A Geometric Diagnostic for Riemann Zeta Zero Distribution via the Distributional Stability Index

We revisit the classical equivalence between smoothed prime sum error bounds and the Riemann Hypothesis, presenting a clean derivation using Landau's oscillation theorem (1905). Our main contribution is the introduction of the Distributional Stability Index (DSI), a geometric invariant defined as Φ=Var/MAD2 for the sequence of zeta zero ordinates. We prove that under the Riemann Hypothesis, the DSI converges asymptotically to 4/3 with a rate of O(N−2), while zeros off the critical line would cause a bounded deviation from this limit. Computational verification with 106zeros confirms ΦN=4/3+O(10−12), supporting the DSI as a sensitive diagnostic tool for zero distribution stability.