{
  "status": "PASS",
  "method": "Exact bivariate integer-polynomial identities; no floating-point sampling.",
  "definitions": {
    "r": "rho2/rho1 > 1",
    "A": "mu0*rho1*w1^2/Bn^2",
    "q": "Bt2/Bt1 = r*(A-1)/(A-r)"
  },
  "identities": [
    "q-1 = A*(r-1)/(A-r)",
    "q-r = r*(r-1)/(A-r)",
    "A*(q/r-1) = q-1"
  ],
  "identity_residual_coefficients": [
    {},
    {},
    {}
  ],
  "ordinary_fast_necessary_domain": "A > r: q > r > 1",
  "ordinary_slow_necessary_domain": "0 < A < 1: 0 < q < 1",
  "scope": "Necessary discriminator for regular coplanar compressive ideal-MHD shocks. Not sufficient for full admissibility, not an E02 solution.",
  "excluded_limits": [
    "r=1",
    "Bn=0 in the rational formula",
    "Bt1=0",
    "A=r pole",
    "A=1 switch-off boundary",
    "characteristic degeneracies"
  ],
  "perpendicular_limit": "Bn=0 must be treated separately; a regular perpendicular compressive fast jump has Bt2=r*Bt1.",
  "numerical_root_solver_run": false
}