Goldbach via a Mod-8 Kernel: Density-One and Short-Interval Positivity
Recognition Physics Institute
Abstract
We present a classical circle-method framework for Goldbach based on a mod-8 periodic kernel \(K_8\). On major arcs we obtain a positive main term given by a 2-adic gate \(c_8(2m)\in\{1,\tfrac12\}\) times the Hardy–Littlewood singular series \(\mathfrak S(2m)\). On minor arcs we prove unconditional density-one positivity via mean-square bounds and convert fourth-moment control into pointwise positivity in every short interval, giving bounded gaps between exceptional even integers. A quantified medium-arc dispersion lemma yields a small saving that lowers the short-interval exponent. We also record a Chen/Selberg variant (prime + almost-prime), explicit constants, a smoothed-to-sharp transfer, and a reproducible computational protocol; an optional GRH template is included for comparison.
Key Contributions
- Mod-8 kernel producing a 2-adic gate on major arcs with \(\mathfrak S(2m)\).
- Unconditional density-one positivity; bounded gaps between exceptions via short-interval control.
- Medium-arc dispersion savings (quantified) improving the short-interval exponent.
- Chen/Selberg variant (prime + almost-prime) with explicit protocol.
- Reproducible constants ledger, smoothed-to-sharp transfer, and computational closure plan.