From Prime Gaps to the Euler Product: A Spectral Sieve via the Divisor Operator

This paper completes the algebraic-spectral framework by providing a direct spectral construction of the Euler product. Using the closed-form formula for individual prime gaps derived from the bounded divisor operator T , we construct a purely prime spectral indicator function θ(x) = ∑n δ(x − pn ). By injecting this sieve kernel into the standard Mellin transform, we first derive the prime zeta function P(s) = ∑p− s n . Then, utilizing the classical Mercator series for ln ζ(s), we rigorously prove that the resulting generating function collapses to the Euler product over primes, free from any composite residuals. This final step closes the circle initiated in five preceding works, establishing a rigorous spectral realization of the Riemann Zeta function and confirming the exact spectral matching and logical consistency at the critical line ℜ(s) = 1/2.