NEW NOTE: chain_half_analysis.tex (4 pages).
Formulates the chain half of the loose conjecture as a tree DP
over the cut tire forest, identifies what's proven vs. open:
PROVEN
- Tree structure (high-side forest): from
cut_tire_tree_structure.tex
- S_3-equivariance of the DP: trivial Lemma in this note
- Per-tire half for spoke-only cut tires (n ≥ 3): Prop 1.13
OPEN / GAPS DISCOVERED
1. Cut tires are NOT in general spoke-only. H_d can have degree-3
vertices (= branch points), making face boundaries non-simple
cycles. Dodecahedron 6-edge cut yields H_1 with one face of
length 20 over only 11 distinct vertices. Prop 1.13's count
2^n + 2(-1)^n applies only to spoke-only tires.
2. OUT-only projection loses S_3 orbit info. The per-tire half
guarantees a full S_3 orbit on the JOINT (in + out) projection,
but restricting to OUT spokes can collapse to |A|=3 (constant
tuples). Empirically observed ~20% of the time on test cases.
Correct DP must track joint projection (analog of
tire_fiber_step2.tex's joint-support tracking).
3. Non-emptiness preservation through the DP is the genuine open
piece (Conj. in this note + Strong per-tire extendibility).
EMPIRICAL TESTS
- chain_dp_test.py: simple cycle DP (assumes spoke-only).
- chain_dp_general.py: handles branched faces via brute-force
3-edge-coloring enumeration (cut off at 12 edges/tire for
tractability).
- chain_dp_debug.py: diagnostic for inspecting H_d face structure.
The general test reveals all three gaps above when run on
Dodecahedron + HM #0. Cross-cut R_0 ∩ R_1 should be non-empty
for both (they are 3-edge-colorable), but the heuristic
parent-finding plus OUT-only projection produce false negatives.
Status table at end of note summarizes what's needed to close.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
203b005336