didericis 984655fd3d 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>

2026-05-22 10:50:13 -04:00

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