From 8f3a1325ee8fc5aacad3c6109198426d696b3a4c Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 02:33:25 -0400 Subject: [PATCH] coloring_nested_tire_graphs: step-2 adjacent-tire compatibility experiment MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit For pairs (T1, T2) sharing a cycle γ, intersect T1's D-projection (inner-spoke pattern from outer tire) with T2's U-projection (outer-spoke pattern from inner tire) on γ. Compatibility = nonempty intersection. Result: 23/23 tested pairs are compatible, spanning k ∈ {3,4,5,6}, both SR/SP models, and a range of chord configurations on each side. Notable: small intersections have clean S_3-orbit structure. Worst tested case (k=6, antipodal-chord T1 vs unchorded SR T2) has intersection of size 6 — exactly the 3! rainbow patterns (a,b,c,b,c,a). This suggests structural rather than accidental overlap, and points to a theorem worth proving. Caveats: 23 cases at k≤6 isn't a proof; longer chains (step 3) require more than pairwise overlap. Co-Authored-By: Claude Opus 4.7 --- .../experiments/tire_fiber_step2.py | 130 +++++++ .../experiments/tire_fiber_step2_data.txt | 27 ++ .../notes/tire_fiber_step2.aux | 6 + .../notes/tire_fiber_step2.log | 324 ++++++++++++++++++ .../notes/tire_fiber_step2.pdf | Bin 0 -> 191120 bytes .../notes/tire_fiber_step2.tex | 217 ++++++++++++ 6 files changed, 704 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/experiments/tire_fiber_step2.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/tire_fiber_step2_data.txt create mode 100644 papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.log create mode 100644 papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.tex diff --git a/papers/coloring_nested_tire_graphs/experiments/tire_fiber_step2.py b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_step2.py new file mode 100644 index 0000000..9025413 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_step2.py @@ -0,0 +1,130 @@ +"""Step 2: For pairs of adjacent tires (T1, T2) sharing a cycle γ, +compute the intersection of T1's D-projection and T2's U-projection +on γ. Compatibility (chain-pigeonhole step) succeeds iff the +intersection is non-empty. + +Conventions +----------- +* T1 is the OUTER tire: γ = B_in^(1). T1's inner-spoke pattern on γ + is its D-projection π_D^(1). +* T2 is the INNER tire: γ = B_out^(2). T2's outer-spoke pattern on γ + is its U-projection π_U^(2). +* Each tire's σ-support is closed under cyclic rotation of its dual + cycle, which induces cyclic rotation on the γ-projection. +* T1 and T2 may walk around γ in opposite orientations, so we check + intersection both forward and with one side reversed. + +Two models per tire: + SR -- Steiner-rich (no chord effect; spoke-only baseline from step 1) + SP -- Steiner-poor (each O-face is one G-face; chord constraints active) +""" +from itertools import product + +from tire_fiber_chords import ( + COLORS, + fiber_distribution, + spoke_only_fiber_distribution, + d_positions_for, + u_positions_for, + projection_support, +) + + +def project_T1_D(gamma_len: int, m_1: int, chords_1, model: str) -> set: + """T1's D-projection on γ. T1 has B_out of length m_1, B_in = γ + of length gamma_len.""" + if model == 'SR': + n = m_1 + gamma_len + fibers = spoke_only_fiber_distribution(n) + else: + fibers, _, _ = fiber_distribution(m_1, gamma_len, chords_1) + d_pos = d_positions_for(m_1, gamma_len) + return projection_support(fibers, d_pos) + + +def project_T2_U(gamma_len: int, k_2: int, chords_2, model: str) -> set: + """T2's U-projection on γ. T2 has B_out = γ of length gamma_len, + B_in of length k_2.""" + if model == 'SR': + n = gamma_len + k_2 + fibers = spoke_only_fiber_distribution(n) + else: + fibers, _, _ = fiber_distribution(gamma_len, k_2, chords_2) + u_pos = u_positions_for(gamma_len, k_2) + return projection_support(fibers, u_pos) + + +def intersect_with_reflection(S1: set, S2: set) -> tuple[set, set]: + """Return (forward intersection, reflection-flipped intersection).""" + forward = S1 & S2 + S2_rev = {s[::-1] for s in S2} + reverse = S1 & S2_rev + return forward, reverse + + +CASES = [ + # (γ, T1=(m_1, chords_1, model1), T2=(k_2, chords_2, model2)) + # γ = 3: no chord constraints possible (C_3 has no chords) + (3, (3, [], 'SR'), (3, [], 'SR')), + (3, (3, [], 'SR'), (4, [(0,2)], 'SP')), + (3, (3, [], 'SP'), (3, [], 'SP')), + (3, (3, [], 'SP'), (4, [(0,2)], 'SP')), + + # γ = 4: SP-feasible only with chord (0,2) on T1 + (4, (4, [], 'SR'), (4, [], 'SR')), + (4, (4, [(0,2)], 'SP'), (4, [(0,2)], 'SP')), + (4, (4, [(0,2)], 'SP'), (4, [], 'SR')), + (4, (4, [], 'SR'), (4, [(0,2)], 'SP')), + (4, (3, [(0,2)], 'SP'), (4, [(0,2)], 'SP')), + (4, (4, [(0,2)], 'SP'), (5, [(0,2)], 'SP')), + (4, (4, [(0,2)], 'SP'), (6, [(0,3)], 'SP')), + (4, (4, [(0,2)], 'SP'), (6, [(0,2),(3,5)], 'SP')), + + # γ = 5 + (5, (5, [(0,2)], 'SP'), (3, [], 'SR')), + (5, (5, [(0,2)], 'SP'), (5, [(0,2)], 'SP')), + (5, (5, [(0,2)], 'SP'), (5, [(0,3)], 'SP')), + (5, (5, [(0,3)], 'SP'), (5, [(0,3)], 'SP')), + (5, (5, [(0,2)], 'SR'), (5, [(0,2)], 'SP')), + + # γ = 6 + (6, (6, [(0,3)], 'SP'), (6, [(0,3)], 'SP')), + (6, (6, [(0,3)], 'SP'), (6, [(0,2),(3,5)], 'SP')), + (6, (6, [(0,2),(3,5)], 'SP'), (6, [(0,2),(3,5)], 'SP')), + (6, (6, [(0,3)], 'SP'), (3, [], 'SR')), + (6, (6, [(0,2),(3,5)], 'SP'), (3, [], 'SR')), + (6, (6, [(0,2),(3,5)], 'SP'), (4, [(0,2)], 'SP')), +] + + +def fmt_cfg(m_or_k: int, chords, model: str) -> str: + ch_str = str(chords) if chords else "—" + return f"({m_or_k}, {ch_str}, {model})" + + +def main(): + print(f"{'γ':>2} {'T1 (m_1, chords_1, model)':<32s} {'T2 (k_2, chords_2, model)':<32s} " + f"{'|S1|':>5s} {'|S2|':>5s} {'3^γ':>5s} {'fwd':>5s} {'rev':>5s} {'compat?':>7s}") + print("-" * 120) + nyes = ntotal = 0 + 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) + compat = "YES" if (forward or reverse) else "NO" + ntotal += 1 + if compat == "YES": + nyes += 1 + print( + f"{gamma:>2} {fmt_cfg(m_1, ch1, model1):<32s} {fmt_cfg(k_2, ch2, model2):<32s} " + f"{len(S1):>5d} {len(S2):>5d} {3**gamma:>5d} " + f"{len(forward):>5d} {len(reverse):>5d} {compat:>7s}" + ) + print("-" * 120) + print(f"{nyes}/{ntotal} compatible.") + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/experiments/tire_fiber_step2_data.txt b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_step2_data.txt new file mode 100644 index 0000000..78cbb0e --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/tire_fiber_step2_data.txt @@ -0,0 +1,27 @@ + γ T1 (m_1, chords_1, model) T2 (k_2, chords_2, model) |S1| |S2| 3^γ fwd rev compat? +------------------------------------------------------------------------------------------------------------------------ + 3 (3, —, SR) (3, —, SR) 27 27 27 27 27 YES + 3 (3, —, SR) (4, [(0, 2)], SP) 27 24 27 24 24 YES + 3 (3, —, SP) (3, —, SP) 6 6 27 6 6 YES + 3 (3, —, SP) (4, [(0, 2)], SP) 6 24 27 6 6 YES + 4 (4, —, SR) (4, —, SR) 81 81 81 81 81 YES + 4 (4, [(0, 2)], SP) (4, [(0, 2)], SP) 36 54 81 36 36 YES + 4 (4, [(0, 2)], SP) (4, —, SR) 36 81 81 36 36 YES + 4 (4, —, SR) (4, [(0, 2)], SP) 81 54 81 54 54 YES + 4 (3, [(0, 2)], SP) (4, [(0, 2)], SP) 36 54 81 36 36 YES + 4 (4, [(0, 2)], SP) (5, [(0, 2)], SP) 36 36 81 12 12 YES + 4 (4, [(0, 2)], SP) (6, [(0, 3)], SP) 36 15 81 6 6 YES + 4 (4, [(0, 2)], SP) (6, [(0, 2), (3, 5)], SP) 36 81 81 36 36 YES + 5 (5, [(0, 2)], SP) (3, —, SR) 36 171 243 18 18 YES + 5 (5, [(0, 2)], SP) (5, [(0, 2)], SP) 36 90 243 36 36 YES + 5 (5, [(0, 2)], SP) (5, [(0, 3)], SP) 36 90 243 36 36 YES + 5 (5, [(0, 3)], SP) (5, [(0, 3)], SP) 36 90 243 36 36 YES + 5 (5, [(0, 2)], SR) (5, [(0, 2)], SP) 243 90 243 90 90 YES + 6 (6, [(0, 3)], SP) (6, [(0, 3)], SP) 36 90 729 36 36 YES + 6 (6, [(0, 3)], SP) (6, [(0, 2), (3, 5)], SP) 36 456 729 36 36 YES + 6 (6, [(0, 2), (3, 5)], SP) (6, [(0, 2), (3, 5)], SP) 216 456 729 216 162 YES + 6 (6, [(0, 3)], SP) (3, —, SR) 36 396 729 6 6 YES + 6 (6, [(0, 2), (3, 5)], SP) (3, —, SR) 216 396 729 108 108 YES + 6 (6, [(0, 2), (3, 5)], SP) (4, [(0, 2)], SP) 216 342 729 108 90 YES +------------------------------------------------------------------------------------------------------------------------ +23/23 compatible. diff --git a/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.aux b/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.aux new file mode 100644 index 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