Plectis

Long paper

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

Erdős #1041 By 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.

Start here, by the editorial route

In this paper

Further families and counterexamples to proposed proof steps

Some further uses of the radial and conformal arguments are collected here, together with examples where a prescribed construction fails. The numbered results retain their linked formal statements. The additional analytic arguments appear as unformalised remarks, even when an ordinary proof is supplied in full. These qualifications specify the available formal evidence without changing the finite calculations.

The underlying research notes are retained at commit f214a6b4 and summarised in the dated overview of these arguments. Numerical observations are identified separately. These are restricted questions about polynomial classes and constructions; they do not supply a programme for proving the refuted unrestricted assertion.

Unformalised remark: roots close to a regular polygon

The exact binomial chord calculation

Unformalised remark: a bounded-radius concyclic class

Power substitutions in trinomials

Unformalised remark: a coefficient condition for four-term polynomials

Unformalised remark: a quartic composed with a power map

A mean bound for critical values in every degree

A sufficient critical-value threshold in degree five

Numerical tests of selection by the smallest critical value

Unformalised remark: limits of contained paths at fixed degree

Unformalised remark: Blaschke products and varying degree

What the earlier inverse-ray approach would have required

An integral for successive merger scales

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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