coloring_nested_tire_graphs: A-irreducibility analysis of smallest strict-Latin SP failure
Joint-support analysis on (γ=6, T_1=(m_1=3, antipodal, SP), T_2=(k_2=3, no chord, SP)): T_1's σ_D space = 18 elements (half of Latin set's 36; saturation- threshold violated by m_1=3 < γ=6). T_2's σ_U space = 84 elements. The two are intrinsically disjoint on γ, AND S_3-closure of T_1's outer-ring colorings is already saturated (all 6 permutations realised) — so abstract Kempe modification on the outside cannot enlarge T_1's γ-support. The failure is A-IRREDUCIBLE in the strict Birkhoff sense. Significance: the SP failure cases aren't 4CT-relevant obstructions but modeling artifacts. SP treats non-triangular O-faces as single G-faces, which is incompatible with maximal-planar G. A faithful maximal-planar G further triangulates these faces, changing the face-connector and enlarging σ-supports. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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"""For a tire T with boundaries (B_out, B_in), compute the two-sided
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joint support Π_T ⊆ {1,2,3}^|B_out| × {1,2,3}^|B_in| — all (σ_out, σ_in)
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pairs realised by some valid edge 3-coloring of T'_{f'}.
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Use this to analyse the "smallest failure case" γ=6 with
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T_1 = (m_1=3, antipodal-chord, SP), T_2 = (γ=6, k_2=3, no chord, SP).
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If the failure is A-irreducible (no Kempe modification of σ_out can
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rescue it), this means: no matter what the OUTSIDE of T_1 ∪ T_2
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forces on its boundary, no joint coloring of T_1 ∪ T_2 exists.
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"""
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from __future__ import annotations
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import numpy as np
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from collections import defaultdict
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from itertools import product
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from tire_fiber_chords import (
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d_positions_for, u_positions_for, compute_faces_from_chords,
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)
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from tire_fiber_chords_fast import (
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OTHER_NP, induced_sigma_vec, proper_cycle_colorings,
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)
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COLORS = (1, 2, 3)
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def joint_support(m: int, k: int, chords):
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"""Returns Π_T ⊆ {1,2,3}^m × {1,2,3}^k: set of (σ_U, σ_D) pairs.
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σ_U is σ projected to U-positions (B_out side);
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σ_D is σ projected to D-positions (B_in side).
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"""
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n = m + k
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d_pos = d_positions_for(m, k)
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u_pos = u_positions_for(m, k)
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o_faces = compute_faces_from_chords(k, chords)
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d_positions_by_face = [
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[d_pos[a] for a in face_edges] for face_edges in o_faces
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]
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c = proper_cycle_colorings(n)
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if c.size == 0:
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return set()
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sigma = induced_sigma_vec(c)
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mask = np.ones(sigma.shape[0], dtype=bool)
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for face in d_positions_by_face:
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if len(face) <= 1:
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continue
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for i in range(len(face)):
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for j in range(i + 1, len(face)):
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mask &= (sigma[:, face[i]] != sigma[:, face[j]])
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sigma_ok = sigma[mask]
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sigma_u = sigma_ok[:, u_pos]
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sigma_d = sigma_ok[:, d_pos]
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pairs = set(zip(map(tuple, sigma_u.tolist()),
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map(tuple, sigma_d.tolist())))
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return pairs
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def report_2sided(label, m, k, chords):
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pairs = joint_support(m, k, chords)
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n = m + k
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print(f"{label}: m={m}, k={k}, chord={chords}")
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print(f" Π_T = {len(pairs)} pairs in {{1,2,3}}^{m} × {{1,2,3}}^{k}")
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# Distinct σ_U and σ_D values
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sus = {p[0] for p in pairs}
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sds = {p[1] for p in pairs}
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print(f" distinct σ_U (U-projection): {len(sus)} (universe 3^{m} = {3**m})")
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print(f" distinct σ_D (D-projection): {len(sds)} (universe 3^{k} = {3**k})")
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return pairs, sus, sds
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def main():
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print("=" * 70)
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print("Joint-support analysis of γ=6 strict-Latin failure case")
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print("=" * 70)
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print()
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# T_1: outer tire with B_out=C_3, B_in=γ=C_6, chord (0,3) (antipodal)
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# In T_1's frame, "U-positions" = B_out spokes (length m_1=3)
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# "D-positions" = γ-side spokes (length 6).
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pi_T1, su_T1, sd_T1 = report_2sided(
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"T_1 (outer tire, γ=6 strict-Latin)",
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m=3, k=6, chords=[(0, 3)]
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)
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print()
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# T_2: inner tire with B_out=γ=C_6, B_in=C_3, no chord
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# In T_2's frame, "U-positions" = γ-side spokes (length 6)
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# "D-positions" = B_in spokes (length 3).
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pi_T2, su_T2, sd_T2 = report_2sided(
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"T_2 (inner tire, k_2=3, no chord)",
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m=6, k=3, chords=[]
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)
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print()
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# Compatibility: ∃ σ_γ such that
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# (σ_out_T1, σ_γ) ∈ Π_T1 (σ_γ is T_1's σ_D, length 6)
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# (σ_γ, σ_in_T2) ∈ Π_T2 (σ_γ is T_2's σ_U, length 6)
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print("Compatibility check:")
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print(f" T_1's σ_D space (= γ from T_1's side): {len(sd_T1)} elements")
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print(f" T_2's σ_U space (= γ from T_2's side): {len(su_T2)} elements")
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# Forward and reverse intersection
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fwd = sd_T1 & su_T2
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su_T2_rev = {s[::-1] for s in su_T2}
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rev = sd_T1 & su_T2_rev
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print(f" Forward γ-overlap: {len(fwd)} elements")
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print(f" Reverse γ-overlap: {len(rev)} elements")
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if not fwd and not rev:
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print()
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print(" → NO σ_γ is shared. No (σ_out, σ_in) pair admits a joint coloring.")
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print(" → A-IRREDUCIBLE: this holds for ANY outer/inner ring colouring.")
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print()
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# For Kempe analysis: list outer-ring colorings σ_out arising from T_1
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print(" Distinct σ_out arising from T_1 (m_1-ring colorings):", len(su_T1))
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print(" Distinct σ_in arising from T_2 (k_2-ring colorings):", len(sd_T2))
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print()
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print(f" {len(su_T1) * len(sd_T2)} possible (σ_out, σ_in) pairs from the outsides.")
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print()
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# Examine the Kempe orbits of σ_out's (on the m_1=3 ring) under abstract
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# Kempe moves. For a length-3 cycle with 6 proper edge 3-colorings,
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# Kempe orbits depend on the structure of the outside graph (unknown).
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# But the ABSTRACT closure includes all colorings reachable by swapping
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# any two colors on any subset that's "valid" in some outside graph.
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# In the maximally permissive case, this is the full S_3-orbit.
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print("Note on Kempe closure (abstract):")
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print(f" σ_out values realised by T_1: {sorted(su_T1)}")
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su_T1_relabel_closure = set()
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from itertools import permutations
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for s in su_T1:
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for perm in permutations([1, 2, 3]):
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mapping = {1: perm[0], 2: perm[1], 3: perm[2]}
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su_T1_relabel_closure.add(tuple(mapping[x] for x in s))
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print(f" S_3-closure of T_1's σ_out: {len(su_T1_relabel_closure)} elements "
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f"({sorted(su_T1_relabel_closure)})")
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# Even if we close under S_3 (color relabel), the image through T_1 is
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# still the SAME γ-projection π_D(T_1) = sd_T1 (up to S_3 acting on
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# colors). So Kempe-by-color-permutation cannot enlarge the γ-side
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# support beyond sd_T1.
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print()
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print("CONCLUSION:")
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print(" Since sd_T1 ∩ su_T2 = ∅ (in both forward and reverse), the failure")
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print(" is A-irreducible: NO Kempe modification of the outer or inner ring")
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print(" colouring can produce a compatible γ-pattern, because the supports")
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print(" on γ from the two sides are intrinsically disjoint.")
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print()
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print(" Implication: any maximal planar G containing this exact SP-modeled")
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print(" tire-pair (T_1, T_2) as described has no 4-colouring under SP.")
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print()
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print(" This is NOT a 4CT counterexample because:")
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print(" - SP isn't a maximal-planar-G model when O has non-triangular faces.")
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print(" - A real G triangulates O's quadrilateral face further, changing")
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print(" the local face-connector structure and admitting more colourings.")
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print(" - The right next question is whether the configuration even ARISES")
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print(" in any maximal planar G with the SP-faithful triangulation.")
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if __name__ == '__main__':
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main()
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