% A converse to the discrete four-vertex theorem (Tabachnikov's Problem 2).
% Standalone arXiv source; compile with pdflatex from this directory.
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% ---------------------------------------------------------------------------
% Notation (draft 2; one meaning per symbol, see Sections 3--5).
%   Z_n indexes vertices P_i and edges [P_i,P_{i+1}] (edge i).
%   kappa_i > 0 curvature at vertex i; r_i = 1/kappa_i (radii only).
%   h_i half-length of edge i, ell_i = 2 h_i; theta_i turning angle;
%   psi_j heading; Gap, Turn closure gap and total turn; D_kappa domain.
%   Frame e_0<e_1<e_2<e_3; readings s_j = kappa_{e_j}, f_j = kappa_{e_j+1};
%   L in R^4 active half-lengths; beta_j block turns; Phi controller;
%   iota slot embedding; cal A active edges, cal Z dormant edges.
%   Four-length system (Sec. 5): readings (1,t,q,l) / (p,q,h,1); Omega the
%   eight walls; split angles alpha_0, alpha_3, gamma_1, gamma_2; S, T pure
%   turns; chi chord equation; eps_1, eps_3 mixed-corner defects;
%   varsigma_a slope; A,B,C,D slope defects; sigma(h,l), phi(p,t,h,l) flag;
%   \hat L(u,v) orthogonal core, Delta = q^2-u^2v^2; rho_1, rho_3 residuals.
%   Peak indices: H global maximum, K highest secondary peak, J repeated max.
% ---------------------------------------------------------------------------
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\title{A converse to the discrete four-vertex theorem}
\author{Kristoffer Josefsson}
\renewcommand{\datename}{\textit{Version:}}
\date{\texttt{\gitversion}}

\begin{document}

\begin{abstract}
Tabachnikov asked whether every cyclic sequence of positive numbers with at
least two local maxima and two local minima is the sequence of radii of the
circles through consecutive vertex triples of a convex polygon whose
circumcenters lie inside the corresponding vertex angles. We prove this for
every $n\ge 4$, with extrema counted as plateaus. This is the sharp form: by
Dahlberg's discrete four-vertex theorem the condition is also necessary for
nonconstant sequences. Locally near every nonconstant admissible sequence the
polygon can be chosen to depend continuously (indeed $C^1$) on the sequence,
but in general not globally: we give an explicit path of admissible hexagon
data along which no continuous choice of realizing polygons exists. The
existence and continuity results are formalized in Lean~4 with Mathlib.
We also prove the corresponding characterization for circles tangent to three
consecutive supporting lines, using a separate tangential-polygon construction
and Wegner's forward theorem.
\end{abstract}

\maketitle

\input{tabachnikov/intro}
\input{tabachnikov/method}
\input{tabachnikov/setup}
\input{tabachnikov/anchor}
\input{tabachnikov/system}
\input{tabachnikov/constructions}
\input{tabachnikov/continuity}
\input{tabachnikov/side-tangent}
\input{tabachnikov/discussion}
\subsection*{Acknowledgements}
The author would like to thank Aristotle, Claude, Codex, Deepseek for helpful
discussions. The proofs have been assisted with the use of AI but the author
takes full responsibility for their correctness.

\appendix
\input{tabachnikov/lean}
\input{tabachnikov/elimination}

\bibliographystyle{amsplain}
\bibliography{tabachnikov/refs}

\end{document}