Resonance Filtering and Structural Inconsistencies in Weil's Explicit Formula: A Function-Theoretic Prime-Zero Duality Approach

This paper presents a rigorous comparative study of structural resonance effects in Weil's explicit formula, investigating the qualitative and quantitative differences between the Riemann zeta function and Dirichlet L-functions under the presence of off-critical-line zeros. By introducing a "resonance probe" test function, we reveal a striking contrast: while Dirichlet L-functions (e.g., L(s, χ₋₄)) maintain structural balance even with non-critical zeros, the Riemann zeta function, when hypothetical off-line zeros (quartets) are introduced, exhibits catastrophic structural imbalance in the explicit formula. 

Our approach provides:

- An analytic framework for quantifying prime-zero duality using function-theoretic and harmonic analysis methods

- A complete Python code appendix for computational verification and reproducibility

- Explicit error estimates for truncations and scaling analysis

**Main findings:**  

- "Resonance filtering" amplifies quartet contributions for ζ(s), but these are exactly balanced in L-functions

- Structural imbalance (explosive quartet effect) appears only in ζ(s), supporting a new structural diagnostic for the Riemann Hypothesis

- The methodology is robust under parameter variations and generalizes to wide classes of test functions

This work offers a new direction in prime-zero duality analysis and proposes a function-theoretic resonance filter as a diagnostic tool for the Riemann Hypothesis.