{
  "paper": "Positive Distribution Shift as a Framework for Understanding Tractable Learning",
  "openreview_id": "DkLQ40hTlt",
  "claim": "Appendix-B printed decoder threshold for Theorem 3.2",
  "exact_identity": "K~Binomial(m,3/(5r)); printed success=P[K>=ceil(3m/(5r))]",
  "logical_consequence": "All-bit success for an all-ones code is at most this one-bit marginal. For r=1 the bound is exact, so the printed decoder cannot attain the usual >=2/3 success probability as epsilon decreases.",
  "rows": 25,
  "all_printed_probabilities_below_two_thirds": true,
  "r1_smallest_epsilon": {
    "r": 1,
    "epsilon": 1e-12,
    "samples_m": 1133,
    "p_one": 0.6,
    "p_zero": 0.4,
    "printed_threshold": 0.6,
    "midpoint_threshold": 0.5,
    "printed_one_bit_success_exact": 0.5080632831256272,
    "midpoint_one_bit_success_exact": 0.9999999999948532,
    "printed_zero_bit_success_exact": 1.0,
    "midpoint_zero_bit_success_exact": 0.9999999999948532,
    "all_ones_success_upper_bound_printed": 0.5080632831256272,
    "printed_exceeds_two_thirds": 0
  },
  "maximum_printed_one_bit_success": 0.5395430622999275,
  "minimum_midpoint_one_bit_success": 0.9895403809911729,
  "asymptotic_limit_printed_one_bit": 0.5,
  "scope": "Exact audit of the printed decoder, not a disproof of a repaired midpoint decoder or of every possible f-PDS universality theorem.",
  "execution": {
    "started_at_utc": "2026-07-22T09:49:56.127198+00:00",
    "completed_at_utc": "2026-07-22T09:49:56.135377+00:00",
    "elapsed_seconds": 0.008177252486348152,
    "python": "3.10.12",
    "platform": "Linux-5.15.0-139-generic-x86_64-with-glibc2.35",
    "packages": {
      "numpy": "2.2.6",
      "scipy": "1.14.0",
      "matplotlib": "3.10.3"
    }
  }
}