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

@@ -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 03:18
+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
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
@@ -193,7 +193,7 @@ File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
 \l__color_backend_stack_int=\count280
 \l__pdf_internal_box=\box52
 )
-No file orbit_decomposition.aux.
+(./orbit_decomposition.aux)
 \openout1 = `orbit_decomposition.aux'.
 
 LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 16.
@@ -210,6 +210,7 @@ LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 16.
 LaTeX Font Info:    ... okay on input line 16.
 LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 16.
 LaTeX Font Info:    ... okay on input line 16.
+
 (/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
 [Loading MPS to PDF converter (version 2006.09.02).]
 \scratchcounter=\count281
@@ -288,26 +289,19 @@ ecomposition[]data.txt\OT1/cmr/m/n/10.95 .
 [1
 
 {/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
-
-LaTeX Warning: Reference `obs:s3-closed' on page 2 undefined on input line 130.
-
-
 [2] [3] (./orbit_decomposition.aux)
 
-LaTeX Warning: There were undefined references.
-
-
 LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right.
 
  ) 
 Here is how much of TeX's memory you used:
- 4573 strings out of 478268
- 74153 string characters out of 5846347
- 373398 words of memory out of 5000000
- 22758 multiletter control sequences out of 15000+600000
+ 4576 strings out of 478268
+ 74235 string characters out of 5846347
+ 373417 words of memory out of 5000000
+ 22759 multiletter control sequences out of 15000+600000
  481067 words of font info for 74 fonts, out of 8000000 for 9000
  1141 hyphenation exceptions out of 8191
- 55i,8n,63p,253b,198s stack positions out of 10000i,1000n,20000p,200000b,200000s
+ 55i,8n,63p,253b,209s stack positions out of 10000i,1000n,20000p,200000b,200000s
 {/usr/local/texlive/2022/texmf-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc
 }</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb>
 </usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb><
@@ -323,7 +317,7 @@ live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texl
 ive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texli
 ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/local/texliv
 e/2022/texmf-dist/fonts/type1/public/cm-super/sfrm1095.pfb>
-Output written on orbit_decomposition.pdf (3 pages, 166513 bytes).
+Output written on orbit_decomposition.pdf (3 pages, 173654 bytes).
 PDF statistics:
  85 PDF objects out of 1000 (max. 8388607)
  51 compressed objects within 1 object stream

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}

papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.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 02:32
+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
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
@@ -289,36 +289,37 @@ tep2[]data.txt\OT1/cmr/m/n/10.95 .
 [1
 
 {/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
-[2] [3] (./tire_fiber_step2.aux) ) 
+[2] [3] [4] (./tire_fiber_step2.aux) ) 
 Here is how much of TeX's memory you used:
- 4569 strings out of 478268
- 74024 string characters out of 5846347
+ 4571 strings out of 478268
+ 74050 string characters out of 5846347
  395361 words of memory out of 5000000
- 22748 multiletter control sequences out of 15000+600000
- 478865 words of font info for 64 fonts, out of 8000000 for 9000
+ 22749 multiletter control sequences out of 15000+600000
+ 479033 words of font info for 65 fonts, out of 8000000 for 9000
  1141 hyphenation exceptions out of 8191
  55i,10n,63p,250b,218s stack positions out of 10000i,1000n,20000p,200000b,200000s
-{/usr/local/texlive/2022/texmf-dist/fonts/enc
-/dvips/cm-super/cm-super-ts1.enc}</usr/local/texlive/2022/texmf-dist/fonts/type
-1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
-/public/amsfonts/cm/cmbx12.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/
-public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
-ublic/amsfonts/cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
-lic/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publi
-c/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
-amsfonts/cm/cmr17.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
-sfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
-nts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
-/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
-cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm
-/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c
-mti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm
-tt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/cm-super/sfrm10
-95.pfb>
-Output written on tire_fiber_step2.pdf (3 pages, 191120 bytes).
+{/usr/local/texlive/2022/texmf-dist/fonts
+/enc/dvips/cm-super/cm-super-ts1.enc}</usr/local/texlive/2022/texmf-dist/fonts/
+type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/t
+ype1/public/amsfonts/cm/cmbx12.pfb></usr/local/texlive/2022/texmf-dist/fonts/ty
+pe1/public/amsfonts/cm/cmitt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/ty
+pe1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
+e1/public/amsfonts/cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
+/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
+ublic/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
+lic/amsfonts/cm/cmr17.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publi
+c/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
+msfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
+onts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
+nts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
+s/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
+cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
+m/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/cm-super/sf
+rm1095.pfb>
+Output written on tire_fiber_step2.pdf (4 pages, 198763 bytes).
 PDF statistics:
- 90 PDF objects out of 1000 (max. 8388607)
- 54 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)
 

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]