coloring_nested_tire_graphs: note the antipodal-chord rainbow conjecture; cross-link from step-2

Promotes the orbit_decomposition finding (rainbow orbit appears in 3
different (T_1, T_2) pairs, all with T_1 = (6, (0,3), SP)) into an
explicit conjecture:

  Conjecture (Obs:antipodal-rainbow-conjecture):
    For T = (m, (0, m/2), SP) (an antipodal-chord SP tire with m even),
    π_D(C(T)) always contains the combined orbit of
    (a, b, c, b, c, ..., b, c, a) under S_3 × C_m, with the a-positions
    at the chord endpoints and b/c alternating elsewhere.

If true, this gives a uniform structural property of antipodal-chord
SP tires: chain pigeonhole on |γ| = m shared cycles reduces to
"π_U of the other tire meets this fixed orbit."  Tested at m = 6 in
3 pairs; the m = 4 direct test (24-element conjectured orbit ⊂
36-element support) is mechanical.

Also adds a forward-pointer paragraph at the end of Obs:rainbow in
tire_fiber_step2.tex referencing orbit_decomposition.tex.

orbit_decomposition.tex: 3 pages -> 3 pages (added Conjecture section
and a "why antipodal?" paragraph).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux

@@ -3,4 +3,7 @@
 \newlabel{obs:orbit-sizes}{{}{1}}
 \newlabel{obs:rainbow-source}{{}{2}}
 \newlabel{obs:universal-orbits}{{}{2}}
+\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}

papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log

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papers/coloring_nested_tire_graphs/notes/orbit_decomposition.tex

@@ -149,6 +149,46 @@ This is a real upgrade on the step-$2$ data:
         topology, not a coincidence of one specific configuration.
 \end{itemize}
 
+\section*{Conjecture suggested by the data}
+
+\begin{obs}[Antipodal-chord rainbow conjecture]
+\label{obs:antipodal-rainbow-conjecture}
+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
+\[
+   \mathrm{Orbit}\bigl(\,(a, b, c, b, c, \dots, b, c, a)\,\bigr)
+\]
+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.
+\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.
+
+\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 $|\gamma| =
+4$ (\texttt{tire\_fiber\_chords.tex}, row ``(4,4) chord $(0,2)$''),
+and the conjectured orbit $(a, b, c, b) \cdot S_3 \times C_4$ has size
+$24$.  Confirming the conjecture at $m = 4$ amounts to checking that
+this $24$-element subset lies inside the $36$-element support; this
+is mechanical.
+
+\paragraph{Why antipodal?}  In the planar dual picture, the antipodal
+chord of $O$ corresponds to the dual edge of a single
+``maximally-separating'' chord in the tire's inner outerplanar graph:
+it splits $\pi_1$ of the annulus most symmetrically.  Any reasonable
+proof of the conjecture would have to exploit this symmetry --- e.g.\
+via reflection invariance on the chord axis.
+
 \section*{Caveats}
 
 \begin{enumerate}

papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.aux

@@ -3,4 +3,4 @@
 \newlabel{obs:containment}{{}{2}}
 \newlabel{obs:rainbow}{{}{2}}
 \newlabel{obs:reflection}{{}{3}}
-\gdef \@abspage@last{3}
+\gdef \@abspage@last{4}

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papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.tex

@@ -140,6 +140,22 @@ antipodal positions are aligned with the antipodal chord
 $(v_0, v_3)$, and the pattern factors through the $S_3$ orbit.  The
 fact that this very small intersection still contains an entire
 $S_3$-orbit is suggestive of structural rather than accidental overlap.
+
+\medskip
+\noindent\emph{Follow-up.}  An $S_3$-orbit decomposition of all $23$
+intersections (\texttt{orbit\_decomposition.tex}) shows: every
+intersection is closed under the diagonal $S_3$ action; every non-
+trivial orbit has size $6$; and the rainbow combined orbit
+$(a, b, c, b, c, a) \cdot (S_3 \times C_6)$ appears in three different
+$(T_1, T_2)$ pairs, all sharing $T_1 = (6, (0, 3), \mathrm{SP})$
+(the antipodal-chord SP tire) but with $T_2$ ranging over chordless
+$\mathrm{SR}$, chordless $\mathrm{SP}$, and two-chord $\mathrm{SP}$
+configurations.  This promotes the observation from
+``one $(T_1, T_2)$'s small intersection happens to be $S_3$-symmetric''
+to ``the antipodal-chord SP tire forces this orbit into every
+$\pi_D$-support, regardless of the other side.''  The candidate
+conjecture is recorded in \texttt{orbit\_decomposition.tex},
+Obs.\ \ref{obs:rainbow} (\emph{loc.\ cit.}).
 \end{obs}
 
 \begin{obs}[Reflection sensitivity]