Tested the candidate induced γ-partition from
worst_case_proof_sketch.tex (Conj t2-induces-partition).
Findings:
1. AT k = k_2 = 6 (antipodal chord, faces 3+3): Candidate
partition (next-D or prev-D) gives Latin ⊆ π_U. ✓
But this is partly coincidental: |π_U| = 90 is so large that
ALL 10 triple-partitions of {0,..,5} have Latin ⊆ π_U.
2. AT k = k_2 = 9 (chords (0,3)(3,6), faces 3+3+3): Candidate
partition FAILS. Only 8 of all 280 triple-partitions of
{0,..,8} have Latin ⊆ π_U, and the candidate is not one of
them. The 8 surviving partitions have no obvious geometric
interpretation.
3. ASYMMETRIC k ≠ k_2 (e.g. k=6, k_2=3): Candidate doesn't
produce a triple-partition at all, and no triple-partition
has Latin ⊆ π_U. Conjecture as stated doesn't cover the
case where the empirical worst-case overlap lives.
Implication: The candidate construction is wrong past k = 6.
Step 3 (prove inclusion) is not the right next move -- we'd
be proving a false statement.
Reassessment of Approach 2: the König-overlap proposition (when
both tires give direct γ-face partitions) is still cleanly proven,
but applies to fewer cases than hoped. The asymmetric pairs that
witness the empirical worst case are not covered.
Both approaches now have known structural obstacles:
- Approach 1 (2-SAT): single open Conjecture 1.5, empirically true.
- Approach 2 (König): natural construction empirically wrong past
k=6, plus asymmetric pairs out of scope.
Honest status: chain pigeonhole has no full proof yet.
Files:
experiments/induced_partition.py
notes/induced_partition_findings.tex (3 pages)
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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