optional diagnostic's four failed exact equalities cannot be cited as successful enclosure calibration. Their actual endpoint discrepancies remain unmeasured in the retained output. If further calibration is required, the coordinator must commission a new, instrumented record; this examination does not silently repair and rerun its frozen keeper.

There is no outstanding native-premise standing read. Original-PDF fidelity, proofs of the external Beltrami theorems, and verification of the ball library are outside the authorized scope and remain the stated external boundaries. The coordinator owns durable filing of the unchanged subject and keeper package, distinct grading, and any additional commissioned work. Private delivery and successful execution alone confer no mathematical tier.