Logarithmic Interaction of Primes and GUE Statistics: A Numerical Study of the Jacobi-Type Operator

We construct a symmetric matrix operator whose diagonal elements are logarithms of prime numbers and whose off-diagonal elements encode a logarithmic resonance interaction between primes. Numerical experiments demonstrate that the spectral gap statistics of this operator closely match those of the Gaussian Unitary Ensemble (GUE), with the proportion of small gaps equal to 0.040, compared to the GUE theoretical value of 0.043. This result suggests a structural connection between the multiplicative architecture of primes and the random matrix statistics conjectured by Montgomery (1973) to describe the non-trivial zeros of the Riemann zeta function. No claim of proof is made; this is a numerical observation motivating further investigation.