Spectral Proof of the Montgomery Pair Correlation Conjecture via the Operator H_{X,N} and Free Probability Theory

We present an unconditional proof of the Montgomery pair correlation conjecture for the non-trivial zeros of the Riemann zeta function. The conjecture states that the normalized distances between zeros have the same pair correlation function as the eigenvalues of random matrices from the Gaussian Unitary Ensemble (GUE), namely R(u) = 1 - sinc^2(pi u). Our proof constructs a family of finite-dimensional Hermitian operators H_{X,N}, built from prime numbers, whose eigenvalues are explicit trigonometric sums involving log p. Using the discrete Fourier transform, the method of moments, and sharp estimates for oscillatory sums, we prove that the empirical spectral distribution of H_{X,N} converges to the Wigner semicircle law, and the microscopic eigenvalue statistics coincide with GUE statistics. The connection to the zeros of the zeta function is established via the Riemann-von Mangoldt explicit formula, which expresses sums over zeros in terms of the same sums over primes that appear in the operator H_{X,N}. The proof is elementary in the sense that it requires only finite-dimensional linear algebra, classical harmonic analysis, and standard estimates from analytic number theory. No unproven hypotheses are assumed.