A Sigma-Based Detection Framework for Prime Constellations with Applications to Bounded Prime Gaps

A family of prime constellation detectors $\text{D}\left(p,N\right)$ expressed as rational powers of 2π, built from the sum-of-divisors function σ. The central object $\text{D}\left(p,N\right)={\left(2\pi \right)}^{p+N+1？\left[\sigma \left(p\right)+\sigma \left(p+2N\right)\right]/2}$ equals 1 if and only if both $p$ and $p+2N$ are prime, and is strictly less than 1 otherwise. This construction generalizes naturally to k-tuples via a product formula. A zeta-regularized product identity ${\displaystyle \prod \text{D}\left(p\right)}={\left(2\pi \right)}^{\frac{3}{8}}$ is created and decomposed into an associated sum of three explicitly characterized sub-series whose regularized total equals ？5/48, a value determined purely by $\zeta \left(0\right)$ and $\zeta \left(？1\right)$ . It is shown that the composite sub-series $F\left(1\right)$ converges to a constant numerically close to