outputs/claim5_sperner_convex.json

{
  "case_bounds_empirical": {
    "blend": {
      "case1_holds": false,
      "case1_hypothesis": "colours 1 and 2 missing among well-positioned samples",
      "case1_max_distance_to_A2A3": 0.5663437172187963,
      "case1_observed_max_Fx": -0.486829762478692,
      "case1_points": 31179,
      "case1_predicted_Fx_upper_bound": -0.6327722283113838,
      "case2_holds": true,
      "case2_hypothesis": "colour 2 missing, colours 1 and 3 present",
      "case2_min_distance_to_A1A2": 0.48245614035087714,
      "case2_observed_max_Fy": -0.3981130511506997,
      "case2_points": 5633,
      "case2_predicted_Fy_upper_bound": -0.3125
    },
    "strict": {
      "case1_holds": true,
      "case1_hypothesis": "colours 1 and 2 missing among well-positioned samples",
      "case1_max_distance_to_A2A3": 0.5663437172187963,
      "case1_observed_max_Fx": -0.6581047325530774,
      "case1_points": 31179,
      "case1_predicted_Fx_upper_bound": -0.6327722283113838,
      "case2_holds": true,
      "case2_hypothesis": "colour 2 missing, colours 1 and 3 present",
      "case2_min_distance_to_A1A2": 0.48245614035087714,
      "case2_observed_max_Fy": -0.3981130511506997,
      "case2_points": 5633,
      "case2_predicted_Fy_upper_bound": -0.3125
    }
  },
  "claim": "Theorem 3.12 (general convex domains)",
  "constant_audit": {
    "arithmetic_slip": "eps/8 = 1/64, not 1/32, when eps = 1/8",
    "case1_lower_bound": 0.5084485602574836,
    "case2_exceeds_1_over_32": false,
    "case2_exceeds_1_over_64": true,
    "case2_lower_bound": 0.01953125,
    "eps_over_8_with_eps_one_eighth": 0.015625,
    "eps_thick": 0.125,
    "k": 16,
    "minimal_k_for_case2_to_reach_eps_over_8": 12,
    "paper_states_eps_prime_le_eps_over_8_eq_1_over_32": true,
    "verdict": "the second case of the proof yields 0.01953, which clears eps/8 = 1/64 but NOT the printed 1/32; the argument goes through with eps' <= 1/64 (or with the printed 1/32 if eps is taken to be 1/4 rather than the 1/8 fixed earlier in the proof)"
  },
  "continuity_finding": {
    "issue": "Rule (18) assigns the colour of the NEAREST side, which is discontinuous across the angle bisectors outside the triangle, so the operator F built from it is discontinuous there -- yet Remark E.5 states F is given by a well-behaved arithmetic circuit and the VI (4) needs a continuous operator for a solution to exist.  Inside the triangle the bit-extraction ramp does make F continuous.",
    "our_fix": "blend the tied sides over a width of a quarter of a grid cell; every blended direction still points into the triangle, so the argument is unaffected, but the case-1 numerical bound loosens by up to about 0.1."
  },
  "corollary_3_13": {
    "identity_holds": true,
    "max_error_perf_gap_minus_c_times_vi_gap": 7.036472099430924e-17,
    "note": "with l(x;z) = 1/2||x-z||^2 and g(x) = x + c F(x), the performative stability gap is exactly c times the VI gap of -F, so an eps-stable point is an (eps/c)-VI solution and rho = Lip(g) <= 1 + c*Lip(F).",
    "scaling_c": 0.001
  },
  "direction_vectors": {
    "a_perp": [
      0.0,
      1.0
    ],
    "a_perp_orthogonal_to_A1A2": true,
    "b_perp": [
      0.8660254037844386,
      -0.5
    ],
    "b_perp_orthogonal_to_A1A3": true,
    "c_perp": [
      -0.8660254037844386,
      -0.5
    ],
    "c_perp_orthogonal_to_A2A3": true,
    "sum_is_zero": true
  },
  "domain_scan": [
    {
      "conclusion_holds_at_paper_eps_1_over_32": true,
      "domain": "disk R=1.00",
      "grid": "400x400",
      "ladder": [
        {
          "conclusion_holds": true,
          "eps": 0.015625,
          "missing_a_colour": 0,
          "n_solutions": 0,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.03125,
          "missing_a_colour": 0,
          "n_solutions": 1,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.0625,
          "missing_a_colour": 0,
          "n_solutions": 4,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.125,
          "missing_a_colour": 0,
          "n_solutions": 20,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.25,
          "missing_a_colour": 0,
          "n_solutions": 89,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": false,
          "eps": 0.5,
          "missing_a_colour": 777,
          "n_solutions": 1065,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": false,
          "eps": 1.0,
          "missing_a_colour": 13212,
          "n_solutions": 13604,
          "outside_triangle": 1748
        }
      ],
      "largest_eps_for_which_the_conclusion_holds": 0.25,
      "min_gap_on_scan": 0.03013057568672911,
      "points_in_X": 124980,
      "refined_minimum": {
        "all_three_colours_among_well_positioned_samples": true,
        "gap": 3.1788522905539787e-06,
        "inside_triangle": true,
        "x": [
          0.8063563389467745,
          0.6983250945871209
        ]
      }
    },
    {
      "conclusion_holds_at_paper_eps_1_over_32": true,
      "domain": "disk R=1.60",
      "grid": "400x400",
      "ladder": [
        {
          "conclusion_holds": true,
          "eps": 0.015625,
          "missing_a_colour": 0,
          "n_solutions": 0,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.03125,
          "missing_a_colour": 0,
          "n_solutions": 0,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.0625,
          "missing_a_colour": 0,
          "n_solutions": 2,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.125,
          "missing_a_colour": 0,
          "n_solutions": 4,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.25,
          "missing_a_colour": 0,
          "n_solutions": 12,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.5,
          "missing_a_colour": 0,
          "n_solutions": 50,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": false,
          "eps": 1.0,
          "missing_a_colour": 1133,
          "n_solutions": 1264,
          "outside_triangle": 0
        }
      ],
      "largest_eps_for_which_the_conclusion_holds": 0.5,
      "min_gap_on_scan": 0.046299369808534,
      "points_in_X": 124980,
      "refined_minimum": {
        "all_three_colours_among_well_positioned_samples": true,
        "gap": 8.80289077447738e-06,
        "inside_triangle": true,
        "x": [
          0.8063562889184213,
          0.6983251457739624
        ]
      }
    },
    {
      "conclusion_holds_at_paper_eps_1_over_32": true,
      "domain": "square (l_inf ball)",
      "grid": "400x400",
      "ladder": [
        {
          "conclusion_holds": true,
          "eps": 0.015625,
          "missing_a_colour": 0,
          "n_solutions": 0,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.03125,
          "missing_a_colour": 0,
          "n_solutions": 1,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.0625,
          "missing_a_colour": 0,
          "n_solutions": 4,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.125,
          "missing_a_colour": 0,
          "n_solutions": 14,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.25,
          "missing_a_colour": 0,
          "n_solutions": 51,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": false,
          "eps": 0.5,
          "missing_a_colour": 2,
          "n_solutions": 207,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": false,
          "eps": 1.0,
          "missing_a_colour": 7908,
          "n_solutions": 8260,
          "outside_triangle": 608
        }
      ],
      "largest_eps_for_which_the_conclusion_holds": 0.25,
      "min_gap_on_scan": 0.031146574428127807,
      "points_in_X": 160000,
      "refined_minimum": {
        "all_three_colours_among_well_positioned_samples": true,
        "gap": 4.734146226581349e-06,
        "inside_triangle": true,
        "x": [
          0.8063565604261106,
          0.6983251384321296
        ]
      }
    },
    {
      "conclusion_holds_at_paper_eps_1_over_32": true,
      "domain": "irregular 7-gon",
      "grid": "400x400",
      "ladder": [
        {
          "conclusion_holds": true,
          "eps": 0.015625,
          "missing_a_colour": 0,
          "n_solutions": 0,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.03125,
          "missing_a_colour": 0,
          "n_solutions": 0,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.0625,
          "missing_a_colour": 0,
          "n_solutions": 2,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.125,
          "missing_a_colour": 0,
          "n_solutions": 11,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": true,
          "eps": 0.25,
          "missing_a_colour": 0,
          "n_solutions": 47,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": false,
          "eps": 0.5,
          "missing_a_colour": 27,
          "n_solutions": 178,
          "outside_triangle": 0
        },
        {
          "conclusion_holds": false,
          "eps": 1.0,
          "missing_a_colour": 5608,
          "n_solutions": 5862,
          "outside_triangle": 167
        }
      ],
      "largest_eps_for_which_the_conclusion_holds": 0.25,
      "min_gap_on_scan": 0.03274938767575923,
      "points_in_X": 116195,
      "refined_minimum": {
        "all_three_colours_among_well_positioned_samples": true,
        "gap": 4.82544928431154e-06,
        "inside_triangle": true,
        "x": [
          0.8063565460207917,
          0.6983253259767227
        ]
      }
    }
  ],
  "geometry_audit": {
    "A1": [
      0.0,
      0.0
    ],
    "A2": [
      1.7320508075688772,
      0.0
    ],
    "A3": [
      0.8660254037844386,
      1.5
    ],
    "circumcentre": [
      0.8660254037844387,
      0.5
    ],
    "circumradius": 1.0,
    "distance_of_A1_from_the_origin": 0.0,
    "distance_of_A2_from_the_origin": 1.7320508075688772,
    "equilateral": true,
    "note": "Definition 3.11 puts the inner ball at the origin, B_{R1}(0) subset X, and the proof takes R1 = 1 with an equilateral triangle on the boundary of that ball -- but it also sets A1 = (0,0), which puts the triangle's circumcentre at (sqrt3/2, 1/2), not at the origin.  The two conventions cannot hold simultaneously; the triangle inscribed in B_1(0) would need |A_i| = 1 for all i.  This is a coordinate slip with no effect on the argument -- we work in the paper's coordinates and place the inner unit ball at the circumcentre.",
    "side_lengths": [
      1.7320508075688772,
      1.7320508075688772,
      1.7320508075688772
    ]
  },
  "lemma_E4": {
    "L": 288,
    "holds": true,
    "k": 16,
    "max_poorly_positioned_samples_observed": 2,
    "mean_poorly_positioned": 0.2650375,
    "n": 3,
    "statement": "with L = (k+2)2^{n+1}, at most two of the k samples are poorly positioned"
  },
  "lipschitz_rescaling": {
    "consistent": true,
    "empirical_Lipschitz_of_F": 28.252269484497358,
    "empirical_Lipschitz_of_F_over_2n": 3.5315336855621697,
    "eps_double_prime": 0.001953125,
    "grid_scale_2_to_the_n": 8,
    "paper_claim": "Lip(F) = O(2^n); F' = F/2^n has Lip = O(1) and eps'' = eps'/2^n = O(2^-n)"
  },
  "seed": 20260725,
  "sperner": {
    "grid": "8 x 8",
    "of_which_yield_a_trichromatic_triangle": 3,
    "recovery_always_possible": true,
    "sperner_lemma_guarantees_at_least_one": true,
    "trichromatic_squares": 3
  },
  "verdict": {
    "case_bounds_hold_with_paper_rule_18": true,
    "conclusion_holds_at_every_eps_up_to": [
      {
        "domain": "disk R=1.00",
        "largest_eps": 0.25
      },
      {
        "domain": "disk R=1.60",
        "largest_eps": 0.5
      },
      {
        "domain": "square (l_inf ball)",
        "largest_eps": 0.25
      },
      {
        "domain": "irregular 7-gon",
        "largest_eps": 0.25
      }
    ],
    "constant_slip": "eps/8 = 1/64, not 1/32, when eps = 1/8",
    "construction_executes_on_all_domains": true,
    "continuity_slip": "rule (18) makes F discontinuous outside the triangle",
    "coordinate_slip": "A1 = (0,0) is incompatible with B_{R1}(0) subset X",
    "lemma_E4_holds": true,
    "min_refined_gap_over_domains": 8.80289077447738e-06
  }
}