Short-Interval Primes: Dispersion, Short-Shift Averaging, and Arithmetic Progressions

We establish a short-interval dispersion bound for the von Mangoldt function Λ in arithmetic progressions on [N, N+H], H = N^θ, θ ∈ (1/2, 1), achieving a power saving of (H/N)^r via moment-vanishing short-shift averaging. Using Kuznetsov and spectral large sieve bounds, we prove unconditionally that [N, N+H] contains three-term and four-term arithmetic progressions (APs) of primes for each fixed θ ∈ (1/2, 1) and large N. Under a short-interval linear-forms hypothesis (SILF_k), the method extends to k-term APs. The dispersion and diagonal-removal components are unconditional; the k-term AP result is conditional on SILF_k.