Newton Flow and Critical-Value Ray Separation

The problem

Numbering and current status follow Bloom’s Erdős problem catalogue [1]. The problem is open. The original source is Problem 5 on printed p. 139 of Erdős–Herzog–Piranian [2]; the preceding paragraph records the known input that one component of the lemniscate contains at least two zeros.

Two recent manuscripts are relevant. The 48-page manuscript posted by shtuka on 24 March 2026 [3] claims the unrestricted statement. Its Proposition 12 (p. 16, with proof continuing through p. 30) supplies the spanning-tree decomposition used in the final proof. The defect was located publicly in the problem’s discussion thread: on 25 March 2026 Tao observed that the invocation of Lemma 8 there is unjustified and that the flow lines need not organise into connected trees, and on 26 March 2026 the manuscript’s author agreed that the statement of Proposition 12 itself, not only its printed proof, is incorrect, and set the strategy aside. Section  records an independent diagnosis of the same failure through the local three-ended saddle model, together with possible repairs. No counterexample to the proposition is exhibited there. Pendyala’s independent June 2026 preprint [4] proves the degree-four case. The quartic theorem does not close the problem.

The object we work with is not the lemniscate directly but the flow that foliates it.

The Newton value equation

Away from the critical set, define the complex Newton field $$N(z)=-\frac{f(z)}{f'(z)} .$$ Let $z(t)$ be differentiable with $z'(t)=N(z(t))$, and put $w(t)=f(z(t))$.

The computation is one line: $w'=f'(z)\,z'=f'(z)\cdot(-f(z)/f'(z))=-f(z)=-w$. The kernel checks it as , together with the differential form of the first integral, : $$\frac{d}{dt}\Bigl(e^{t}f(z(t))\Bigr)=0, \qquad\text{equivalently}\qquad f(z(t))=e^{-t}f(z(0)) .$$ Observe what this says about the geometry. The value moves radially inward at exponential rate and never changes argument. The lemniscate $\{|f|<1\}$ is therefore forward-invariant, and the flow lines are exactly the preimages of rays from the origin.

This is checked in consumer form: the kernel accepts the implication from the hypothesis of distinct rays to the absence of a connection. It is the statement the topology needs, and it is a genuine sharpening of the criterion used in the literature.

Arguments, not moduli

It is tempting to arrange a generic perturbation so that the critical values are pairwise distinct, or that their moduli are pairwise distinct, and to conclude that saddle connections are excluded. Neither is enough.

Two distinct critical values can lie on one ray, and two critical values with distinct moduli certainly can: the ray records the argument, and the modulus is exactly the coordinate the flow contracts. By Corollary  the invariant that excludes connections is the argument. What a perturbation must therefore achieve is pairwise distinct critical-value , which is a condition on $n-1$ points modulo the circle rather than on their positions in the plane.

The cost of that condition is also checked, and it is small.

So each pair of critical values contributes a one-real-parameter forbidden locus in the translation plane, given in closed form (). A finite union of such loci has empty interior, which is the shape one wants for an avoidance argument. Turning that into a perturbation of $f$ is not immediate: the translation model must be replaced by an actual perturbation of the roots that keeps them inside $\mathbb{D}$ and preserves the length slack. That is the first open producer of §.

A proof gap in the unrestricted argument

The March manuscript’s Proposition 12 claims the following load-bearing statement. For $u=-\log|f|$, a connected component $V$ of $\{u>c\}$ carrying $m\ge2$ simple zeros, and the stated regularity and Morse hypotheses, it constructs, for every $\varepsilon>0$, an embedded spanning tree $G_\varepsilon\subset V$ with $$\operatorname{len}(G_\varepsilon) \le\frac1{2\pi}\int_{2\alpha}^{\infty}P_V(t)\,dt+\varepsilon. \tag{5.1}\label{eq:prop12-bound}$$ The final theorem uses this estimate, so the issue below cannot be bypassed by calling the proposition auxiliary.

At an interior index-one critical point $p$, the proof invokes a Morse chart $$u=\mu+x^2-y^2$$ and replaces the saddle by a three-ended neighbourhood having one connected lower cross-section and two connected upper cross-sections. That local model is false as written. Because $p\in V$ and $V$ is open, a sufficiently small closed disc around $p$ lies entirely in $V$. In that disc the full Morse chart has four sectors: two components of $u>\mu$ and two components of $u<\mu$. A global component argument cannot delete one local sector from a disc already contained in $V$.

This diagnoses a proof step, not the proposition’s statement. A repair might cut an adjoining regular annulus along a separatrix or regular flow arc before forming the block, retain a four-pronged saddle neighbourhood and change the assembly, or replace the local construction by the ray-cut decomposition proposed below. Any repair must prove that its connector cost can be made arbitrarily small uniformly in the attachment points and that the repaired blocks still assemble to an embedded tree satisfying . The shorter descriptions of the same three-ended block do not repair the four-sector topology.

Corollary  supplies one independent input for a different route: distinct critical-value arguments exclude saddle-to-saddle Newton connections. It does not itself prove the compact planar decomposition, classify all orbit endpoints or provide the metric gluing estimate. Those are separate problems below.

Finite evidence

A search was run over random monic polynomials with roots in the unit disc. For each sample the region $\{|f|<1\}$ was rasterised and shortest grid paths were computed between every pair of roots. The best upper bounds obtained were $$\begin{array}{lrr} \text{degree } 5, & 500 \text{ trials}: & 1.1052648928\\ \text{degree } 6, & 1500 \text{ trials}: & 0.8450414343\\ \text{degree } 8, & 1500 \text{ trials}: & 0.6203916714\\ \text{degree } 10, & 1500 \text{ trials}: & 0.4303640486 . \end{array}$$ No counterexample candidate was found, and the measured values sit well below the threshold $2$. These are grid distances for the sampled configurations: upper bounds on those samples, not bounds over the family. The apparent decrease with degree is a property of the sample, and we draw no conjecture from it. Future searches should target named failure modes—near-degenerate saddles, thin necks, boundary-critical configurations and almost-connected separatrices—and report those diagnostics. Raster paths remain candidate finders, never continuous certificates.

Complements and further questions

The dependency chain has five separate gates. A proof, a minimally corrected hypothesis set, or an explicit polynomial or planar counterexample is a decisive answer to any one of them.

Guide to the formal sources

The public ErdosProblems.Erdos1041.NewtonFlowRaySeparation module contains the checked source for this note. The search of § is scripts/search_counterexample.py in the source package. The declaration table below is pinned to the shared formal-source commit used throughout this problem-note series.