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

Guide to the results

The main constructions have different hypotheses. The table locates their conclusions and distinguishes the formal results from the additional analytic remarks. The detailed evidence associations are in the evidence record.

Family or mechanism	Conclusion and qualification
Degree seven	The fixed polynomial has no connected two-root subset of Hausdorff measure at most $2$ in its strict lemniscate.
Trinomials	Every root-to-origin segment is contained; any two roots are joined with length below $2$ .
Low critical values	Unformalised area-growth criterion with cutoff $13 / 25$ and scaled level $(25 / 13) μ$ . Its recorded rational comparison also covers $μ \leq 529 / 1000$ ; the analytic reduction remains required.
Separated simple value	Unformalised square-root, Bergman and capacity construction. The chosen critical point must be simple and its value isolated as stated.
Collinear and sparse families	Formal segment constructions for the specified root geometry or coefficient patterns. The stronger $2 \sqrt{1 - \| h \|^{2}}$ cyclic-fibre estimate remains an ordinary argument.
Critical-value means	Formal exponent $2 / (n - 1)$ . The sharp $4 / (n - 1)$ refinement and higher exponents under complex power-sum cancellations remain unformalised. No path selection follows.
Perimeter and prescribed paths	The Cassini example refutes the stated spanning-tree coefficient. The degree-eight example refutes the proposed Gamma-function constant, while a uniform perimeter bound is still open. The implication from $β σ^{1 / n}$ to a path of length at most $β ρ$ in $K_{μ}$ needs the unformalised first-critical-level split. Several prescribed straight-segment rules fail.

The inverse-ray estimate in the earlier note is retained only as a historical sufficient condition. It does not follow from these value, segment or perimeter calculations.

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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