Title: A Numerical-Spectral Approach to Robin's Criterion via Prime-Gap Jacobi Matrices

We construct an explicit real symmetric Jacobi matrix D generated from prime gaps and the maximal divisor-density function σ(n)/n, and relate it in a direct, non-metaphorical way to Robin's criterion---a known equivalent form of the Riemann Hypothesis. We verify numerically the positivity and boundary properties of D for the first $N=3000$ prime gaps (up to the prime $27457$), and correct a boundary artifact in the definition of the first diagonal entry. We document the experimental observation that the largest eigenvalue of D stays within a ratio of $1.02$ to $1.11$ of Robin's bound eγlnln p, and we prove two rigorous analytic theorems: (i) a uniform lower bound Dk,k≥ 7/4 on all diagonal entries, and (ii) an explicit upper bound λₘₐₓ(DN)<eγlnln(pN+2)+√2 derived from Weyl's inequality and the Gershgorin circle theorem. We emphasise that this paper does not approach a proof of the Riemann Hypothesis; its actual contribution is a precisely scoped numerical--diagnostic tool for Robin's criterion, correct, extensible, and testable.