Resolve n=21 boundary: all four open Holton-McKay duals are bridge-derived

Backward bridge-switch search (sharded over valid parity partitions) found
an Even Level Graph witness for each of the four previously-open duals:
  dual 0: partition 12, witness orbit 9458
  dual 3: partition  9, witness orbit  388
  dual 4: partition 23, witness orbit 3842
  dual 5: partition 12, witness orbit 165668
So all four are bridge-derived level graphs, hence valid derived level
graphs. Combined with the two duals that are Even Level Graphs outright,
the disjunction is now confirmed for ALL SIX critical iso classes at n=21
-- the first nontrivial test of the conjecture passes.

Why it worked where exhaustion failed: a witness, when it exists, tends to
sit in a SMALL orbit (here a few hundred to ~1.7e5 states) reachable
quickly, while other parity partitions of the same triangulation have
orbits >1e6. We only need one good partition. The bridge restriction both
shrinks orbits ~100x and guarantees validity, so any ELG found in a
backward orbit is an immediate witness.

- Update paper n=21 subsection to report the resolution.
- Add shard_hunt.py (partition-sharded parallel witness hunt).

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

papers/even_level_graph_generators/paper.tex

@@ -365,19 +365,26 @@ source vertex), hence trivially valid derived level graphs. So the
 disjunction holds for them through the derived-level-graph disjunct --
 the first instances where that disjunct does work the intertwining-tree
 disjunct cannot.
-\item The remaining four are not Even Level Graphs for any source. A
-bounded backward $E/O$-orbit search (tens of thousands of states, a
-handful of source labellings) found no Even Level Graph in their
-orbits, but this is far too shallow relative to the orbit size at
-$n = 21$ to be conclusive; their status as derived level graphs is
-open.
+\item The remaining four are not Even Level Graphs for any source, and
+their full $E/O$-orbits ($\sim\!10^8$ states per source labelling) are
+far too large to exhaust. Restricting to \emph{bridge switches}
+(Definition~\ref{def:bridge-switch}) shrinks the relevant orbits by
+roughly two orders of magnitude and, crucially, keeps every reachable
+triangulation valid. A backward bridge-switch search over the valid
+parity partitions found an Even Level Graph witness for each of the
+four, so all four are \emph{bridge-derived level graphs}
+(Definition~\ref{def:bridge-derived-level-graph}) and hence valid
+derived level graphs. The witnessing orbits are small -- between a few
+hundred and $\sim\!1.7\times 10^5$ states -- even though other parity
+partitions of the same triangulations have orbits exceeding $10^6$;
+finding one good partition suffices.
 \end{itemize}
-Thus at $n = 21$ the disjunction is confirmed for two of the six
-critical iso classes and undetermined for the other four. Settling
-those four -- equivalently, deciding $E/O$-orbit reachability from an
-Even Level Graph -- is the first genuinely open instance of the
-conjecture, and calls for either a better reachability algorithm or a
-structural invariant of $E/O$-orbits.
+Thus at $n = 21$ the disjunction is confirmed for all six critical iso
+classes: two are Even Level Graphs outright, and the other four are
+bridge-derived level graphs. The bridge-switch restriction is what made
+the search tractable -- it both shrinks the orbit and guarantees
+validity, so any Even Level Graph located in a backward orbit is an
+immediate witness.
 
 \begin{thebibliography}{9}