files / problems / union-closed / compute / q2 / certs / ceiling.json

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{
  "statement": "any 2-sample bit protocol, product coupling of (S,T), family {b,1}: first-crossing \u2264 min f, f(b)=1-(1-b)h(b)",
  "b_star": "0.296493923569337569030703383782",
  "b_star_float": 0.29649392356933757,
  "h_b": "0.876962778407473904290489498992",
  "ceiling": "0.383051356586825582501646873002",
  "ceiling_float": 0.3830513565868256,
  "critical_eq_residual": 1.0542197943230523e-81,
  "g_second_deriv": "-7.3589709725847554085",
  "g_second_deriv_negative": true,
  "f_at_thresh": "0.383099297681388324963411",
  "f_at_half": "0.5",
  "f_at_phi": "0.407046616515146088734638",
  "thresh": "0.292893218813452475599156",
  "phi": "0.381966011250105151795413",
  "why_half_is_not_this_class": "f(1/2)=1/2, but the minimizer on the saturation interval is at b*\u22480.2965, where f=0.383051\u2026",
  "claimed_c": 0.38304,
  "claimed_below_ceiling": true,
  "beats_repo_0_38285": true,
  "beats_liu": true,
  "half_above_ceiling": true,
  "note": "this is a ceiling for the hypothesis class, not a new frequency constant.  0.38304 is already below it."
}