Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections

Will Cook

Long paper

Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections

Erdős #1041 By Will Cook 29 September 2026 Authorship, AI use and citation

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Summary. The record gives the rational estimates and geometric proof for ani's degree-seven polynomial: every connected set in its strict unit lemniscate joining two roots has Hausdorff measure greater than two. It develops contained paths for trinomials, collinear roots and two sparse families. Ordinary analytic arguments for critical-value criteria retain their unformalised inputs beside the statements. Critical-value moments, counterexamples to prescribed paths and questions about flow-strip geometry complete the account. The one-polynomial formal statement and correspondence with the historical wording remain distinct.

This paper covers the exact geometry of ani's one-polynomial counterexample, proved positive families and critical-value arguments with their unformalised inputs identified; formal-statement status and historical correspondence remain owned by the claims registry.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository.

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In this paper

Four points without a matching or numerical split.

The preceding weighted proof with $m = 4$ and $w_{j} = 1 / 4$ gives

\sum_{j = 1}^{4} {(\prod_{k = 1}^{4} | 1 - \overset{―}{c_{k}} c_{j} |)}^{1 / 4} \leq 4, | c_{j} | \leq 1,

with equality if and only if all four centres vanish. Equality in this linear form forces equality in the quadratic inequality, by the variance identity above; Lean proves that equality case. In particular, the inequality is strict as soon as one centre is nonzero, even with repeated centres or centres on the unit circle.

This four-point consequence needs neither the former $21 / 25$ split nor a perfect-matching Hölder step. The scalar stationary-point inequalities from that approach are not a proof of its missing steps. The two restricted arguments above remain independent alternatives.

The mean inequality gives $μ \leq R^{n}$ . In the unit-disc normalisation $R = 1$ , this does not force the smaller quintic threshold $μ < 1 / M_{5}$ below. For $f (z) = z^{n} - 1$ , every critical value has modulus one, so no uniformly smaller bound holds on the closed-disc class. These modulus bounds still require a separate path selection or metric comparison before they yield an inverse-ray length estimate.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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