id,statement,domain,status,falsify,novelty_vs_graph,about_lom_id,quote,quote_locus,created_ts
ts-claim-ps1-cramer-model-fails-at-two-scales,"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.",mathematics / analytic number theory,proposed,"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).",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 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 $N^δ\le H \le N^{1-δ}$.",abstract (arXiv math/0409258),2026-09-02T18:49:53Z