From f17dfdabd162737b77bb0ab6408c8f27e90b178e Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 03:29:44 -0400 Subject: [PATCH] coloring_nested_tire_graphs: note the antipodal-chord rainbow conjecture; cross-link from step-2 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- .../notes/orbit_decomposition.aux | 3 + .../notes/orbit_decomposition.log | 24 +++----- .../notes/orbit_decomposition.pdf | Bin 166513 -> 173654 bytes .../notes/orbit_decomposition.tex | 40 +++++++++++++ .../notes/tire_fiber_step2.aux | 2 +- .../notes/tire_fiber_step2.log | 53 +++++++++--------- .../notes/tire_fiber_step2.pdf | Bin 191120 -> 198763 bytes .../notes/tire_fiber_step2.tex | 16 ++++++ 8 files changed, 96 insertions(+), 42 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux index 1433a06..ba1cbd9 100644 --- a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux +++ b/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} diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log index c3e3616..65e6984 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 03:18 +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 } < @@ -323,7 +317,7 @@ live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb> -Output written on orbit_decomposition.pdf (3 pages, 166513 bytes). +Output written on orbit_decomposition.pdf (3 pages, 173654 bytes). 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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]