Investigated the 8 surviving triple-partitions of γ at k=k_2=9
(chord (0,3),(3,6) on B_in^(2)). Found a clean structural
description.
CLASSIFICATION of γ-edges by T_2's face structure:
For each O^(2)-face F_i, 2 γ-edges are "internal" to F_i
(their adjacent D-triangles are both in F_i).
For each adjacent face pair (F_i, F_{i+1}), 1 γ-edge is
"boundary" between them.
Total: 2r internal + r boundary = 3r γ-edges = |γ| when k=k_2.
STRUCTURAL DESCRIPTION (Prop face-pair-connection):
Latin ⊆ π_U(T_2) iff the partition has the following form:
- One block per cyclically-adjacent face pair (F_i, F_{i+1}).
- Each block = 1 boundary edge δ_{i,i+1} + 1 internal of F_i
+ 1 internal of F_{i+1}.
- For each face F_i, its 2 internal γ-edges are distributed
one per block (the two blocks involving F_i).
Count: 2^r partitions (each face has 2 choices of how to split
its internals across its 2 adjacent blocks).
AT k = k_2 = 9 (r = 3 faces): 2^3 = 8 partitions, matching the
empirical survivors.
WHY NAIVE CANDIDATES FAIL: The next-D and prev-D candidates from
worst_case_proof_sketch.tex group BOTH internals of one face into
one block (e.g., {0,1,2} = both Internal_{F_A} + δ_{AB}, no internal
F_B). This violates the "one internal per face per block" rule.
IMPLICATION: The König-lift approach can be RESCUED by replacing
the naive candidate F~_2 with any of the 2^r face-pair-connection
partitions. Apply König's theorem on bipartite face-incidence
graph of F_1 vs this new F~_2.
NEXT STEP: prove Prop face-pair-connection for all r, then apply
König lift. This is more leveraged than re-tackling the naive
construction.
Files:
experiments/k9_surviving_analysis.py
notes/k9_surviving_partitions.tex (3 pages)
Note also updates notes/induced_partition_findings.tex to point at
the new structural description.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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