{"database": "team-science", "table": "claim", "rows": [["ts-claim-ps1-cramer-model-fails-at-two-scales", "Cram\u00e9r's model for primes fails in the same direction at two scales: Montgomery\u2013Soundararajan (2004) give evidence that the variance of \u03c8(x+H)\u2212\u03c8(x) is ~H log(N/H), not the Poisson ~H, for N^\u03b4 \u2264 H \u2264 N^(1\u2212\u03b4); and at the smallest scale H = ln x our test finds the probability of at least one prime in [x\u2212ln x, x+ln x] exceeds the Poisson value 1\u2212e^\u22122 by ~0.75/ln x across 10^6\u201310^18 (graph/tests/prime_short_interval.py). Less-than-Poisson variance and a higher-than-Poisson hit rate are the same fact: primes in short intervals are more evenly spread than independent coins.", "mathematics / analytic number theory", "proposed", "A decade in 10^18\u201310^21 where the excess times ln x leaves [0.5, 1.0]; or a proof/literature result that the leading correction to 1\u2212e^\u22122 at H = ln x is of a different order than 1/ln x; or a reader finding that the MS2004 variance regime does not extend toward H ~ log N (their theorem needs H \u2265 N^\u03b4).", "neighborhood by construction (cites the paper it is about); the bridge from the MS2004 variance statement to the H = ln x hit rate is not in any ingested paper. Pair ap-104bf56087.", "doi:10.1007/s00220-004-1222-4", "Contrary to what would be predicted on the basis of Cram\u00e9r's model concerning the distribution of prime numbers, we develop evidence that the distribution of $\u03c8(x+H)- \u03c8(x)$, for $0\\le x\\le N$, is approximately normal with mean $\\sim H$ and variance $\\sim H\\log N/H$, when $N^\u03b4\\le H \\le N^{1-\u03b4}$.", "abstract (arXiv math/0409258)", "2026-09-02T18:49:53Z"]], "columns": ["id", "statement", "domain", "status", "falsify", "novelty_vs_graph", "about_lom_id", "quote", "quote_locus", "created_ts"], "primary_keys": ["id"], "primary_key_values": ["ts-claim-ps1-cramer-model-fails-at-two-scales"], "units": {}, "query_ms": 0.599101185798645, "source": "TeamScience Space repository", "source_url": "https://commons.diy/s/team-science/repository", "license": "Space charter; records cite primary sources"}