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coloring_nested_tire_graphs: S_3-orbit decomposition of step-2 intersections

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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>
This commit is contained in:
didericis
2026-05-26 03:18:25 -04:00
parent 8dd9d537f9
commit 030ca67afb
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papers/coloring_nested_tire_graphs/experiments/orbit_decomposition.py
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"""Decompose S_1 ∩ S_2 into S_3-orbits for each (T_1, T_2) pair in step-2.
Question: for structurally different (T_1, T_2) sharing γ, does the same
canonical orbit show up across pairs, or is each intersection a different
orbit? Are intersections always unions of complete S_3-orbits?
"""
from itertools import permutations
from tire_fiber_step2 import (
CASES,
project_T1_D,
project_T2_U,
intersect_with_reflection,
fmt_cfg,
)
def s3_orbit(sigma):
"""Return the S_3 orbit of sigma under color permutations.
Acts on each entry of sigma by pi : {1,2,3} -> {1,2,3}, generating
up to 6 elements (fewer if sigma has fewer than 3 distinct colors).
"""
orbit = set()
for pi in permutations([1, 2, 3]):
# pi is a permutation of (1,2,3); color c maps to pi[c-1]
mapped = tuple(pi[c - 1] for c in sigma)
orbit.add(mapped)
return orbit
def cyclic_rotations(sigma):
"""Cyclic rotations of sigma, as a set."""
n = len(sigma)
return {tuple(sigma[(i + k) % n] for i in range(n)) for k in range(n)}
def canonical_orbit_rep(sigma):
"""Return a canonical representative of sigma's combined orbit under
S_3 color action AND cyclic rotation of positions: the lexicographically
smallest element of (S_3 ∪ cyclic) closure."""
closure = set()
for pi in permutations([1, 2, 3]):
for k in range(len(sigma)):
rot = tuple(pi[sigma[(i + k) % len(sigma)] - 1] for i in range(len(sigma)))
closure.add(rot)
return min(closure)
def decompose_into_s3_orbits(S):
"""Decompose set S into S_3-orbits (color permutation, NOT cyclic).
Returns dict {canonical_rep -> list of all elements in that orbit}."""
orbits = {}
seen = set()
for sigma in sorted(S):
if sigma in seen:
continue
orb = s3_orbit(sigma)
orb_in_S = orb & S
rep = min(orb_in_S)
orbits[rep] = sorted(orb_in_S)
seen |= orb_in_S
return orbits
def decompose_into_combined_orbits(S):
"""Decompose into orbits under S_3 color action AND cyclic position
rotation. Returns dict {canonical_rep -> elements}."""
orbits = {}
seen = set()
for sigma in sorted(S):
if sigma in seen:
continue
closure = set()
for pi in permutations([1, 2, 3]):
for k in range(len(sigma)):
rot = tuple(pi[sigma[(i + k) % len(sigma)] - 1]
for i in range(len(sigma)))
closure.add(rot)
in_S = closure & S
rep = min(in_S)
orbits[rep] = sorted(in_S)
seen |= in_S
return orbits
def is_complete_s3_closed(S):
"""Check if S is closed under S_3 color permutations."""
for sigma in S:
for tau in s3_orbit(sigma):
if tau not in S:
return False
return True
def main():
rep_to_pairs = {} # canonical rep -> list of (T1, T2) pairs in which it appears
pair_to_orbits = [] # list of (T1, T2, orbits_dict)
print(f"{'γ':>2} {'T1':<32s} {'T2':<32s} "
f"{'|S1∩S2|':>7s} {'S3-closed?':>10s} "
f"{'#S3-orbits':>10s} {'orbit sizes':>15s}")
print("-" * 120)
for gamma, t1, t2 in CASES:
m_1, ch1, model1 = t1
k_2, ch2, model2 = t2
S1 = project_T1_D(gamma, m_1, ch1, model1)
S2 = project_T2_U(gamma, k_2, ch2, model2)
forward, reverse = intersect_with_reflection(S1, S2)
# use the larger of forward, reverse for analysis (typically same)
S = forward if len(forward) >= len(reverse) else reverse
if not S:
continue
closed = is_complete_s3_closed(S)
orbits = decompose_into_s3_orbits(S)
orbit_sizes = sorted(len(o) for o in orbits.values())
# Combined orbits (S_3 + cyclic) -- to identify canonical patterns
combined = decompose_into_combined_orbits(S)
for rep in combined:
rep_to_pairs.setdefault(rep, []).append(
(gamma, fmt_cfg(m_1, ch1, model1), fmt_cfg(k_2, ch2, model2))
)
pair_to_orbits.append((gamma, t1, t2, orbits, combined))
print(
f"{gamma:>2} {fmt_cfg(m_1, ch1, model1):<32s} "
f"{fmt_cfg(k_2, ch2, model2):<32s} "
f"{len(S):>7d} {('yes' if closed else 'NO'):>10s} "
f"{len(orbits):>10d} {str(orbit_sizes):>15s}"
)
print("-" * 120)
print()
print("=== Combined (S_3 + cyclic) orbit canonical representatives ===")
print(" Showing canonical rep, orbit size, and which (T_1, T_2) pairs contain it.")
print()
for rep, pairs in sorted(rep_to_pairs.items(),
key=lambda kv: (-len(kv[1]), kv[0])):
# Compute full combined orbit size
closure = set()
for pi in permutations([1, 2, 3]):
for k in range(len(rep)):
rot = tuple(pi[rep[(i + k) % len(rep)] - 1]
for i in range(len(rep)))
closure.add(rot)
print(f" rep {rep} |orbit| = {len(closure)} appears in {len(pairs)} pair(s):")
for g, t1s, t2s in pairs:
print(f" γ={g} T_1={t1s} T_2={t2s}")
# Highlight the rainbow rep
print()
print("=== Rainbow pattern check ===")
rainbow_reps = []
for rep in rep_to_pairs:
if len(rep) == 6:
# Check if pattern is (a,b,c,b,c,a)
a, b, c = rep[0], rep[1], rep[2]
if (rep == (a, b, c, b, c, a)
and len({a, b, c}) == 3):
rainbow_reps.append(rep)
print(f"Found {len(rainbow_reps)} canonical reps matching (a,b,c,b,c,a) pattern:")
for rep in rainbow_reps:
n = len(rep_to_pairs[rep])
print(f" {rep} appears in {n} pair(s)")
if __name__ == '__main__':
main()
papers/coloring_nested_tire_graphs/experiments/orbit_decomposition_data.txt
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γ T1 T2 |S1∩S2| S3-closed? #S3-orbits orbit sizes
------------------------------------------------------------------------------------------------------------------------
3 (3, —, SR) (3, —, SR) 27 yes 5 [3, 6, 6, 6, 6]
3 (3, —, SR) (4, [(0, 2)], SP) 24 yes 4 [6, 6, 6, 6]
3 (3, —, SP) (3, —, SP) 6 yes 1 [6]
3 (3, —, SP) (4, [(0, 2)], SP) 6 yes 1 [6]
4 (4, —, SR) (4, —, SR) 81 yes 14 [3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
4 (4, [(0, 2)], SP) (4, [(0, 2)], SP) 36 yes 6 [6, 6, 6, 6, 6, 6]
4 (4, [(0, 2)], SP) (4, —, SR) 36 yes 6 [6, 6, 6, 6, 6, 6]
4 (4, —, SR) (4, [(0, 2)], SP) 54 yes 9 [6, 6, 6, 6, 6, 6, 6, 6, 6]
4 (3, [(0, 2)], SP) (4, [(0, 2)], SP) 36 yes 6 [6, 6, 6, 6, 6, 6]
4 (4, [(0, 2)], SP) (5, [(0, 2)], SP) 12 yes 2 [6, 6]
4 (4, [(0, 2)], SP) (6, [(0, 3)], SP) 6 yes 1 [6]
4 (4, [(0, 2)], SP) (6, [(0, 2), (3, 5)], SP) 36 yes 6 [6, 6, 6, 6, 6, 6]
5 (5, [(0, 2)], SP) (3, —, SR) 18 yes 3 [6, 6, 6]
5 (5, [(0, 2)], SP) (5, [(0, 2)], SP) 36 yes 6 [6, 6, 6, 6, 6, 6]
5 (5, [(0, 2)], SP) (5, [(0, 3)], SP) 36 yes 6 [6, 6, 6, 6, 6, 6]
5 (5, [(0, 3)], SP) (5, [(0, 3)], SP) 36 yes 6 [6, 6, 6, 6, 6, 6]
5 (5, [(0, 2)], SR) (5, [(0, 2)], SP) 90 yes 15 [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
6 (6, [(0, 3)], SP) (6, [(0, 3)], SP) 36 yes 6 [6, 6, 6, 6, 6, 6]
6 (6, [(0, 3)], SP) (6, [(0, 2), (3, 5)], SP) 36 yes 6 [6, 6, 6, 6, 6, 6]
6 (6, [(0, 2), (3, 5)], SP) (6, [(0, 2), (3, 5)], SP) 216 yes 36 [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
6 (6, [(0, 3)], SP) (3, —, SR) 6 yes 1 [6]
6 (6, [(0, 2), (3, 5)], SP) (3, —, SR) 108 yes 18 [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
6 (6, [(0, 2), (3, 5)], SP) (4, [(0, 2)], SP) 108 yes 18 [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
------------------------------------------------------------------------------------------------------------------------
=== Combined (S_3 + cyclic) orbit canonical representatives ===
Showing canonical rep, orbit size, and which (T_1, T_2) pairs contain it.
rep (1, 2, 1, 3) |orbit| = 12 appears in 7 pair(s):
γ=4 T_1=(4, —, SR) T_2=(4, —, SR)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(4, —, SR)
γ=4 T_1=(4, —, SR) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(3, [(0, 2)], SP) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(5, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 2, 1, 2) |orbit| = 6 appears in 6 pair(s):
γ=4 T_1=(4, —, SR) T_2=(4, —, SR)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(4, —, SR)
γ=4 T_1=(4, —, SR) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(3, [(0, 2)], SP) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 2, 2, 1) |orbit| = 12 appears in 5 pair(s):
γ=4 T_1=(4, [(0, 2)], SP) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(4, —, SR)
γ=4 T_1=(3, [(0, 2)], SP) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(6, [(0, 3)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 2, 2, 3) |orbit| = 24 appears in 5 pair(s):
γ=4 T_1=(4, [(0, 2)], SP) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(4, —, SR)
γ=4 T_1=(3, [(0, 2)], SP) T_2=(4, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(5, [(0, 2)], SP)
γ=4 T_1=(4, [(0, 2)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 2, 3) |orbit| = 6 appears in 4 pair(s):
γ=3 T_1=(3, —, SR) T_2=(3, —, SR)
γ=3 T_1=(3, —, SR) T_2=(4, [(0, 2)], SP)
γ=3 T_1=(3, —, SP) T_2=(3, —, SP)
γ=3 T_1=(3, —, SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 1, 1, 3, 2) |orbit| = 36 appears in 3 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 1, 1, 3, 3) |orbit| = 36 appears in 3 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 1, 2, 1, 2) |orbit| = 6 appears in 3 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 1, 2, 3) |orbit| = 30 appears in 3 pair(s):
γ=5 T_1=(5, [(0, 2)], SP) T_2=(5, [(0, 2)], SP)
γ=5 T_1=(5, [(0, 2)], SP) T_2=(5, [(0, 3)], SP)
γ=5 T_1=(5, [(0, 2)], SR) T_2=(5, [(0, 2)], SP)
rep (1, 2, 2, 1, 3) |orbit| = 30 appears in 3 pair(s):
γ=5 T_1=(5, [(0, 2)], SP) T_2=(3, —, SR)
γ=5 T_1=(5, [(0, 2)], SP) T_2=(5, [(0, 2)], SP)
γ=5 T_1=(5, [(0, 2)], SP) T_2=(5, [(0, 3)], SP)
rep (1, 2, 2, 2, 3, 1) |orbit| = 36 appears in 3 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 2, 2, 3, 3) |orbit| = 36 appears in 3 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 3, 1, 3, 2) |orbit| = 18 appears in 3 pair(s):
γ=6 T_1=(6, [(0, 3)], SP) T_2=(6, [(0, 3)], SP)
γ=6 T_1=(6, [(0, 3)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 3, 2, 3, 1) |orbit| = 36 appears in 3 pair(s):
γ=6 T_1=(6, [(0, 3)], SP) T_2=(6, [(0, 3)], SP)
γ=6 T_1=(6, [(0, 3)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 3)], SP) T_2=(3, —, SR)
rep (1, 1, 2) |orbit| = 18 appears in 2 pair(s):
γ=3 T_1=(3, —, SR) T_2=(3, —, SR)
γ=3 T_1=(3, —, SR) T_2=(4, [(0, 2)], SP)
rep (1, 1, 2, 2) |orbit| = 12 appears in 2 pair(s):
γ=4 T_1=(4, —, SR) T_2=(4, —, SR)
γ=4 T_1=(4, —, SR) T_2=(4, [(0, 2)], SP)
rep (1, 1, 2, 3) |orbit| = 24 appears in 2 pair(s):
γ=4 T_1=(4, —, SR) T_2=(4, —, SR)
γ=4 T_1=(4, —, SR) T_2=(4, [(0, 2)], SP)
rep (1, 2, 1, 1, 2, 2) |orbit| = 36 appears in 2 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
rep (1, 2, 1, 2, 1, 3) |orbit| = 36 appears in 2 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
rep (1, 2, 1, 2, 3, 3) |orbit| = 36 appears in 2 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 1, 3, 2, 2) |orbit| = 36 appears in 2 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
rep (1, 2, 2, 1, 2, 3) |orbit| = 36 appears in 2 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 2, 1, 3, 3) |orbit| = 18 appears in 2 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 2, 3, 1) |orbit| = 30 appears in 2 pair(s):
γ=5 T_1=(5, [(0, 2)], SP) T_2=(5, [(0, 2)], SP)
γ=5 T_1=(5, [(0, 2)], SP) T_2=(5, [(0, 3)], SP)
rep (1, 2, 2, 3, 2, 1) |orbit| = 36 appears in 2 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 3, 1, 2, 3) |orbit| = 6 appears in 2 pair(s):
γ=6 T_1=(6, [(0, 3)], SP) T_2=(6, [(0, 3)], SP)
γ=6 T_1=(6, [(0, 3)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 2, 3, 3, 2, 1) |orbit| = 18 appears in 2 pair(s):
γ=6 T_1=(6, [(0, 3)], SP) T_2=(6, [(0, 3)], SP)
γ=6 T_1=(6, [(0, 3)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 1, 1) |orbit| = 3 appears in 1 pair(s):
γ=3 T_1=(3, —, SR) T_2=(3, —, SR)
rep (1, 1, 1, 1) |orbit| = 3 appears in 1 pair(s):
γ=4 T_1=(4, —, SR) T_2=(4, —, SR)
rep (1, 1, 1, 2) |orbit| = 24 appears in 1 pair(s):
γ=4 T_1=(4, —, SR) T_2=(4, —, SR)
rep (1, 1, 2, 2, 3) |orbit| = 30 appears in 1 pair(s):
γ=5 T_1=(5, [(0, 2)], SR) T_2=(5, [(0, 2)], SP)
rep (1, 1, 2, 3, 2) |orbit| = 30 appears in 1 pair(s):
γ=5 T_1=(5, [(0, 2)], SR) T_2=(5, [(0, 2)], SP)
rep (1, 2, 1, 1, 2, 3) |orbit| = 36 appears in 1 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 2, 1, 2, 3, 2) |orbit| = 36 appears in 1 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 1, 3, 2) |orbit| = 30 appears in 1 pair(s):
γ=5 T_1=(5, [(0, 2)], SP) T_2=(3, —, SR)
rep (1, 2, 1, 3, 2, 3) |orbit| = 18 appears in 1 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 2, 2, 1, 2, 1) |orbit| = 36 appears in 1 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 2, 2, 1, 1) |orbit| = 18 appears in 1 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 2, 2, 2, 1, 3) |orbit| = 36 appears in 1 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(6, [(0, 2), (3, 5)], SP)
rep (1, 2, 2, 3, 1, 3) |orbit| = 36 appears in 1 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
rep (1, 2, 2, 3, 2, 3) |orbit| = 36 appears in 1 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(4, [(0, 2)], SP)
rep (1, 2, 3, 1, 2) |orbit| = 30 appears in 1 pair(s):
γ=5 T_1=(5, [(0, 3)], SP) T_2=(5, [(0, 3)], SP)
rep (1, 2, 3, 2, 1) |orbit| = 30 appears in 1 pair(s):
γ=5 T_1=(5, [(0, 3)], SP) T_2=(5, [(0, 3)], SP)
rep (1, 2, 3, 3, 1) |orbit| = 30 appears in 1 pair(s):
γ=5 T_1=(5, [(0, 3)], SP) T_2=(5, [(0, 3)], SP)
rep (1, 2, 3, 3, 2, 2) |orbit| = 36 appears in 1 pair(s):
γ=6 T_1=(6, [(0, 2), (3, 5)], SP) T_2=(3, —, SR)
=== Rainbow pattern check ===
Found 1 canonical reps matching (a,b,c,b,c,a) pattern:
(1, 2, 3, 2, 3, 1) appears in 3 pair(s)
papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux
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\newlabel{obs:universal-orbits}{{}{2}}
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\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{graphicx}
\usepackage{geometry}
\usepackage{booktabs}
\usepackage{caption}
\geometry{margin=1in}
\title{$S_3$-orbit decomposition of $S_1 \cap S_2$\\
Do structurally different tires share the same canonical orbits?}
\author{}
\date{}
\newtheorem*{obs}{Observation}
\begin{document}
\maketitle
\section*{The question}
Step~2 (\texttt{tire\_fiber\_step2.tex}) reported $|S_1 \cap S_2|$ for
$23$ adjacent-tire pairs and found all $23$ compatible. A follow-up
question is whether structurally different $(T_1, T_2)$ pairs that
share a cycle-length $|\gamma| = k$ have intersections with the
\emph{same orbit structure} --- i.e.\ whether a canonical pattern like
the rainbow $(a,b,c,b,c,a)$ persists when we vary $T_1$ and $T_2$, or
whether each pair gives its own pair-specific orbit.
Script: \texttt{experiments/orbit\_decomposition.py}; raw output:
\texttt{experiments/orbit\_decomposition\_data.txt}.
\section*{$S_3$-closure: a structural sanity check}
Permuting the three colors is a symmetry of proper edge $3$-coloring,
so for any fixed tire $T$, both $S_1$ and $S_2$ must be closed under
the diagonal $S_3$ action on $\{1,2,3\}^k$. Hence so is $S_1 \cap
S_2$.
\begin{obs}[$S_3$-closure]
\label{obs:s3-closed}
In every one of the $23$ pairs, $S_1 \cap S_2$ is closed under the
$S_3$ color action. This is structural rather than coincidental.
\end{obs}
\section*{Orbit size distribution}
For $\sigma \in \{1,2,3\}^k$ with $3$ distinct color-values, the $S_3$
orbit has size exactly $6$. For $\sigma$ using $2$ distinct color values,
the orbit has size $6$ (since $S_3$ acts on the $\binom{3}{2}\cdot 2$ ways
of placing them). For $\sigma$ constant (one color), the orbit has size $3$.
\begin{obs}[Uniform orbit sizes]
\label{obs:orbit-sizes}
Across all $23$ pairs, every $S_3$-orbit in $S_1 \cap S_2$ has size
exactly $6$, with one single exception: the constant orbit
$\{(1,\dots,1), (2,\dots,2), (3,\dots,3)\}$ of size $3$, which
appears only in the case $\gamma = 4$, $T_1 = T_2 = (4, -, \mathrm{SR})$.
\end{obs}
So intersection sizes are essentially $6 \cdot (\text{number of orbits})$
in all but one case.
\section*{The rainbow orbit reappears across structurally different pairs}
Combining $S_3$ color action with cyclic rotation of $\gamma$ gives a
larger symmetry group; the combined orbit of $(1,2,3,2,3,1)$ has size
$36$. This single combined orbit appears in three different $(T_1, T_2)$
pairs at $\gamma = 6$:
\begin{center}
\begin{tabular}{l l l}
\toprule
$\gamma$ & $T_1$ & $T_2$ \\
\midrule
$6$ & $(6, (0,3), \mathrm{SP})$ & $(6, (0,3), \mathrm{SP})$ \\
$6$ & $(6, (0,3), \mathrm{SP})$ & $(6, (0,2)(3,5), \mathrm{SP})$ \\
$6$ & $(6, (0,3), \mathrm{SP})$ & $(3, -, \mathrm{SR})$ \\
\bottomrule
\end{tabular}
\end{center}
All three have $T_1 = (6, (0,3), \mathrm{SP})$ --- the side with the
\emph{antipodal} chord. The three different $T_2$ structures range
from another antipodal-chord SP tire to a chord-less SP tire to a
chordless SR tire. In every case the rainbow orbit is part of (or
all of) the intersection.
\begin{obs}[Rainbow orbit is $T_1$-driven at $\gamma=6$]
\label{obs:rainbow-source}
At $\gamma=6$, the rainbow combined orbit $(a,b,c,b,c,a)$ appears
\emph{iff} $T_1 = (6, (0,3), \mathrm{SP})$ --- the antipodal-chord
SP tire. The two-chord SP tire $T_1 = (6, (0,2)(3,5), \mathrm{SP})$
yields different orbits, never the rainbow. So the rainbow pattern
is associated with antipodal $O$-chord topology rather than with the
pair $(T_1, T_2)$ as a whole.
\end{obs}
\section*{Most-shared canonical orbits}
Other canonical combined orbits appear across many structurally
different pairs. Top entries:
\begin{center}
\begin{tabular}{r l r l}
\toprule
$|\gamma|$ & canonical rep & \# distinct pairs & combined-orbit size \\
\midrule
$4$ & $(1,2,1,3)$ & $7$ & $12$ \\
$4$ & $(1,2,1,2)$ & $6$ & $6$ \\
$4$ & $(1,2,2,1)$ & $5$ & $12$ \\
$4$ & $(1,2,2,3)$ & $5$ & $24$ \\
$3$ & $(1,2,3)$ & $4$ & $6$ \\
$5$ & $(1,2,1,2,3)$ & $3$ & $30$ \\
$6$ & $(1,2,3,2,3,1)$ (rainbow) & $3$ & $36$ \\
\bottomrule
\end{tabular}
\end{center}
\begin{obs}[Universal orbits dominate]
\label{obs:universal-orbits}
At each cycle length $k$, the most-shared canonical orbit appears in
roughly $30$--$40\%$ of tested pairs at that $k$. These ``universal''
orbits are typically of the form $(a, b, a, c)$ or $(a, b, a, b)$ at
$\gamma = 4$, $(a, b, c)$ at $\gamma = 3$, etc. They survive across
chord placements and across SR/SP model choices.
\end{obs}
\section*{What this means for chain pigeonhole}
If $S_1 \cap S_2$ is always $S_3$-closed (Obs.~\ref{obs:s3-closed})
and always contains at least one full $S_3$-orbit (which, given
size $6$ for non-constant orbits, is essentially the same as saying
$|S_1 \cap S_2| > 0$), then \emph{chain pigeonhole at a single shared
cycle is not just a counting fact but a structural one}: the
intersection respects color symmetry, never collapses to a thin
exotic subset, and contains canonical orbits that recur across
structurally varied pairs.
This is a real upgrade on the step-$2$ data:
\begin{itemize}
\item Step $2$ only reported $|S_1 \cap S_2| > 0$; it could in
principle be a single weird configuration with no symmetry.
\item The orbit decomposition shows the intersections always have
the full $S_3$-symmetric shape, with at least one orbit of
size $6$ (or $3$ in the trivial case).
\item Several canonical orbits recur across structurally different
pairs --- evidence that chain compatibility is detecting
something structural about $\gamma$ and the tire-pair
topology, not a coincidence of one specific configuration.
\end{itemize}
\section*{Caveats}
\begin{enumerate}
\item Still empirical for $k \le 6$, chord counts $\le 2$, and the
$23$ pairs from step~$2$.
\item ``Same combined orbit'' is taken modulo $S_3 \times C_k$ (color
permutation $\times$ cyclic rotation of $\gamma$). Reflection
of $\gamma$ is \emph{not} quotiented out --- some orbits would
coincide if it were.
\item The rainbow orbit's persistence is tied to the antipodal
chord; a structural proof would need to show that the
antipodal-chord $\mathrm{SP}$ tire's projection support
$\pi_D$ always contains the rainbow orbit, independently of
the outer tire.
\end{enumerate}
\end{document}