{
  "assessment": "verified",
  "claim": 3,
  "convex_combinations_checked": 570,
  "destructive_control": {
    "effect": "the middle atom is never active, so the parameterization is not lean",
    "minimum_domination_margin_on_129_points": 0.25,
    "mutation": "raise only the y=0 intercept of r(y)=y\u00b2 by 1/2"
  },
  "exact_witness_inequalities_checked": 14250,
  "finite_Y": [
    "-1",
    "-1/2",
    "0",
    "1/2",
    "1"
  ],
  "kernel": "Phi(x,y)=xy",
  "lean_parameterizations": 20,
  "literal_claim": "Theorem 4 (Section V.C) proves that the lean subset of finitely \u1ef8-convex functions forms a convex parameter space, which is what allows bilevel objectives to be rewritten as single-level problems solvable with standard first-order optimization (Section I).",
  "unordered_pairs": 190,
  "witness_identity": "at x=2ay_i+b, line_i-line_j=a(y_i-y_j)^2 >= 0"
}