Spectral Realization of the Riemann-von Mangoldt Law via a Bounded Divisor Operator

This paper completes the spectral bridge initiated in two preceding works. Starting from the bounded Toeplitz-Catalan operator T defined on the primeexponent space, we show that the spectral parameter tk, inversely proportional to the eigenvalues (λk = 1/tk), rigorously reproduces the Riemann-von Mangoldt law for the zeros of the Zeta function. Crucially, by mapping the trace coefficients to the unbounded inverse operator T −1 , we derive the full spectral density, including the oscillatory term governed by prime powers. By demonstrating that the trace of T −1 contains the exact oscillatory contribution of the primes, we establish a rigorous spectral realization of the statistical laws underlying the distribution of prime numbers. Furthermore, we derive a closed-form spectral formula for the mean prime gaps via a localized Fourier transform dualization, proving that the distribution of primes is a deterministic consequence of the algebraic structure of the divisor network.