diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux index ba1cbd9..276d5d1 100644 --- a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux +++ b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux @@ -6,4 +6,4 @@ \newlabel{obs:antipodal-rainbow-conjecture}{{}{3}} \@writefile{toc}{\contentsline {paragraph}{Direct test.}{3}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Why antipodal?}{3}{}\protected@file@percent } -\gdef \@abspage@last{3} +\gdef \@abspage@last{4} diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log index 65e6984..7a156e9 100644 --- a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log +++ b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 03:29 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 03:50 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -289,38 +289,44 @@ ecomposition[]data.txt\OT1/cmr/m/n/10.95 . 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PDF statistics: - 85 PDF objects out of 1000 (max. 8388607) - 51 compressed objects within 1 object stream + 98 PDF objects out of 1000 (max. 8388607) + 59 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 1 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.pdf b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.pdf index b605e1f..0d19fd4 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.pdf and b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.tex b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.tex index ba383f0..2b3539b 100644 --- a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.tex +++ b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.tex @@ -151,28 +151,71 @@ This is a real upgrade on the step-$2$ data: \section*{Conjecture suggested by the data} -\begin{obs}[Antipodal-chord rainbow conjecture] +\begin{obs}[Antipodal-chord rainbow conjecture --- \emph{refuted as +originally stated}; refined below] \label{obs:antipodal-rainbow-conjecture} +\emph{Original (incorrect) statement.} Let $T = (m, (0, m/2), \mathrm{SP})$ be a Steiner-poor tire whose inner outerplanar graph $O$ is a cycle of length $m$ together with a -single antipodal chord (so $m$ is even). Conjecture: the projection -support $\pi_D(\mathcal{C}(T))$ on the $|\gamma| = m$ inner-side -spokes always contains the combined orbit +single antipodal chord (so $m$ is even). Conjecture (originally, +without an outer-boundary precondition): the projection support +$\pi_D(\mathcal{C}(T))$ on the $|\gamma| = m$ inner-side spokes +always contains the combined orbit \[ \mathrm{Orbit}\bigl(\,(a, b, c, b, c, \dots, b, c, a)\,\bigr) \] -under $S_3 \times C_m$ (color permutation $\times$ cyclic rotation), -where the pattern has length $m$ and the $a$-positions are exactly -the two chord endpoints, with $b$ and $c$ alternating elsewhere. -At $m = 6$ this is the rainbow orbit of size $36$ that -Obs.~\ref{obs:rainbow-source} witnessed. +under $S_3 \times C_m$, with $a$-positions at the two chord endpoints +and $b, c$ alternating elsewhere. At $m = 6$ this is the rainbow +orbit of size $36$ that Obs.~\ref{obs:rainbow-source} witnessed. + +\medskip +\noindent +\emph{The original statement is false:} the +counterexample-search log +(\texttt{experiments/counterexample\_search.log}) records the pair +\[ + T_1 = (m_1 = 3,\ (0, 3),\ \mathrm{SP}), + \qquad + T_2 = (k_2 = 3,\ -,\ \mathrm{SP}), + \qquad + |\gamma| = 6, +\] +where $T_1$ is the antipodal-chord SP tire at $\gamma = 6$ but with +\emph{small outer boundary} $m_1 = 3$. Direct computation gives +$|\pi_D(\mathcal{C}(T_1))| = 18$. The conjectured rainbow combined +orbit has size $36$, so the orbit cannot fit inside $\pi_D$; in fact +only $6$ of the $36$ rainbow-orbit elements lie in $\pi_D$, and the +literal generator pattern $(1, 2, 3, 2, 3, 1)$ is itself among the +$30$ \emph{missing} elements. Furthermore, $\pi_U^{(2)}(\mathcal{C}(T_2))$ +has $|\pi_U| = 84$, but $\pi_D(\mathcal{C}(T_1)) \cap +\pi_U(\mathcal{C}(T_2)) = \emptyset$ in both orientations, so this is +also a (strict-Latin) counterexample to step-$2$ compatibility. + +\medskip +\noindent +\emph{Refined statement (now consistent with all data).} +The original conjecture omitted an outer-boundary precondition. The +correct conjecture is: +\begin{quote} +\textbf{If additionally $m_1 = |B_{\mathrm{out}}^{(1)}| \geq m$ +(i.e.\ the outer boundary is at least as long as the inner +boundary), then $\pi_D(\mathcal{C}(T))$ contains the antipodal +rainbow combined orbit.} +\end{quote} +The threshold $m_1 \geq m$ is the same saturation threshold from +step~$1$: when the dual cycle of $T$ has length $n_1 = m_1 + m +\geq 2m$, the spoke projection saturates the full Latin set on +$\gamma$. Under this precondition all $23$ pairs of the original +step-$2$ tests and all $44$ strict-Latin pairs in the search log +with $m_1 \geq m$ \emph{and} $k_2 \geq m$ confirm the property. \end{obs} -If true, this is a uniform structural property of the antipodal-chord -SP tire, independent of the outer boundary length. The chain -pigeonhole step at $|\gamma| = m$ on such a tire reduces to -``$\pi_U$ of the other tire intersects this fixed orbit,'' a much -smaller compatibility claim. +If the refined statement holds, this is a uniform structural property +of the antipodal-chord SP tire \emph{with sufficient outer cycle +length}. The chain pigeonhole step at $|\gamma| = m$ on such a tire +reduces to ``$\pi_U$ of the other tire intersects this fixed orbit,'' +a much smaller compatibility claim --- but only once both sides clear +the outer-boundary threshold. \paragraph{Direct test.} At $m = 4$ ($\theta(1, 2, 2) = K_4 - e$) the antipodal-chord SP tire's $\pi_D$ support has size $36$ at $|\gamma| =