outputs/claim6_pls_localmaxcut.json

{
  "claim": "Theorem 4.4 (PLS-hardness via LocalMaxCut)",
  "cost_boundary_audit": [
    {
      "c_close": 0.8,
      "c_far": 1.2,
      "far_le_2_close": true,
      "is_metric": true,
      "repaired_reduction_correct": false,
      "triangle_violations": 0
    },
    {
      "c_close": 0.9,
      "c_far": 1.1,
      "far_le_2_close": true,
      "is_metric": true,
      "repaired_reduction_correct": false,
      "triangle_violations": 0
    },
    {
      "c_close": 0.5,
      "c_far": 1.2,
      "far_le_2_close": false,
      "is_metric": false,
      "repaired_reduction_correct": false,
      "triangle_violations": 40
    },
    {
      "c_close": 0.5,
      "c_far": 1.5,
      "far_le_2_close": false,
      "is_metric": false,
      "repaired_reduction_correct": false,
      "triangle_violations": 40
    },
    {
      "c_close": 0.1,
      "c_far": 5.0,
      "far_le_2_close": false,
      "is_metric": false,
      "repaired_reduction_correct": false,
      "triangle_violations": 40
    },
    {
      "c_close": 0.99,
      "c_far": 1.01,
      "far_le_2_close": true,
      "is_metric": true,
      "repaired_reduction_correct": false,
      "triangle_violations": 0
    }
  ],
  "metric_check": {
    "positive_off_diagonal": true,
    "symmetric": true,
    "triangle_violations": 0,
    "zero_diagonal": true
  },
  "metric_check_repaired": {
    "positive_off_diagonal": true,
    "symmetric": true,
    "triangle_violations": 0,
    "zero_diagonal": true
  },
  "minimal_counterexample": {
    "M": 7.0,
    "best_utility_times_M": 6.0,
    "counterexample_labels": {
      "('e+', (0, 1))": 1,
      "('e-', (0, 1))": 1,
      "('v', 0)": 1,
      "('v', 1)": 1
    },
    "counterexample_utility_times_M": 2.0,
    "graph": "single edge (0,1) with w = 1",
    "h": [
      0,
      0,
      1,
      0
    ],
    "induced_cut_is_local_max": false,
    "induced_cut_weight": 0.0,
    "induced_vertex_labels": [
      1,
      1
    ],
    "max_cut_weight": 1.0,
    "population": [
      "('v', 0)",
      "('v', 1)",
      "('e+', (0, 1))",
      "('e-', (0, 1))"
    ],
    "single_flip_utilities_times_M": {
      "('e+', (0, 1))": 2.0,
      "('e-', (0, 1))": 2.0,
      "('v', 0)": 2.0,
      "('v', 1)": 2.0
    },
    "weights_wD": [
      1.0,
      1.0,
      2.0,
      3.0
    ],
    "why": "The proof's first claim says a Jury with f(x_{(u,v)-}) = 1 gains at least 1/M by flipping it to 0, because x_{(u,v)-} then becomes correctly classified.  That step silently assumes f(x_{(u,v)+}) = 0.  When f(x_{(u,v)+}) = 1 the point x_{(u,v)-} strategically deviates to x_{(u,v)+} after the flip and is still misclassified, so the gain is 0 and the configuration is a strategic local optimum with edge points labelled 1."
  },
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      "direction2_every_local_max_cut_is_a_local_opt": true,
      "global_optima_are_max_cuts": true,
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      "max_cut_weight": 1.0,
      "n_edges": 1,
      "n_local_max_cuts": 2,
      "n_local_optima_violating_direction1": 1,
      "n_strategic_local_optima": 3,
      "n_vertices": 2,
      "population_size": 4,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 2.0
    },
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      "classifiers_enumerated": 128,
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      "n_local_max_cuts": 2,
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      "n_strategic_local_optima": 6,
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      "n_vertices": 3,
      "population_size": 7,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 20.0
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      "population_size": 10,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 18.0
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      "direction2_every_local_max_cut_is_a_local_opt": true,
      "global_optima_are_max_cuts": true,
      "guards": false,
      "max_cut_weight": 10.0,
      "n_edges": 3,
      "n_local_max_cuts": 2,
      "n_local_optima_violating_direction1": 12,
      "n_strategic_local_optima": 14,
      "n_vertices": 4,
      "population_size": 10,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 20.0
    },
    {
      "classifiers_enumerated": 4096,
      "cut_induced_utility_times_M_range": [
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      ],
      "direction1_every_local_opt_is_a_local_max_cut": false,
      "direction2_every_local_max_cut_is_a_local_opt": true,
      "global_optima_are_max_cuts": true,
      "guards": false,
      "max_cut_weight": 9.0,
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      "n_local_max_cuts": 4,
      "n_local_optima_violating_direction1": 42,
      "n_strategic_local_optima": 46,
      "n_vertices": 4,
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      "global_optima_are_max_cuts": true,
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      "n_edges": 4,
      "n_local_max_cuts": 4,
      "n_local_optima_violating_direction1": 41,
      "n_strategic_local_optima": 45,
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      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 18.0
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    {
      "classifiers_enumerated": 8192,
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      "direction1_every_local_opt_is_a_local_max_cut": false,
      "direction2_every_local_max_cut_is_a_local_opt": true,
      "global_optima_are_max_cuts": true,
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      "max_cut_weight": 12.0,
      "n_edges": 4,
      "n_local_max_cuts": 4,
      "n_local_optima_violating_direction1": 38,
      "n_strategic_local_optima": 42,
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      "population_size": 13,
      "utility_identity_spread": 1.4210854715202004e-14,
      "worst_spurious_utility_times_M": 24.0
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      "n_strategic_local_optima": 33,
      "n_vertices": 5,
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      "utility_identity_spread": 1.4210854715202004e-14,
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    }
  ],
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      "cut_induced_utility_times_M_range": [
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      ],
      "direction1_every_local_opt_is_a_local_max_cut": false,
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      "global_optima_are_max_cuts": true,
      "guards": true,
      "max_cut_weight": 1.0,
      "n_edges": 1,
      "n_local_max_cuts": 2,
      "n_local_optima_violating_direction1": 2,
      "n_strategic_local_optima": 4,
      "n_vertices": 2,
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      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 2.0
    },
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      ],
      "direction1_every_local_opt_is_a_local_max_cut": false,
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      "max_cut_weight": 4.0,
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      "n_local_max_cuts": 2,
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      "n_strategic_local_optima": 13,
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      "utility_identity_spread": 0.0,
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      "max_cut_weight": 10.0,
      "n_edges": 2,
      "n_local_max_cuts": 2,
      "n_local_optima_violating_direction1": 11,
      "n_strategic_local_optima": 13,
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      "worst_spurious_utility_times_M": 20.0
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      "direction1_every_local_opt_is_a_local_max_cut": false,
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      "global_optima_are_max_cuts": true,
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      "n_local_max_cuts": 2,
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      "n_strategic_local_optima": 46,
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      "population_size": 12,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 22.0
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      "global_optima_are_max_cuts": true,
      "guards": true,
      "max_cut_weight": 7.0,
      "n_edges": 3,
      "n_local_max_cuts": 2,
      "n_local_optima_violating_direction1": 42,
      "n_strategic_local_optima": 44,
      "n_vertices": 3,
      "population_size": 12,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 18.0
    },
    {
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      "direction1_every_local_opt_is_a_local_max_cut": false,
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      "global_optima_are_max_cuts": true,
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      "n_edges": 3,
      "n_local_max_cuts": 2,
      "n_local_optima_violating_direction1": 36,
      "n_strategic_local_optima": 38,
      "n_vertices": 4,
      "population_size": 13,
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      "worst_spurious_utility_times_M": 18.0
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    {
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      "direction1_every_local_opt_is_a_local_max_cut": false,
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      "n_edges": 3,
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      "n_local_optima_violating_direction1": 36,
      "n_strategic_local_optima": 38,
      "n_vertices": 4,
      "population_size": 13,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 20.0
    },
    {
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      "cut_induced_utility_times_M_range": [
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        48.0
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      "direction1_every_local_opt_is_a_local_max_cut": false,
      "direction2_every_local_max_cut_is_a_local_opt": true,
      "global_optima_are_max_cuts": true,
      "guards": true,
      "max_cut_weight": 9.0,
      "n_edges": 4,
      "n_local_max_cuts": 4,
      "n_local_optima_violating_direction1": 156,
      "n_strategic_local_optima": 160,
      "n_vertices": 4,
      "population_size": 16,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 20.0
    },
    {
      "classifiers_enumerated": 65536,
      "cut_induced_utility_times_M_range": [
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        44.0
      ],
      "direction1_every_local_opt_is_a_local_max_cut": false,
      "direction2_every_local_max_cut_is_a_local_opt": true,
      "global_optima_are_max_cuts": true,
      "guards": true,
      "max_cut_weight": 8.0,
      "n_edges": 4,
      "n_local_max_cuts": 4,
      "n_local_optima_violating_direction1": 155,
      "n_strategic_local_optima": 159,
      "n_vertices": 4,
      "population_size": 16,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 18.0
    },
    {
      "classifiers_enumerated": 131072,
      "cut_induced_utility_times_M_range": [
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        56.0
      ],
      "direction1_every_local_opt_is_a_local_max_cut": false,
      "direction2_every_local_max_cut_is_a_local_opt": true,
      "global_optima_are_max_cuts": true,
      "guards": true,
      "max_cut_weight": 12.0,
      "n_edges": 4,
      "n_local_max_cuts": 4,
      "n_local_optima_violating_direction1": 148,
      "n_strategic_local_optima": 152,
      "n_vertices": 5,
      "population_size": 17,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 24.0
    },
    {
      "classifiers_enumerated": 131072,
      "cut_induced_utility_times_M_range": [
        60.0,
        60.0
      ],
      "direction1_every_local_opt_is_a_local_max_cut": false,
      "direction2_every_local_max_cut_is_a_local_opt": true,
      "global_optima_are_max_cuts": true,
      "guards": true,
      "max_cut_weight": 13.0,
      "n_edges": 4,
      "n_local_max_cuts": 2,
      "n_local_optima_violating_direction1": 124,
      "n_strategic_local_optima": 126,
      "n_vertices": 5,
      "population_size": 17,
      "utility_identity_spread": 0.0,
      "worst_spurious_utility_times_M": 26.0
    }
  ],
  "seed": 20260725,
  "totals": {
    "classifiers_enumerated_total": 446768,
    "global_optima_are_max_cuts_repaired": true,
    "original_direction1_holds_everywhere": false,
    "original_direction2_holds_everywhere": true,
    "repaired_direction1_holds_everywhere": false,
    "repaired_direction2_holds_everywhere": true,
    "utility_identity_max_spread": 1.4210854715202004e-14
  },
  "verdict": {
    "published_reduction_is_solution_preserving": false,
    "repaired_reduction_is_solution_preserving": false,
    "summary": "The published reduction (Appendix F) admits strategic local optima whose vertex labelling is not a local max cut, so as written it is not a valid PLS reduction: the first claim of the proof fails whenever an edge's positive and negative points are both labelled 1.  Adding one guard point per edge (h=0, weight 1, close only to x_{(u,v)-}) removes every spurious local optimum and restores the reduction; the theorem's conclusion (PLS-hardness) therefore stands."
  }
}