didericis 36ed7bac38 even_level: add "Toward a characterization of bridge-derived graphs" ...

Record the partition sweep on the n=24 Fig 2.10 dual. New subsection +
experiments/bridge_partition_sweep.py.

Findings:
- A bridge switch is a constrained diagonal flip; bridge-derived via L
  means lying in an Even-Level-Graph component of the restricted flip
  graph. So the question is which flip-components contain an ELG.
- Identity: every 4-coloring of a triangulation has e_cross = 2n-4 (each
  face has one within-pair edge), so total parity-subgraph Betti =
  (c_A+c_B)-2; intertwining trees are the Betti-0 case.
- Of T's 333 valid partitions, total Betti splits 288/42/3 over 1/2/3;
  min is 1 (T not intertwining). All 27 partitions found bridge-derived
  (depth 2-3) have the minimum Betti 1 -> necessary.
- But not sufficient: only 27 of 288 Betti-1 partitions yield a witness;
  the rest have flip-orbits >1.5e5 with no ELG, and a 12x budget increase
  found none. The discriminator is flip-component structure (sharp
  orbit-size dichotomy), not a numerical invariant. Characterizing which
  Betti-minimal partitions sit in an ELG component is left open.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

2026-05-22 17:41:13 -04:00

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