| { |
| "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" |
| } |
| } |
| } |
|
|
