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

Reframe the constraint floor honestly as a conjecture

Browse Source 
Section 4 no longer states the floor as a proven Proposition. Now: prove
interior-free disks attain 2^(n-2) (ear-peeling) and the un-stacking
lemma, state |Phi(D)| >= 2^(n-2) as a Conjecture, and give an honest
status remark -- holds for the Apollonian class, reduces to the
irreducible case, empirically strict (5/4), but |Phi| is NOT monotone
(the earlier freedom-positive monotonicity claim was wrong) and both
natural elementary proofs provably fail. Soften the note's observation to
match.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-06-17 21:32:57 -04:00
parent c482bc5633
commit 163e453464
7 changed files with 104 additions and 99 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.log
+2 -2
View File
@@ -1,4 +1,4 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 20:27
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 21:32
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
@@ -291,7 +291,7 @@ onts/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
s/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
/symbols/msbm10.pfb>
Output written on boundary_restriction_structure.pdf (3 pages, 217886 bytes).
Output written on boundary_restriction_structure.pdf (3 pages, 218332 bytes).
PDF statistics:
104 PDF objects out of 1000 (max. 8388607)
62 compressed objects within 1 object stream
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
+5 -4
View File
@@ -102,10 +102,11 @@ realises $5$ sequences against the fan's $4$, since each interior vertex
contributes two faces but only one constraint. 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.
The interior-free triangulation already attains $2^{n-2}$, no search disk
beats it, and deep nesting only approaches this value from above ---
suggesting it is a floor, with a single trivial tire already maximally
constraining. Whether $2^{n-2}$ is a genuine lower bound for \emph{all}
disks is the Conjecture below; it is \emph{not} a proven theorem.
\end{obs}
The achievability is transparent: in a fan from $v_0$,