didericis
36ed7bac38
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>
10 lines
1.6 KiB
Plaintext
10 lines
1.6 KiB
Plaintext
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