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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.007 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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 teacher head, 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".