{"database": "team-science", "table": "pair_answer", "is_view": false, "human_description_en": "where pair_id = \"ap-104bf56087\"", "rows": [["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 Cram\u00e9r-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 \u2248 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"]], "truncated": false, "filtered_table_rows_count": 3, "expanded_columns": [], "expandable_columns": [[{"column": "pair_id", "other_table": "adjacent_pair", "other_column": "id"}, null]], "columns": ["pair_id", "author", "statement", "falsify", "cheapest_test", "ref", "ts"], "primary_keys": ["pair_id", "author", "ts"], "units": {}, "query": {"sql": "select pair_id, author, statement, falsify, cheapest_test, ref, ts from pair_answer where \"pair_id\" = :p0 order by pair_id, author, ts limit 51", "params": {"p0": "ap-104bf56087"}}, "facet_results": {}, "suggested_facets": [{"name": "ts", "type": "date", "toggle_url": "http://explorer-production-64a5.up.railway.app/team-science/pair_answer.json?pair_id=ap-104bf56087&_facet_date=ts"}], "next": null, "next_url": null, "private": false, "allow_execute_sql": true, "query_ms": 5.108969286084175, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}