coloring_nested_tire_graphs: examine chain pigeonhole on cut tires

New note cut_tire_chain_pigeonhole.tex (3 pages) walking through the
chain pigeonhole argument applied to the cut-tire framework:

Setup: minimum counterexample G' to 4CT, 6-edge cut splits into
G'_0, G'_1, depth labelling gives chains of cut tires on each side.

Argument shape:
  (1) Minimality ⇒ both G'_i have proper 3-edge-colorings.
  (2) Restrict to depth-0 pendants → boundary configurations σ_0, σ_1.
  (3) Glue iff σ_0 = σ_1; counterexample ⇒ no such matching.
  (4) Layered description: each cut tire has inner/outer projection
      constraints; adjacent tires share layers (outer of T_d =
      inner of T_{d+1}).
  (5) Chain pigeonhole: if each π_in is "large enough,"
      R_0 ∩ R_1 ≠ ∅, contradiction.

What this needs to be a proof:
  (a) Chain well-definedness: each H_d has ≥ 1 face, adjacencies
      clean, no degenerate cases.
  (b) Quantitative chain pigeonhole at each layer (= the rainbow
      conjecture or König-lift conjecture from existing notes).
  (c) Cut-tire-specific issues: H_d not cubic, face boundaries may
      not be simple cycles, transfer of primal-tire results requires
      verification.

Empirical chain on G'_1 of Holton-McKay #0: chain length 7,
irregular face structure across depths.  No counterexample yet but
no proof either.

Net assessment: structurally sound reformulation, but inherits all
open conjectures from the existing approaches plus new technical
issues.

Concrete next step: extend tire_fiber_step2-style pairwise compatibility
check to the cut-tire setting and see if R_0 ∩ R_1 = ∅ empirically
on the Holton-McKay graphs.

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

papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.aux

@@ -0,0 +1,8 @@
+\relax 
+\@writefile{toc}{\contentsline {paragraph}{(Step 1) Reduction by minimality.}{1}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{(Step 2) The induced cut configuration.}{1}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{(Step 3) Gluing.}{1}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{(Step 4) Layered description of $\mathcal  {R}_i$ via cut tires.}{1}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{(Step 5) Chain pigeonhole at the cut.}{2}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{Concrete next steps.}{4}{}\protected@file@percent }
+\gdef \@abspage@last{4}

papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.log

@@ -0,0 +1,315 @@
+This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  26 MAY 2026 15:37
+entering extended mode
+ restricted \write18 enabled.
+ %&-line parsing enabled.
+**cut_tire_chain_pigeonhole.tex
+(./cut_tire_chain_pigeonhole.tex
+LaTeX2e <2021-11-15> patch level 1
+L3 programming layer <2022-02-24>
+(/usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
+Document Class: article 2021/10/04 v1.4n Standard LaTeX document class
+(/usr/local/texlive/2022/texmf-dist/tex/latex/base/size11.clo
+File: size11.clo 2021/10/04 v1.4n Standard LaTeX file (size option)
+)
+\c@part=\count185
+\c@section=\count186
+\c@subsection=\count187
+\c@subsubsection=\count188
+\c@paragraph=\count189
+\c@subparagraph=\count190
+\c@figure=\count191
+\c@table=\count192
+\abovecaptionskip=\skip47
+\belowcaptionskip=\skip48
+\bibindent=\dimen138
+)
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
+Package: amsmath 2021/10/15 v2.17l AMS math features
+\@mathmargin=\skip49
+
+For additional information on amsmath, use the `?' option.
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
+Package: amstext 2021/08/26 v2.01 AMS text
+
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
+File: amsgen.sty 1999/11/30 v2.0 generic functions
+\@emptytoks=\toks16
+\ex@=\dimen139
+))
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
+Package: amsbsy 1999/11/29 v1.2d Bold Symbols
+\pmbraise@=\dimen140
+)
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
+Package: amsopn 2021/08/26 v2.02 operator names
+)
+\inf@bad=\count193
+LaTeX Info: Redefining \frac on input line 234.
+\uproot@=\count194
+\leftroot@=\count195
+LaTeX Info: Redefining \overline on input line 399.
+\classnum@=\count196
+\DOTSCASE@=\count197
+LaTeX Info: Redefining \ldots on input line 496.
+LaTeX Info: Redefining \dots on input line 499.
+LaTeX Info: Redefining \cdots on input line 620.
+\Mathstrutbox@=\box50
+\strutbox@=\box51
+\big@size=\dimen141
+LaTeX Font Info:    Redeclaring font encoding OML on input line 743.
+LaTeX Font Info:    Redeclaring font encoding OMS on input line 744.
+\macc@depth=\count198
+\c@MaxMatrixCols=\count199
+\dotsspace@=\muskip16
+\c@parentequation=\count266
+\dspbrk@lvl=\count267
+\tag@help=\toks17
+\row@=\count268
+\column@=\count269
+\maxfields@=\count270
+\andhelp@=\toks18
+\eqnshift@=\dimen142
+\alignsep@=\dimen143
+\tagshift@=\dimen144
+\tagwidth@=\dimen145
+\totwidth@=\dimen146
+\lineht@=\dimen147
+\@envbody=\toks19
+\multlinegap=\skip50
+\multlinetaggap=\skip51
+\mathdisplay@stack=\toks20
+LaTeX Info: Redefining \[ on input line 2938.
+LaTeX Info: Redefining \] on input line 2939.
+)
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
+Package: amssymb 2013/01/14 v3.01 AMS font symbols
+
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
+Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
+\symAMSa=\mathgroup4
+\symAMSb=\mathgroup5
+LaTeX Font Info:    Redeclaring math symbol \hbar on input line 98.
+LaTeX Font Info:    Overwriting math alphabet `\mathfrak' in version `bold'
+(Font)                  U/euf/m/n --> U/euf/b/n on input line 106.
+))
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
+Package: amsthm 2020/05/29 v2.20.6
+\thm@style=\toks21
+\thm@bodyfont=\toks22
+\thm@headfont=\toks23
+\thm@notefont=\toks24
+\thm@headpunct=\toks25
+\thm@preskip=\skip52
+\thm@postskip=\skip53
+\thm@headsep=\skip54
+\dth@everypar=\toks26
+)
+(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
+Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR)
+
+(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
+Package: keyval 2014/10/28 v1.15 key=value parser (DPC)
+\KV@toks@=\toks27
+)
+(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
+Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR)
+
+(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
+Package: trig 2021/08/11 v1.11 sin cos tan (DPC)
+)
+(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
+File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
+)
+Package graphics Info: Driver file: pdftex.def on input line 107.
+
+(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
+File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex
+))
+\Gin@req@height=\dimen148
+\Gin@req@width=\dimen149
+)
+(/usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
+Package: geometry 2020/01/02 v5.9 Page Geometry
+
+(/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
+Package: ifvtex 2019/10/25 v1.7 ifvtex legacy package. Use iftex instead.
+
+(/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
+Package: iftex 2022/02/03 v1.0f TeX engine tests
+))
+\Gm@cnth=\count271
+\Gm@cntv=\count272
+\c@Gm@tempcnt=\count273
+\Gm@bindingoffset=\dimen150
+\Gm@wd@mp=\dimen151
+\Gm@odd@mp=\dimen152
+\Gm@even@mp=\dimen153
+\Gm@layoutwidth=\dimen154
+\Gm@layoutheight=\dimen155
+\Gm@layouthoffset=\dimen156
+\Gm@layoutvoffset=\dimen157
+\Gm@dimlist=\toks28
+)
+(/usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
+Package: booktabs 2020/01/12 v1.61803398 Publication quality tables
+\heavyrulewidth=\dimen158
+\lightrulewidth=\dimen159
+\cmidrulewidth=\dimen160
+\belowrulesep=\dimen161
+\belowbottomsep=\dimen162
+\aboverulesep=\dimen163
+\abovetopsep=\dimen164
+\cmidrulesep=\dimen165
+\cmidrulekern=\dimen166
+\defaultaddspace=\dimen167
+\@cmidla=\count274
+\@cmidlb=\count275
+\@aboverulesep=\dimen168
+\@belowrulesep=\dimen169
+\@thisruleclass=\count276
+\@lastruleclass=\count277
+\@thisrulewidth=\dimen170
+)
+(/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
+File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
+\l__color_backend_stack_int=\count278
+\l__pdf_internal_box=\box52
+)
+(./cut_tire_chain_pigeonhole.aux)
+\openout1 = `cut_tire_chain_pigeonhole.aux'.
+
+LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 18.
+LaTeX Font Info:    ... okay on input line 18.
+LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 18.
+LaTeX Font Info:    ... okay on input line 18.
+LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 18.
+LaTeX Font Info:    ... okay on input line 18.
+LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 18.
+LaTeX Font Info:    ... okay on input line 18.
+LaTeX Font Info:    Checking defaults for TS1/cmr/m/n on input line 18.
+LaTeX Font Info:    ... okay on input line 18.
+LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 18.
+LaTeX Font Info:    ... okay on input line 18.
+LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 18.
+LaTeX Font Info:    ... okay on input line 18.
+
+(/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
+[Loading MPS to PDF converter (version 2006.09.02).]
+\scratchcounter=\count279
+\scratchdimen=\dimen171
+\scratchbox=\box53
+\nofMPsegments=\count280
+\nofMParguments=\count281
+\everyMPshowfont=\toks29
+\MPscratchCnt=\count282
+\MPscratchDim=\dimen172
+\MPnumerator=\count283
+\makeMPintoPDFobject=\count284
+\everyMPtoPDFconversion=\toks30
+) (/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
+Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
+Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4
+85.
+
+(/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
+File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
+e
+))
+*geometry* driver: auto-detecting
+*geometry* detected driver: pdftex
+*geometry* verbose mode - [ preamble ] result:
+* driver: pdftex
+* paper: <default>
+* layout: <same size as paper>
+* layoutoffset:(h,v)=(0.0pt,0.0pt)
+* modes: 
+* h-part:(L,W,R)=(72.26999pt, 469.75502pt, 72.26999pt)
+* v-part:(T,H,B)=(72.26999pt, 650.43001pt, 72.26999pt)
+* \paperwidth=614.295pt
+* \paperheight=794.96999pt
+* \textwidth=469.75502pt
+* \textheight=650.43001pt
+* \oddsidemargin=0.0pt
+* \evensidemargin=0.0pt
+* \topmargin=-37.0pt
+* \headheight=12.0pt
+* \headsep=25.0pt
+* \topskip=11.0pt
+* \footskip=30.0pt
+* \marginparwidth=59.0pt
+* \marginparsep=10.0pt
+* \columnsep=10.0pt
+* \skip\footins=10.0pt plus 4.0pt minus 2.0pt
+* \hoffset=0.0pt
+* \voffset=0.0pt
+* \mag=1000
+* \@twocolumnfalse
+* \@twosidefalse
+* \@mparswitchfalse
+* \@reversemarginfalse
+* (1in=72.27pt=25.4mm, 1cm=28.453pt)
+
+LaTeX Font Info:    Trying to load font information for U+msa on input line 19.
+
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
+File: umsa.fd 2013/01/14 v3.01 AMS symbols A
+)
+LaTeX Font Info:    Trying to load font information for U+msb on input line 19.
+
+
+(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
+File: umsb.fd 2013/01/14 v3.01 AMS symbols B
+) [1
+
+{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
+LaTeX Font Info:    Font shape `OT1/cmtt/bx/n' in size <10.95> not available
+(Font)              Font shape `OT1/cmtt/m/n' tried instead on input line 140.
+
+Overfull \hbox (0.83797pt too wide) in paragraph at lines 147--153
+[]\OT1/cmr/bx/n/10.95 K^^?onig-lift con-jec-ture (\OT1/cmtt/m/n/10.95 worst[]ca
+se[]proof[]sketch.tex\OT1/cmr/bx/n/10.95 , Conj \OT1/cmr/bx/it/10.95 t2-induces
+-partition\OT1/cmr/bx/n/10.95 ): \OT1/cmr/m/n/10.95 ad-
+ []
+
+[2]
+Overfull \hbox (3.26193pt too wide) in paragraph at lines 224--228
+[]\OT1/cmr/m/n/10.95 Inherits all the open con-jec-tures from \OT1/cmtt/m/n/10.
+95 rainbow[]proof.tex \OT1/cmr/m/n/10.95 and \OT1/cmtt/m/n/10.95 worst[]case[]p
+roof[]sketch.tex
+ []
+
+[3] [4] (./cut_tire_chain_pigeonhole.aux) ) 
+Here is how much of TeX's memory you used:
+ 3257 strings out of 478268
+ 48521 string characters out of 5846347
+ 350622 words of memory out of 5000000
+ 21443 multiletter control sequences out of 15000+600000
+ 479884 words of font info for 69 fonts, out of 8000000 for 9000
+ 1141 hyphenation exceptions out of 8191
+ 55i,7n,62p,242b,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/f
+onts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fo
+nts/type1/public/amsfonts/cm/cmbx12.pfb></usr/local/texlive/2022/texmf-dist/fon
+ts/type1/public/amsfonts/cm/cmbxti10.pfb></usr/local/texlive/2022/texmf-dist/fo
+nts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fon
+ts/type1/public/amsfonts/cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts
+/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/t
+ype1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
+e1/public/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/p
+ublic/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ
+ic/amsfonts/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/a
+msfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
+fonts/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
+nts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
+ts/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/cm-supe
+r/sfrm1095.pfb>
+Output written on cut_tire_chain_pigeonhole.pdf (4 pages, 211212 bytes).
+PDF statistics:
+ 103 PDF objects out of 1000 (max. 8388607)
+ 62 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/cut_tire_chain_pigeonhole.tex

@@ -0,0 +1,252 @@
+\documentclass[11pt]{article}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{graphicx}
+\usepackage{geometry}
+\usepackage{booktabs}
+\geometry{margin=1in}
+
+\title{Chain pigeonhole on cut tires:\\
+       a sketch and honest assessment}
+\author{}
+\date{}
+
+\newtheorem*{prop}{Proposition}
+\newtheorem*{obs}{Observation}
+\newtheorem*{conj}{Conjecture}
+\newtheorem*{lem}{Lemma}
+
+\begin{document}
+\maketitle
+
+\section*{Setup}
+
+Take $G'$ the cubic planar dual of a maximal planar graph $G$.
+Suppose $G'$ is a minimum counterexample to the $4$-colour theorem ---
+no proper $3$-edge-colouring of $G'$ exists, but every smaller cubic
+planar graph admits one.
+
+By cyclic edge-connectivity ($G$ internally $6$-connected implies $G'$
+cyclically $6$-edge-connected), pick a $6$-edge cut $C \subseteq
+E(G')$ partitioning $V(G')$ into $S$ and $V \setminus S$, both
+non-trivial.  Form $G'_0$ and $G'_1$ as in
+\texttt{cut\_depth\_label.tex} by removing $C$ and attaching pendant
+edges at degree-$2$ vertices.
+
+For the cleanest setting, assume $C$ is a \emph{matching cut} (each
+boundary vertex on each side has exactly $1$ cut edge, so each side
+attaches $6$ pendants).  Apply the BFS depth labelling: pendants get
+depth $0$; each edge adjacent (sharing a vertex) to a depth-$d$ edge
+gets depth $d + 1$.
+
+For each $d > 0$, the cut tires at $(d, f)$ (Definition in
+\texttt{cut\_depth\_label.tex}) layer $G'_i$ concentrically around
+the cut.
+
+\section*{The argument, step by step}
+
+\paragraph{(Step 1) Reduction by minimality.}
+Each $G'_i$ has $|S_i| + 6 < |V(G')|$ vertices (assuming
+$|V \setminus S_i| > 6$, which holds in the matching-cut case for any
+non-degenerate cut).  By minimality of $G'$, each $G'_i$ is properly
+$3$-edge-colourable.  Let $\chi_i : E(G'_i) \to \{1, 2, 3\}$ be any
+such colouring.
+
+\paragraph{(Step 2) The induced cut configuration.}
+The $6$ depth-$0$ pendant edges of $G'_i$ correspond bijectively to
+the $6$ cut edges of $C$ (each pendant replaces a cut edge).  Let
+$\sigma_i := \chi_i|_{\text{depth-}0\text{ edges of } G'_i} \in
+\{1, 2, 3\}^6$, indexed by the cut edges.  This is the
+\emph{boundary configuration} at the cut.
+
+\paragraph{(Step 3) Gluing.}
+A proper $3$-edge-colouring of $G'$ exists iff there exists a single
+colouring of $C$ that extends to both sides, i.e.\ iff some
+$\sigma \in \{1, 2, 3\}^6$ is achievable as both $\sigma_0$ for some
+$\chi_0$ and $\sigma_1$ for some $\chi_1$.  Let
+\[
+   \mathcal{R}_i := \{\sigma_i : \chi_i \text{ a proper edge $3$-colouring of } G'_i\}.
+\]
+Then $G'$ has a proper $3$-edge-colouring iff
+$\mathcal{R}_0 \cap \mathcal{R}_1 \ne \emptyset$.  Assuming $G'$ is a
+counterexample, this intersection is empty.
+
+\paragraph{(Step 4) Layered description of $\mathcal{R}_i$ via cut tires.}
+Each cut tire $T_d^{(i, f)}$ has its own ``ring projection''
+constraints.  Define:
+\begin{itemize}
+  \item $\pi_{\mathrm{in}}(T_d^{(i, f)})$: the projection of $\chi_i$
+        onto the depth-$(d-1)$ inner spokes of the cut tire.  For
+        $d = 1$, this is exactly $\sigma_i$ restricted to those
+        pendants whose boundary vertex sits on $f$'s boundary.
+  \item $\pi_{\mathrm{out}}(T_d^{(i, f)})$: projection onto
+        depth-$(d+1)$ outer spokes.
+\end{itemize}
+
+Adjacent cut tires share layers: outer spokes of $T_d$ are inner
+spokes of $T_{d+1}$ (when their faces are appropriately adjacent in
+the embedding).  So the chain of cut tires at depths $1, 2, \dots,
+d_{\max}$ has consistency constraints
+$\pi_{\mathrm{out}}(T_d) = \pi_{\mathrm{in}}(T_{d+1})$
+along each chain.
+
+\paragraph{(Step 5) Chain pigeonhole at the cut.}
+$\mathcal{R}_i$ is determined by the chain of cut tires on side $i$:
+a $\sigma_i$ is achievable iff there exists a consistent sequence
+of cut-tire colourings from the deep interior outward to the cut
+that projects to $\sigma_i$ on the depth-$0$ pendants.
+
+Chain pigeonhole says: if at each step $d$, the cut tire's
+achievable inner-projection set $\pi_{\mathrm{in}}(T_d)$ is
+sufficiently large (containing enough $S_3$-orbits), then
+$\mathcal{R}_0 \cap \mathcal{R}_1 \ne \emptyset$, contradicting that
+$G'$ is a counterexample.
+
+\section*{What this needs to be a proof}
+
+The argument above sketches the \emph{shape} of a proof.  The
+non-trivial parts are:
+
+\subsection*{(a) Chain consistency: well-definedness of the chain}
+
+For the chain $T_1 \to T_2 \to \dots \to T_{d_{\max}}$ to be
+well-defined, each $T_d$ must have $\ge 1$ face, and adjacent tires
+must share layers cleanly.  Obstacles:
+\begin{itemize}
+  \item The depth-$d$ subgraph $H_d$ may have no faces (if $H_d$ is a
+        tree or empty).  The empirical example
+        (\texttt{cut\_depth\_label.tex}) shows $H_d$ has $1$--$3$
+        faces at each depth $1 \le d \le 7$, but this is not
+        guaranteed in general.
+  \item Multiple faces at the same depth mean the chain forks; chain
+        pigeonhole becomes a tree-pigeonhole.
+  \item Face boundary walks need not be simple cycles --- they can
+        revisit vertices, as in the existing tire definition's
+        treatment of cut-vertices.
+\end{itemize}
+
+\subsection*{(b) Quantitative chain pigeonhole}
+
+Even with a clean chain, we need a quantitative argument that
+$|\pi_{\mathrm{in}}(T_d)|$ is large enough at each step to force
+non-empty intersection with the adjacent tire.  This is the
+\emph{same} chain pigeonhole question studied in
+\texttt{rainbow\_proof.tex} and
+\texttt{worst\_case\_proof\_sketch.tex}, now applied to the
+cut-tire chain.
+
+The two open conjectures that would close this step are:
+\begin{itemize}
+  \item \textbf{Rainbow conjecture
+        (\texttt{rainbow\_proof.tex}, Conj 1.5):} for the antipodal-chord
+        SP case, the inner-spoke projection support equals the
+        perms-per-half set $\mathcal{P}_m$.  For cut tires this would
+        mean each cut tire's $\pi_{\mathrm{in}}$ saturates a known
+        $S_3$-symmetric set.
+  \item \textbf{König-lift conjecture
+        (\texttt{worst\_case\_proof\_sketch.tex}, Conj
+        \emph{t2-induces-partition}):} adjacent tires induce
+        $\gamma$-face partitions whose K\"onig lifts give a Latin
+        intersection of size $\ge 6$.  The
+        face-pair-connection refinement
+        (\texttt{k9\_surviving\_partitions.tex}) corrects the naive
+        candidate partition.
+\end{itemize}
+
+Neither is fully proved.  See
+\texttt{two\_approaches\_comparison.tex} for the comparison.
+
+\subsection*{(c) Cut-tire-specific issues}
+
+Cut tires differ from the tires of \texttt{paper.tex} in important
+ways:
+\begin{itemize}
+  \item Cut tires are derived from the \emph{depth labelling} on
+        $G'_i$, not from a primal level structure on $G$.  The
+        correspondence to primal tires (Defs 1.15--1.17) is by
+        analogy, not by direct identification.
+  \item The depth-$d$ subgraph $H_d$ is generally \emph{not} cubic
+        and may not even be connected.  Its faces may behave
+        differently from $G'$'s faces.
+  \item The cut tire's $T'_{\mathrm{ann}}$-analogue (= face boundary
+        of $f$ in $H_d$) is a closed walk in $H_d$, not a cycle of
+        $G'$.  Its structure depends on the depth labelling.
+\end{itemize}
+
+So even with the rainbow/K\"onig conjectures proved for primal tires,
+their transfer to cut tires requires verification.
+
+\section*{Empirical check: chain length and tire structure}
+
+For the example tire chain on $G'_1$ of Holton-McKay \#0
+($d = 1, \ldots, 7$):
+\begin{center}
+\small
+\begin{tabular}{c|cccc}
+\toprule
+$d$ & \# faces in $H_d$ & largest face length & inner spokes & outer spokes \\
+\midrule
+$1$ & $2$  & $12$ & $5$ & $4$ \\
+$2$ & $2$  & $7$  & $4$ & $3$ \\
+$3$ & $3$  & $2$  & $2$ & $2$ \\
+$4$ & $2$  & $8$  & $2$ & $5$ \\
+$5$ & $2$  & $14$ & $4$ & $6$ \\
+$6$ & $1$  & $12$ & $7$ & $1$ \\
+\bottomrule
+\end{tabular}
+\end{center}
+
+\noindent
+Observations:
+\begin{itemize}
+  \item Chain length $\le 7$ in this example.
+  \item Face counts and sizes vary irregularly --- depth $3$ has
+        three small ($2$-edge) faces; depth $5$ has a face of length
+        $14$.
+  \item Total inner $+$ outer spokes at each depth ranges from
+        $4$ to $10$.
+\end{itemize}
+
+This irregularity is the structural obstacle to clean chain
+pigeonhole: a uniform bound on $|\pi_{\mathrm{in}}|$ across depths
+seems unlikely.  The actual chain pigeonhole would need to handle
+depth-by-depth structure.
+
+\section*{Net assessment}
+
+The chain pigeonhole argument on cut tires is \emph{structurally
+sound but technically open}.  It:
+\begin{itemize}
+  \item Gives a clean reformulation of the 4CT reducibility
+        question in terms of cut-derived layered structure.
+  \item Maps directly onto the existing
+        \texttt{paper.tex} tire framework via the
+        depth-as-distance-to-cut analogy.
+  \item Inherits all the open conjectures from
+        \texttt{rainbow\_proof.tex} and
+        \texttt{worst\_case\_proof\_sketch.tex} (chain pigeonhole at
+        each layer, intersection non-emptiness, etc.).
+  \item Adds new technical issues specific to the cut-tire setting
+        (depth subgraphs $H_d$ may degenerate, irregular face
+        structure across depths).
+\end{itemize}
+
+\paragraph{Concrete next steps.}
+\begin{enumerate}
+  \item Verify that the cut-tire chain is well-defined on the
+        $6$ Holton-McKay graphs and a few other test cases (each
+        $G'_i$ has cut tires at every depth $d$ in some range; no
+        empty $H_d$).
+  \item Compute $\pi_{\mathrm{in}}(T_d^{(i)})$ for each cut tire and
+        check the rainbow $S_3$-orbit appears at the cut layer ($d = 1$).
+  \item Check pairwise compatibility at the cut between $G'_0$ and
+        $G'_1$: do $\mathcal{R}_0$ and $\mathcal{R}_1$ overlap?  If
+        empirically yes for all $6$ Holton-McKay graphs, that's
+        evidence; if no for some, that's a falsification of the
+        chain argument as currently stated.
+\end{enumerate}
+
+Step 3 is the cleanest empirical test --- it's just an extension of
+the step-$2$ pairwise compatibility analysis
+(\texttt{tire\_fiber\_step2.tex}) to the cut-tire / 4CT setting.
+
+\end{document}