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

What the earlier inverse-ray approach would have required

The earlier proposed sufficient condition concerned curved inverse images of rays, not straight segments. On a generic ray-separated class let $L_{f} (c)$ be the length of the two-root descent arc at an admissible critical point, where $| f (c) | \leq 1$ . Then

min_{c admissible} L_{f} (c) \leq 2

would imply the existence of a short curve joining two roots by the stated compactness argument. The unrestricted infimum over all contained curves can be smaller; equality of these two optimisation problems is not assumed. The obstruction to two straight segments through a critical point does not by itself prove or disprove this curved-arc estimate. Examples in which another critical point gives a shorter curve do not prove the estimate universally. The critical value bound, containment and length bound must hold for the same curve. The displayed universal inequality is a historical sufficient condition, not an open assertion: together with the stated compactness and scaling argument it would contradict Theorem 1.2. The reduction remains useful on separately specified polynomial classes.

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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