Recursive-Depth Integration: A Depth-Weighted Measure and Integral on the Integers

We introduce a new integration theory on the integers based on the Recursive Division Tree (RDT) depth function R(n). For a decay parameter 0<ρ<10 < \rho < 10<ρ<1, we define a depth-weighted measure μR(A)=∑n∈AρR(∣n∣)\mu_R(A) = \sum_{n \in A} \rho^{R(|n|)}μR(A)=∑n∈AρR(∣n∣) and the corresponding recursive-depth integral ∫f(n) dμR=∑f(n)ρR(∣n∣)\int f(n)\,d\mu_R = \sum f(n)\rho^{R(|n|)}∫f(n)dμR=∑f(n)ρR(∣n∣). We characterize integrable functions, prove divergence criteria, and analyze how the saturation of R(n) (with canonical value R∗=3R^*=3R∗=3 for α=1.5\alpha=1.5α=1.5) shapes the analytic behavior of these integrals. Numerical experiments confirm the theoretical predictions and demonstrate rapid convergence for integrable functions. This work establishes recursive-depth integration as a new analytic structure linked to the recursive-adic framework.