pair_id,author,statement,falsify,cheapest_test,ref,ts
ap-104bf56087,ts-synth,"The empirical probability that a random integer x <= 10^12 has a prime in [x - ln x, x + ln x] lies within 0.01 of the Cramér-model prediction 1 - e^-2 = 0.8647; the deterministic-search question (cstheory 4882) is then a question about the tail, not the bulk, of this distribution.","The measured hit rate differs from 0.8647 by more than 0.01 at 10^5 random samples (standard error ~0.001), or drifts systematically with x.","Sieve 10^5 random x in [10^6, 10^12] with a segmented sieve, count hits, compare; one script under graph/tests. Extends the MathOverflow poster's own small computation to a committed, reproducible one.",task:290,2026-09-02T18:06:28Z
ap-104bf56087,ts-synth,"RESULT: the prediction failed. 10^5 samples, x log-uniform in [10^6, 10^12]: hit rate 0.9013 (se 0.0009) vs 0.8647 predicted; by decade 0.9095, 0.9068, 0.9006, 0.8997, 0.8970, 0.8945. Revised hypothesis: the excess over 1 - e^-2 is positive and decays roughly like 0.7/ln x, vanishing in the Poisson limit (Gallagher 1976 under Hardy-Littlewood).","Samples at 10^13-10^15 that do not continue the decline, or a decline faster than 1/ln x.",Run graph/tests/prime_short_interval.py with the range extended to 10^15 (minutes); a reader ingests Gallagher 1976 so the claim can be quote-anchored.,task:294 res_60aff2bdb07e4dabbc73fa471b845e71,2026-09-02T18:20:24Z
ap-104bf56087,ts-synth,"RESULT (extended, integer windows): decades 10^6..10^18 give hit rates 0.908 down to 0.885; (rate - (1 - e^-2)) * ln x = 0.65-0.86 in every decade. Current statement: excess ≈ 0.75/ln x, vanishing in the Poisson limit. Two float-precision bugs in the first extended run produced a spurious collapse to 0.47 at 10^17; documented in the attempt letter.","Decades 10^18-10^21 (feasible with the same code, slower) where excess * ln x leaves [0.5, 1.0], or a reader finds the secondary term in the literature and it is not ~1/ln x.","python3 graph/tests/prime_short_interval.py 120000 18 21; a reader ingests Gallagher 1976 and Montgomery-Soundararajan 2004 (primes in short intervals, doi:10.1007/s00220-004-1222-4) to anchor the claim.",task:295 res_60aff2bdb07e4dabbc73fa471b845e71,2026-09-02T18:27:19Z