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\hypersetup{
  pdftitle={Newton flow and critical-value ray separation for Erdős Problem 1041},
  pdfsubject={Erdős Problem Notes: Problem 1041},
  pdfkeywords={lemniscate, Newton flow, saddle connection, Morse theory, Reeb decomposition, Lean 4},
}

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\runninghead{Erd\H{o}s \#1041: Newton flow and ray separation}
\title{Newton Flow and Critical-Value Ray Separation}
\subtitle{A checked dynamical input for Erd\H{o}s Problem~\#1041}
\author{Will Cook}
\date{July 2026}

\begin{document}
\maketitle
\noteprovenance

\begin{abstract}
\noindent
Let $f(z)=\prod_{i}(z-z_i)$ be monic with every root in the open unit disc.
Erd\H{o}s, Herzog, and Piranian asked whether two roots can always be joined by
a curve of length less than $2$ contained in the open lemniscate
$\{|f|<1\}$.  The problem is open.  A manuscript posted in March 2026 claims
the result for all such $f$; its Proposition~12 is a load-bearing spanning-tree
decomposition whose printed proof contains an invalid local saddle block.  At
an interior Morse saddle a sufficiently small chart lies wholly in the open
component and contains all four sectors, so the asserted one-lower/two-upper
three-ended neighbourhood does not follow.  The proposition itself is not
refuted and may admit a cut-annulus or four-pronged repair.  An independent
June 2026 preprint proves the degree-four case.  This note isolates and checks a
dynamical input that could support an alternative decomposition.

Along a trajectory of the complex Newton field $-f/f'$ the polynomial value
satisfies $w'=-w$, so $f(z(t))=e^{-t}f(z(0))$ and $e^{t}f(z(t))$ is a first
integral.  Both are checked by the Lean kernel.  The consequence is a
separation criterion: the endpoints of any finite Newton-flow connection lie on
one oriented ray from the origin, so critical values on distinct rays admit no
saddle-to-saddle connection.  The criterion is sharper than the one the
manuscript uses.  Distinct critical values, and even distinct critical-value
moduli, do not exclude saddle connections; distinct critical-value
\emph{arguments} do.  The obstruction to arranging distinct arguments by
perturbation is also made explicit: for each pair of critical values the set of
common translations causing a ray collision is a one-real-parameter locus, given
in closed form.

What remains is the planar topology and the length bookkeeping.  Neither is
proved here, and Erd\H{o}s \#1041 is not settled.
\end{abstract}

\externalreviewobject{The paper checks exponential Newton-value decay,
parameterises positive-ray collisions, constructs arbitrarily small
translations avoiding all collisions in a finite family, and proves root
retention under a quantified constant perturbation.}{These are dynamical and
perturbative inputs.  They do not repair the global topology and metric gluing
needed for a curve of length less than $2$, so Problem~\#1041 remains open.}{Comparator
selects the finite-family small-translation theorem and the quantified
root-retention theorem.  The literature diagnosis and finite computation
remain separate evidence classes.}

\section{The problem}\label{sec:problem}

\begin{problem}[Erd\H{o}s \#1041]\label{res:problem}
Let $f(z)=\prod_{i=1}^{n}(z-z_i)$ be monic with $z_i\in\dbar$, the open unit
disc.  Show that two of the roots can be joined by a curve of length less than
$2$ lying in the open lemniscate $E=\{z\in\mathbb{C}:|f(z)|<1\}$.
\end{problem}

\noindent Numbering and current status follow Bloom's Erd\H{o}s problem
catalogue \cite{bloom}.
The problem is open.  The original source is Problem~5 on printed p.~139 of
Erd\H{o}s--Herzog--Piranian \cite[p.~139]{ehp1958}; 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
\texttt{shtuka} on 24 March 2026
\cite[Theorem~1, p.~1]{march2026} 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~\ref{sec:gap}
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 \cite[Thm.~1, p.~1]{june2026} 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.

\statusboundary

\begin{table}[t]
\centering
\small
\begin{tabular}{@{}
  >{\raggedright\arraybackslash}p{0.34\linewidth}
  >{\raggedright\arraybackslash}p{0.20\linewidth}
  >{\raggedright\arraybackslash}p{0.38\linewidth}@{}}
\toprule
Statement & Status & Exact boundary \\
\midrule
Erd\H{o}s \#1041 & Open & No proof is claimed. \\
Newton value equation $w'=-w$ & Checked & Away from critical points, along any trajectory tangent to $-f/f'$. \\
Exponential first integral & Checked & $\tfrac{d}{dt}\bigl(e^{t}f(z(t))\bigr)=0$. \\
Ray separation of critical values & Checked (consumer form) & Endpoints of a finite connection share one oriented ray; distinct rays exclude a connection. \\
Ray-collision locus & Checked & $\beta=(ra-b)/(1-r)$, $r>0$, $r\ne1$: one real parameter per pair. \\
Quartic case & Cited & Proved in \cite[Thm.~1, p.~1]{june2026}; does not extend to general degree. \\
Unrestricted proof of \cite[Theorem~1, p.~1]{march2026} & Proof gap & Proposition~12 uses a false three-ended local saddle block; located publicly by Tao (25 March 2026), conceded by the author at statement level (26 March 2026).  No counterexample is exhibited. \\
Constant-translation ray separation and root retention & Checked & After critical-value injectivity, arbitrary small ray avoidance and an explicit unit-disc margin. \\
Coefficient perturbation and slack stability & Open & Must first create injective critical values and preserve the component, collars and length budget. \\
Reeb decomposition and length fan-in & Open & The two surviving producers. \\
Random search to degree $10$ & Verified finite instances & Upper bounds for sampled configurations only. \\
\bottomrule
\end{tabular}
\end{table}

\section{The Newton value equation}\label{sec:newton}

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 $w(t)=f(z(t))$.

\begin{theorem}[value equation]\label{res:value}
$w'(t)=-w(t)$.
\end{theorem}

\noindent The computation is one line: $w'=f'(z)\,z'=f'(z)\cdot(-f(z)/f'(z))=-f(z)=-w$.
The kernel checks it as \pdecl{newtonFlow_value_hasDerivAt}, together with the
differential form of the first integral,
\pdecl{newtonFlow_scaledValue_hasDerivAt_zero}:
\[
  \frac{d}{dt}\Bigl(e^{t}f(z(t))\Bigr)=0,
  \qquad\text{equivalently}\qquad
  f(z(t))=e^{-t}f(z(0)) .
\]
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.

\begin{corollary}[ray separation]\label{res:ray}
The endpoints of any finite Newton-flow connection lie on the same oriented ray
from $0$.  Consequently, if two critical values lie on distinct positive rays,
no Newton-flow saddle connection joins the corresponding critical points.
\end{corollary}

\noindent 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.

\section{Arguments, not moduli}\label{sec:arguments}

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~\ref{res:ray} the
invariant that excludes connections is the argument.  What a perturbation must
therefore achieve is pairwise distinct critical-value \emph{arguments}, 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.

\begin{theorem}[ray-collision locus]\label{res:locus}
Let $a\ne b$ be complex.  Every common translation $\beta$ for which $a+\beta$
and $b+\beta$ lie on the same positive ray has the form
\[
  \beta=\frac{ra-b}{1-r},
  \qquad r\in\mathbb{R}_{>0},\ r\ne1 .
\]
\end{theorem}

\noindent So each pair of critical values contributes a one-real-parameter
forbidden locus in the translation plane, given in closed form
(\pdecl{translated_samePositiveRay_parameterization}).  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 $\dbar$ and preserves the length slack.  That is the first
open producer of \S\ref{sec:open}.

\section{A proof gap in the unrestricted argument}\label{sec:gap}

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~\eqref{eq:prop12-bound}.
The shorter descriptions of the same three-ended block do not repair the
four-sector topology.

Corollary~\ref{res:ray} 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.

\section{Finite evidence}\label{sec:finite}

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.

\section{Complements and further questions}\label{sec:open}

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.

\subsection*{1. Repair or refute the saddle block}

\begin{problem}[corrected local saddle assembly]\label{prob:saddle1041}
Under the hypotheses of Proposition~12, replace the invalid three-ended local
model by a four-pronged block or by a block formed after a specified annular
cut.  Prove that for every $\eta>0$ its connector has total Euclidean length
below $\eta$, uniformly over the selected attachment points, and that the
corrected blocks assemble to an embedded tree satisfying
\[
 \operatorname{len}(G_\varepsilon)
 \le\frac1{2\pi}\int_{2\alpha}^{\infty}P_V(t)\,dt+\varepsilon;
\]
or give a polynomial or harmonic planar counterexample to that statement.
\end{problem}

The full-disc model $x^2-y^2$, an annulus in which lower branches rejoin, two
saddles joined by a separatrix and simultaneous saddle levels are mandatory
tests.  Repeating the one-lower/two-upper assertion does not answer the
problem.

\subsection*{2. The compact ray-cut decomposition}

Let $p$ have simple roots, simple nonzero critical points and let
$u=-\log|p|$.  For regular values $c<T$, take
\[
 M_{c,T}=\overline{V\cap\{c<u<T\}},
\]
and assume explicitly that this is a compact genus-zero surface, its lower
boundary is one smooth Jordan curve, its upper boundary consists of $m$ smooth
root curves, all interior critical points are nondegenerate index-one saddles,
the normalised gradient field
\[
 X=\frac{\nabla u}{|\nabla u|^2}=-\frac{p}{p'}
\]
is transverse to the level boundaries, and no maximal $X$-trajectory has two
saddle endpoints.

\begin{problem}[finite strip decomposition after ray cuts]\label{prob:reeb1041}
For every $\eta>0$, construct disjoint Morse neighbourhoods $N_j$ with
$\sum_j\operatorname{diam}N_j<\eta$ and a finite set of complete separatrix or
regular-flow cuts so that every component of the complement is flow-diffeomorphic
to a rectangle
\[
 [a_S,b_S]\times[0,1],\qquad u(\Phi_S(t,s))=t,
\]
with connected level sections and no uncut annular component.  Give an explicit
finite bound $E(s,m)$ for the number of strips, or exhibit the minimal missing
hypothesis or a counterexample.
\end{problem}

No particular formula such as $2s+1$ is presumed.  Boundary tangencies,
simultaneous levels, branch reunion through an annulus and non-Hausdorff orbit
spaces must be handled rather than suppressed.

\subsection*{3. Metric fan-in without losing the coefficient}

For a strip $S$, write
\[
 \Gamma_t^S=S\cap\{u=t\},\qquad
 P_S(t)=\mathcal H^1(\Gamma_t^S),
\]
and let $k_S$ be its number of root ends.  The flux identity
\[
 \int_{\Gamma_t^S}|\nabla u|\,ds=2\pi k_S
\]
gives the average trajectory estimate
\[
 \int_{\Gamma_{t_0}^S}\operatorname{len}(\gamma_x)\,d\mu_{t_0}(x)
 =
 \frac1{2\pi k_S}\int_{a_S}^{b_S}P_S(t)\,dt.
\]

\begin{problem}[additive-error strip gluing]\label{prob:metric1041}
Assuming Problem~\ref{prob:reeb1041}, select trajectories in all strips and
connect them through the saddle and root neighbourhoods so that, for every
$\eta>0$, the resulting embedded tree contains all $m$ roots and obeys
\[
 \operatorname{len}(G)
 \le\frac1{2\pi}\int_{2\alpha}^{\infty}P_V(t)\,dt+\eta.
\tag{6.1}\label{eq:metric-fanin1041}
\]
The total saddle, annular-cut and root-cap cost must be below $\eta$ without a
multiplicative loss in $1/(2\pi)$.
\end{problem}

For the final strict inequality, use the actual collar slack
\[
 q=\frac1{2\pi}\int_{\alpha}^{2\alpha}P_V(t)\,dt>0
\]
and give budgets that keep the perturbation, tree error and transfer cost below
fixed fractions of $q$.  Independently choosing a shortest trajectory in each
strip is not enough unless the attachment mismatch is controlled.

\subsection*{4. Coefficient perturbation and stability}

The constant-translation stage is no longer open.  Once a finite critical-value
family is injective, Lean proves an arbitrarily small translation making every
value nonzero and pairwise positive-ray separated
(\pdecl{exists_small_translation_separating_arguments}).  It also proves the
explicit root-retention estimate; a shift below $\varepsilon$ keeps all roots
in the unit disc when
\[
 ((n+1)\varepsilon)^{1/n}+\rho<1
\]
(\pdecl{constant_perturbation_roots_in_unitDisk}).  A constant translation
cannot separate initially equal critical values.

\begin{problem}[two-stage generic perturbation with slack]\label{prob:perturb1041}
For fixed root discs, compact collar $K$, regular levels
$\alpha/2,\alpha,2\alpha,5\alpha/2$ and collar slack $q>0$, prove that an
arbitrarily small
\[
 g_{\lambda,\beta}(z)=f(z)+\lambda z+\beta
\]
can be chosen so that:

\begin{enumerate}
\item $f+\lambda z$ has simple relevant critical points and injective complex
critical values;
\item the checked constant translation $\beta$ makes those values nonzero and
pairwise ray-separated;
\item no critical point enters the protected collar or truncation boundary;
\item the relevant component remains in $K$, contains exactly the corresponding
perturbed roots and has slack $q_g>q/2$; and
\item root displacement and straight-line transfer back to the original roots
consume less than a prescribed fraction of $q/m$.
\end{enumerate}

If the one-coefficient perturbation $\lambda z$ cannot ensure all five
properties, give the smallest additional lower-coefficient direction that can,
or an explicit obstruction.
\end{problem}

Finite planar avoidance and the subsequent constant translation must not be
relisted as missing; coefficient genericity, component stability and slack
stability are the open content.

\subsection*{5. The global Newton-flow claim ceiling}

\begin{problem}[relative global Newton-flow theorem]\label{prob:globalflow1041}
On a compact regular band
\[
 M_{a,b}=\{z:a\le-\log|p(z)|\le b\},
\]
prove real-time existence and the identities
\[
 p(z(t))=e^{-(t-t_0)}p(z(t_0)),\qquad
 u(z(t))=u(z(t_0))+t-t_0
\]
throughout every maximal orbit; classify both limiting endpoints among the
regular boundary, root ends and finite saddle set; and exclude recurrent,
accumulating and non-Hausdorff orbit phenomena.  Under pairwise ray separation
of the critical values, decide whether the flow is Morse--Smale relative to the
boundary or whether the strongest valid conclusion is only absence of
saddle-to-saddle connections.
\end{problem}

The checked algebra proves the pointwise value equation and the endpoint-ray
consumer.  It does not supply global solution theory or an orbit-space graph.
A positive stronger theorem must give the graph and a finite edge bound; a
negative answer should exhibit the simplest ray-separated polynomial carrying
the remaining pathology.

Erd\H{o}s~\#1041 remains open.  The source now publicly verifies the Newton
kernel, finite ray avoidance and quantitative constant-translation root
control; the coefficient perturbation, corrected planar decomposition and
metric gluing remain the exact unresolved producers.

\section*{Statements and declarations}

\paragraph{Declaration of generative AI use.} Every word of this manuscript was
generated by agents based on large language models operating within Will Cook's
private research system for artificial intelligence. The formal proofs and
repository software were likewise drafted and revised by the agents through
that system under Cook's direction. Cook set the objectives and acceptance
criteria, selected and reviewed the public claims, and approved the published
version. Cook assumes responsibility for the accuracy, interpretation, and
presentation of the work. Generative systems are production tools, not authors,
and supply no independent authority.

Lean does not authorise the exposition, the citation choices, or the
interpretation, for which the author remains responsible.  This manuscript is
authored exposition, not Lean proof authority.  The checked
core is the Newton value equation, the exponential first integral, the consumer
form of ray separation, the finite planar-avoidance theorem, quantitative
constant-translation root retention, and the ray-collision parameterisation.  The
decomposition and length statements of \S\ref{sec:open} are not proved.  The
diagnosis of Proposition~12 in \S\ref{sec:gap} concerns its printed local
saddle construction; it does not refute the proposition's statement.  The
search results of \S\ref{sec:finite} are computations.

\appendix
\section{Guide to the formal sources}\label{app:sources}

The public \texttt{ErdosProblems.Erdos1041.NewtonFlowRaySeparation} module
contains the checked source for this note.  The search of \S\ref{sec:finite} is
\texttt{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.

\begin{itemize}[leftmargin=*]
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{34}{newtonFlowVector}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{38}{derivative_mul_newtonFlowVector}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{50}{newtonFlow_value_hasDerivAt}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{64}{newtonFlow_scaledValue_hasDerivAt_zero}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{77}{SamePositiveRay}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{80}{samePositiveRay_refl}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{84}{samePositiveRay_symm}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{92}{samePositiveRay_trans}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{107}{translated_samePositiveRay_parameterization}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{127}{realAffineLine}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{130}{realAffineLine_eq_affineSpan}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{147}{isClosed_realAffineLine}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{152}{dense_compl_realAffineLine}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{162}{exists_small_avoiding_finite_realAffineLines}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{179}{samePositiveRay_imp_mem_realAffineLine}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{197}{exists_small_translation_separating_arguments}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{230}{norm_lt_one_of_near_root}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{257}{constant_perturbation_root_near_original}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{287}{constant_perturbation_roots_in_unitDisk}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{306}{samePositiveRay_of_real_exp_decay}
\item \lref{Erdos1041/NewtonFlowRaySeparation.lean}{315}{no_newtonConnection_of_not_samePositiveRay}
\end{itemize}

\begin{thebibliography}{9}
\bibitem{bloom} T.~F. Bloom, \emph{Erd\H{o}s Problems}, problem~1041.
  \url{https://www.erdosproblems.com/1041}
\bibitem{ehp1958} P.~Erd\H{o}s, F.~Herzog, and G.~Piranian,
  \emph{Metric properties of polynomials}, J. Analyse Math. \textbf{6}
  (1958), 125--148. \url{https://doi.org/10.1007/BF02790232}
\bibitem{march2026} \texttt{shtuka},
  \emph{A Short Path Joining Two Zeros Inside a Polynomial Lemniscate},
  manuscript posted 24 March 2026, 48~pp.
  \url{https://shtuka123.github.io/1041/main.pdf}.
  The file at this URL has since been replaced by a shorter partial version
  that no longer contains Proposition~12; the durable public record of the
  March version, its defect, and the author's 26 March 2026 concession is the
  discussion thread at
  \url{https://www.erdosproblems.com/forum/thread/1041}.
\bibitem{june2026} V.~S. Pendyala,
  \emph{A Degree-Four Lemniscate Path Theorem},
  arXiv:2606.24875v1 (2026).
  \url{https://arxiv.org/abs/2606.24875},
  doi:\href{https://doi.org/10.48550/arXiv.2606.24875}
  {10.48550/arXiv.2606.24875}.
\end{thebibliography}

\end{document}
