A Poisson--Zeta Smoothing Approach to an RH-Conditional Prime-Interval Bound with Exponent 0.51

This paper develops a real-variable Poisson--Zeta smoothing framework for studying prime-counting functions and the distribution of primes in short intervals. We introduce the positive prime-density function \(\mathcal{D}_K(x)=\frac{K^Kx^{K-1}}{\Gamma(K)}\sum_p p^{-K}e^{-Kx/p},\) which also admits a representation involving real values of the Riemann zeta function through Mobius inversion. Assuming the Riemann Hypothesis (RH), we establish that, for all sufficiently large \(x\), \(\mathcal{D}_{K(x)}(x)\geq\frac{1}{2\log x},\) where \(K(x)=\left\lceil 1024e^2x^{0.98}\log x\right\rceil.\) Using gamma-kernel estimates, Stieltjes integration by parts, and the classical RH-conditional error bound for the Chebyshev function, we prove that every sufficiently large interval \(\left(x-\frac12x^{0.51},\,x+\frac12x^{0.51}\right)\) contains at least one prime. The method connects the localization properties of a positive smoothing kernel with prime-counting error estimates and provides a framework for investigating weighted prime-distribution inequalities. The resulting prime-interval bound is conditional on RH and does not improve existing RH-conditional results, including the stronger bounds of order \(O(\sqrt{x}\log x)\) established by Carneiro, Milinovich, and Soundararajan. The contribution is an alternative proof framework based on Poisson--Zeta smoothing. Establishing unconditional density lower bounds or sharper estimates within this framework remains an open research direction.