Adds orbit_decomposition note + script answering: do structurally
different (T_1, T_2) pairs share the same canonical orbits in
S_1 ∩ S_2?
Findings across the 23 step-2 pairs:
1. EVERY intersection is closed under S_3 color permutation
(structural sanity check, follows from color-symmetry of
proper edge 3-coloring).
2. EVERY S_3-orbit has size exactly 6, with one exception: the
constant orbit {(c,c,c,c) : c ∈ {1,2,3}} of size 3 at γ=4,
T_1=T_2=(4,-,SR). So |S_1 ∩ S_2| = 6 × (# S_3-orbits) almost
always.
3. The RAINBOW orbit (a,b,c,b,c,a) at γ=6 appears in 3 different
(T_1, T_2) pairs -- all with T_1 = (6, (0,3), SP). The two-
chord SP tire (6, (0,2)(3,5), SP) never produces the rainbow
orbit. So rainbow is associated with the antipodal-chord
topology, not the pair as a whole.
4. Other canonical orbits recur across structurally different
pairs. E.g. (1,2,1,3) at γ=4 appears in 7 of 12 tested
γ=4 pairs.
This upgrades the step-2 finding: the intersection isn't just
non-empty -- it has full S_3-symmetric structure, contains at
least one size-6 orbit in essentially all cases, and shares
canonical orbits across varied (T_1, T_2) pairings.
Files:
experiments/orbit_decomposition.py
experiments/orbit_decomposition_data.txt
notes/orbit_decomposition.tex (3 pages)
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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