We introduce an ultra-minimal two-parameter diagnostic model to probe the correlation structure of strictly increasing sequences exhibiting near-logarithmic growth. The model, fixed a priori with no refitting or normalization, reliably discriminates between fast-growth (unit-root) regimes, weakly correlated slow-growth regimes, and an intermediate regime characterized by nontrivial long-range correlations. Synthetic controls confirm the model's sensitivity. Application to the imaginary parts of the nontrivial zeros of the Riemann zeta function using Odlyzko's high-precision data (first 100,000 zeros) reveals a numerically rigid feedback coefficient φ_ζ ≈ 0.87-0.88, bounded away from both limiting cases with statistical significance p < 10^(-100). While acknowledging deliberate simplification of the asymptotic growth rate, the observed stabilization persists across five orders of magnitude (N = 20-100,000), establishing a robust and falsifiable empirical correlation signature detectable by this minimalist framework.