outputs/claim6.json

{
  "assessment": "verified",
  "claim": 6,
  "couplings_exhausted": 40320,
  "destructive_control": {
    "mutation": "reverse the coupling",
    "optimality_gap": 0.21428571428571427,
    "surplus": 0.14285714285714285
  },
  "dual_constraints_checked": 64,
  "dual_potentials": "phi(x)=x\u00b2/2 and psi(y)=y\u00b2/2; phi is the Y-transform of psi",
  "dual_value": 0.35714285714285715,
  "exact_duality_gap": "0/1",
  "grid": [
    0.0,
    0.14285714285714285,
    0.2857142857142857,
    0.42857142857142855,
    0.5714285714285714,
    0.7142857142857143,
    0.8571428571428571,
    1.0
  ],
  "literal_claim": "Kantorovich dual solutions for optimal transport are characterized as \u1ef8-convex functions in Section IV, so gradients of the learned parametrization directly yield optimal transport maps via a diffeomorphism condition.",
  "map_mismatches": 0,
  "minimum_dual_slack": 0.0,
  "optimal_permutation": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7
  ],
  "primal_value": 0.35714285714285715,
  "surplus": "Phi(x,y)=xy",
  "twist_gradient": "nabla_x Phi(x,y)=y is a diffeomorphism; inverse applied to nabla phi(x)=x yields gamma(x)=x"
}