{
  "sparse_active_set": [
    {
      "ratio": 1,
      "condition": 1.0000000000000009,
      "cg_matvecs": 28,
      "pg_steps": 2,
      "pivots": 13,
      "support": 3,
      "sum_error": 2.220446049250313e-16,
      "min_x": 0.0,
      "kkt_residual": 3.3306690738754696e-16,
      "objective_gap": 0.0
    },
    {
      "ratio": 10,
      "condition": 9.991008991008988,
      "cg_matvecs": 266,
      "pg_steps": 49,
      "pivots": 13,
      "support": 3,
      "sum_error": 0.0,
      "min_x": 0.0,
      "kkt_residual": 4.440892098500626e-16,
      "objective_gap": 0.0
    },
    {
      "ratio": 100,
      "condition": 99.9010989010986,
      "cg_matvecs": 291,
      "pg_steps": 125,
      "pivots": 13,
      "support": 3,
      "sum_error": 0.0,
      "min_x": 0.0,
      "kkt_residual": 1.1213252548714081e-14,
      "objective_gap": -1.7763568394002505e-15
    },
    {
      "ratio": 1000,
      "condition": 999.0019980020072,
      "cg_matvecs": 329,
      "pg_steps": 380,
      "pivots": 13,
      "support": 3,
      "sum_error": 0.0,
      "min_x": 0.0,
      "kkt_residual": 1.5963896871085126e-12,
      "objective_gap": 0.0
    },
    {
      "ratio": 10000,
      "condition": 9990.01098901406,
      "cg_matvecs": 366,
      "pg_steps": 1437,
      "pivots": 13,
      "support": 3,
      "sum_error": 0.0,
      "min_x": 0.0,
      "kkt_residual": 1.6560419702216223e-11,
      "objective_gap": 1.1368683772161603e-13
    }
  ],
  "isolated_cg": [
    {
      "condition": 1,
      "sqrt_condition": 1.0,
      "cg_iterations": 1,
      "relative_residual": 0.0
    },
    {
      "condition": 10,
      "sqrt_condition": 3.1622776601683795,
      "cg_iterations": 16,
      "relative_residual": 1.6851536577794748e-13
    },
    {
      "condition": 100,
      "sqrt_condition": 10.0,
      "cg_iterations": 19,
      "relative_residual": 4.811742637911966e-11
    },
    {
      "condition": 1000,
      "sqrt_condition": 31.622776601683793,
      "cg_iterations": 23,
      "relative_residual": 6.986908700724487e-13
    },
    {
      "condition": 10000,
      "sqrt_condition": 100.0,
      "cg_iterations": 27,
      "relative_residual": 5.6553155353353794e-12
    }
  ],
  "quantitative_predictions": [
    {
      "prediction": "CG work is sublinear in condition number, with ideal upper-bound scaling O(sqrt(kappa)); PG work grows more steeply.",
      "observed": "On router sweep, CG 2->55 while PG 2->3239 as kappa 1->9990; log-log slopes are reported below."
    },
    {
      "prediction": "Strictly positive inactive KKT margins trigger active-set pivots and yield sparse feasible x.",
      "observed": "Sparse sweep has support 3, nonnegative coefficients, simplex error and KKT residual reported per ratio."
    },
    {
      "prediction": "CG residual reaches tolerance without violating SPD stability.",
      "observed": "Standalone diagonal SPD sweep reports residual <=1e-10 for every condition."
    }
  ],
  "slopes": {
    "router_cg_vs_sqrt_kappa": 0.23250280700867262,
    "router_pg_vs_kappa": 0.6602910727031096
  }
}