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Back the 2^(n-2) floor with validated diverse-disk search

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The stacked-only search missed non-stacked disks, and cocircular boundary
points gave degenerate Delaunay (invalid disks, spurious sub-floor |Phi|).
Add floor_diverse_disks.py: 1700+ validated disks per n (convex non-
cocircular boundary, face-count and boundary-edge checks) confirm min|Phi|
= 2^(n-2). Note records that interior structure tends to ENLARGE Phi
(wheel 5 vs fan 4) and that depth adds two faces per one constraint.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-06-17 02:22:41 -04:00
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4 changed files with 113 additions and 9 deletions
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papers/heawood_restrictions_on_nested_tire_graph_duals/notes/boundary_restriction_structure.log
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\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:
A search over $1700{+}$ \emph{validated} triangulated disks per $n$
(boundary points in convex but non-cocircular position, random interior
points, Delaunay; each checked to have $2k+n-2$ faces and all $n$
boundary edges present), together with deep-stacked single-apex chains up
to $8$ interior vertices, never beat $2^{n-2}$, and the interior-free
triangulation already attains it. (Note: cocircular boundary points
produce degenerate Delaunay outputs --- invalid disks missing a boundary
edge --- which spuriously report sub-floor values; these are excluded by
the validity check.) Counterintuitively, adding interior structure tends
to \emph{enlarge} $\Phi$: e.g.\ on the $4$-cycle the central-apex wheel
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