didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Add Heawood boundary-restriction experiments and findings note

Browse Source 
Experiments probing the cluster restriction set R_K / Phi: R_K is a Z/3
zonotope (not a GF(3) subspace), the "richness" invariant is an artifact
of non-shrinking annuli, the interface gluing always works on interior
cycles (forced by 4CT), and the maximal constraint achievable on an
n-cycle is a floor of 2^(n-2) -- already reached by the trivial tire.
Note boundary_restriction_structure.tex writes these up.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-06-17 02:12:54 -04:00
parent 351ae0cdfe
commit 60c9f1d3a8
8 changed files with 1087 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.aux
+2
View File
@@ -0,0 +1,2 @@
\relax
\gdef \@abspage@last{2}
papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.log
+296
View File
@@ -0,0 +1,296 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 02:12
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
**boundary_restriction_structure.tex
(./boundary_restriction_structure.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
)
No file boundary_restriction_structure.aux.
\openout1 = `boundary_restriction_structure.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}] [2]
(./boundary_restriction_structure.aux) )
Here is how much of TeX's memory you used:
3254 strings out of 478268
48506 string characters out of 5846347
347668 words of memory out of 5000000
21442 multiletter control sequences out of 15000+600000
480359 words of font info for 70 fonts, out of 8000000 for 9000
1141 hyphenation exceptions out of 8191
55i,8n,62p,247b,208s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/font
s/type1/public/amsfonts/cm/cmbx12.pfb></usr/local/texlive/2022/texmf-dist/fonts
/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/
type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/t
ype1/public/amsfonts/cm/cmmi12.pfb></usr/local/texlive/2022/texmf-dist/fonts/ty
pe1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type
1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/
public/amsfonts/cm/cmr12.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pu
blic/amsfonts/cm/cmr17.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/cmss8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
sfonts/cm/cmsy10.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/amsfont
s/symbols/msbm10.pfb>
Output written on boundary_restriction_structure.pdf (2 pages, 184212 bytes).
PDF statistics:
91 PDF objects out of 1000 (max. 8388607)
54 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/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.pdf
BIN
View File
Binary file not shown.
papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.tex
+139
View File
@@ -0,0 +1,139 @@
\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{graphicx}
\usepackage{geometry}
\usepackage{booktabs}
\geometry{margin=1in}
\title{Heawood boundary restriction sets:\\
zonotope structure and the $2^{n-2}$ constraint floor}
\author{}
\date{}
\newtheorem*{obs}{Observation}
\newtheorem*{prop}{Proposition}
\newtheorem*{conj}{Conjecture}
\newtheorem*{lem}{Lemma}
\begin{document}
\maketitle
This note records the empirical structure of the Heawood boundary
restriction sets studied in \texttt{paper.tex}, and a clean
\emph{maximal-constraint} result. All claims below are backed by the
experiments in \texttt{experiments/} (filenames given inline). Sequences
live in $(\mathbb{Z}/3)^{\,\cdot}$, displayed in $\{0,1,-1\}$.
\section*{Setup}
Fix a triangulated disk $D$ with boundary cycle $C = (v_0,\dots,v_{n-1})$.
A Heawood face-labelling is $\lambda : \{\text{faces of }D\} \to \{+1,-1\}$,
with induced vertex value $\lambda^{*}(v) = \sum_{f \ni v}\lambda(f) \bmod 3$.
The achievable outer set is
\[
\Phi(D) \;=\; \bigl\{\, (\lambda^{*}(v_0),\dots,\lambda^{*}(v_{n-1}))
\;:\; \lambda \in \{+1,-1\}^{F(D)},\;
\lambda^{*}(w) \equiv 0 \ \forall\ \text{interior } w \,\bigr\}.
\]
This is exactly the value the recursive transfer operator produces at
$C$ (interior consistency $=$ all descendant gluings performed; boundary
deferred). Crucially $\Phi(D)$ depends \emph{only} on the disk
triangulation, not on any BFS/tire-tree labelling.
\section*{1. The restriction sets are zonotopes, not subspaces}
(\texttt{probe\_RK\_structure.py}.) Writing $\lambda = \mathbf{1}+b$ with
$b \in \{0,1\}^F$, the labelling map is $\lambda \mapsto M\mathbf{1}+Mb
\pmod 3$, a linear image of the Boolean cube ($M$ the face/vertex
incidence matrix). Over $3655$ cluster restriction sets $R_{\mathsf K}$:
none was an affine $\mathrm{GF}(3)$ subspace; the map is usually
injective, so $|R_{\mathsf K}| = 2^{|F|}$ (a power of $2$ inside the
column space of size $3^{\operatorname{rank} M}$); the nowhere-zero
constraint $\lambda \neq 0$ shrank the set below the full linear image in
\emph{every} case. The only surviving linear structure is
$R_{\mathsf K} \subseteq \operatorname{col}(M)$ (cokernel relations such
as $\sum_v \lambda^{*}(v) \equiv 0$). So $\Phi$ is a $\mathbb{Z}/3$
zonotope: a projected cube, sign-closed but not closed under addition.
\section*{2. ``Richness'' is not a self-similar invariant}
(\texttt{transfer\_operator.py}, \texttt{branch\_invariant.py}.) In a
homogeneous same-$n$ spoke-only chain the operator saturates: $\Phi$ has
full single-position marginals (every interface vertex independently
attains all of $\{0,1,-1\}$), and the alternating tire reaches the
\emph{entire} space $3^n$. This is an artifact of non-shrinking annuli
with no interior constraints. On genuine triangulations the marginal
fullness holds for only ${\sim}8\%$ of regions: depth (not branching)
shrinks $\Phi$, e.g.\ a region with $|C|=10$ realised only $|\Phi|=400$
of $3^{10}\approx 59000$. Only non-emptiness and sign-closure survive,
both of which are automatic / equivalent to $4$CT. Hence no abundance
(counting) pigeonhole: a working invariant must tolerate \emph{small}
$\Phi$.
\section*{3. The maximal-constraint floor}
(\texttt{maximally\_constrain.py}.) Minimising $|\Phi(D)|$ over disks with
a fixed boundary $n$-cycle:
\begin{center}
\begin{tabular}{ccccc}
\toprule
$n$ & $4$ & $5$ & $6$ & $7$\\
\midrule
$\min |\Phi|$ (search) & $4$ & $8$ & $16$ & $32$\\
fan, $0$ interior vertices & $4$ & $8$ & $16$ & $32$\\
$2^{\,n-2}$ & $4$ & $8$ & $16$ & $32$\\
\bottomrule
\end{tabular}
\end{center}
A search over random and deep-stacked disks (up to $8$ interior vertices)
never beat $2^{n-2}$, and the interior-free triangulation already
attains it. Thus:
\begin{obs}
The outer $n$-cycle cannot be constrained below $2^{n-2}$ achievable
sequences, and no nested structure is needed to reach the floor: a single
trivial tire is already maximal. Deep nesting only approaches the floor
from above.
\end{obs}
The achievability is transparent: in a fan from $v_0$,
\[
\sigma_1 = \lambda_1,\quad
\sigma_i = \lambda_{i-1}+\lambda_i \ (1<i<n-1),\quad
\sigma_{n-1} = \lambda_{n-2},\quad
\sigma_0 = \textstyle\sum_j \lambda_j ,
\]
so $(\lambda_1,\dots,\lambda_{n-2})$ is recoverable from $\sigma$ and the
map is injective onto $2^{n-2}$ sequences. The lower bound over
\emph{all} disks is the substance:
\begin{conj}[Boundary degrees of freedom]
For every triangulated disk $D$ with boundary $n$-cycle,
$|\Phi(D)| \ge 2^{n-2}$. Equivalently, the $n-2$ binary degrees of
freedom carried by the boundary-incident faces survive every interior
Heawood constraint (which relates only interior-incident faces).
\end{conj}
The minimal set is itself a sign-closed zonotope of size $2^{n-2}$, hull
dimension $n-2$, not a $\mathrm{GF}(3)$ subspace --- the same fingerprint
as $\S1$.
\section*{Consequence for the pigeonhole}
Even a maximally-constraining child still presents $2^{n-2}$ outer
options --- exponential in the interface length $n$. So the gluing
problem has the least slack at \emph{short} interfaces ($n=4$ leaves $4$
options, $n=3$ leaves $2$), and is easy at long ones. The crux of the
Heawood programme therefore lives entirely at short level cycles, exactly
where the medial programme's $N(k)$ bound concentrates.
\medskip
\noindent\emph{Meta-remark.} Because $4$CT holds, every actual
triangulation glues, so no experiment can exhibit an obstruction (pair or
chain). The experiments measure \emph{structure} (zonotope type,
constraint floor), not proof difficulty; the difficulty is localised, not
removed.
\end{document}