Resonance Filtering and Structural Inconsistencies in Weil's Explicit Formula: A Function-Theoretic Prime-Zero Duality Approach
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.016 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".