PALEY-WIENER TENSION OPERATORS, BANDWIDTH INDUCTION, AND THE STRUCTURE OF THE RIEMANN HYPOTHESIS
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2026
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| _version_ | 1866901033960079360 |
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| author | Mellor, Victoria |
| author_facet | Mellor, Victoria |
| contents | <p>For each bandwidth parameter A > 0, the Weil explicit formula defines a tension operator TA on L 2 ([−A, A]) whose positive semi-definiteness is equivalent to the Riemann Hypothesis. I develop a bandwidth induction framework for studying this positivity. The key tool is a bandwidth monotonicity theorem: the Weil quadratic form QA(ϕ) is invariant under bandwidth increase for functions supported in the smaller interval. This yields a base case (TA ⪰ 0 for A < 1 2 log 2, where no primes contribute), a core-wing decomposition at each prime threshold, and a reduction of positivity to three conditions: the induction hypothesis, wing positivity, and a cross-term bound. We prove that the cross-term bound is equivalent to the positive semi-definiteness of the Weil bilinear form, and therefore equivalent to the Riemann Hypothesis itself. I then pursue a de Branges propagation approach to circumvent this circularity. I prove that the derivative dQp A/dA of the prime quadratic form is a rank-at-most-two perturbation concentrated at the boundary values ϕ(±A) (the de Branges Hamiltonian structure) and establish<br>a shimmer bound on the prime boundary functional via the large sieve. For A beyond a finite threshold A ∗ , the archimedean boundary term (growing as e A/2 ) unconditionally dominates the prime boundary functional, yielding a finite window reduction: positivity propagates upward from A ∗ by Gronwall’s inequality and downward by bandwidth monotonicity. Numerical<br>computation of the true rupture ratio—the generalized eigenvalue supϕ Q p (ϕ)/Q∞(ϕ)—reveals that the prime and archimedean eigenspaces are substantially misaligned: the true ratio is 3–7 times smaller than the crude operator norm ratio, remaining below 9.1% for all A ≤ 8. Verification across A ∈ [0.25, 12.0] using 300 zeta zeros confirms all conditions with over 90%<br>headroom at every scale.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18920297 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | PALEY-WIENER TENSION OPERATORS, BANDWIDTH INDUCTION, AND THE STRUCTURE OF THE RIEMANN HYPOTHESIS Mellor, Victoria <p>For each bandwidth parameter A > 0, the Weil explicit formula defines a tension operator TA on L 2 ([−A, A]) whose positive semi-definiteness is equivalent to the Riemann Hypothesis. I develop a bandwidth induction framework for studying this positivity. The key tool is a bandwidth monotonicity theorem: the Weil quadratic form QA(ϕ) is invariant under bandwidth increase for functions supported in the smaller interval. This yields a base case (TA ⪰ 0 for A < 1 2 log 2, where no primes contribute), a core-wing decomposition at each prime threshold, and a reduction of positivity to three conditions: the induction hypothesis, wing positivity, and a cross-term bound. We prove that the cross-term bound is equivalent to the positive semi-definiteness of the Weil bilinear form, and therefore equivalent to the Riemann Hypothesis itself. I then pursue a de Branges propagation approach to circumvent this circularity. I prove that the derivative dQp A/dA of the prime quadratic form is a rank-at-most-two perturbation concentrated at the boundary values ϕ(±A) (the de Branges Hamiltonian structure) and establish<br>a shimmer bound on the prime boundary functional via the large sieve. For A beyond a finite threshold A ∗ , the archimedean boundary term (growing as e A/2 ) unconditionally dominates the prime boundary functional, yielding a finite window reduction: positivity propagates upward from A ∗ by Gronwall’s inequality and downward by bandwidth monotonicity. Numerical<br>computation of the true rupture ratio—the generalized eigenvalue supϕ Q p (ϕ)/Q∞(ϕ)—reveals that the prime and archimedean eigenspaces are substantially misaligned: the true ratio is 3–7 times smaller than the crude operator norm ratio, remaining below 9.1% for all A ≤ 8. Verification across A ∈ [0.25, 12.0] using 300 zeta zeros confirms all conditions with over 90%<br>headroom at every scale.</p> |
| title | PALEY-WIENER TENSION OPERATORS, BANDWIDTH INDUCTION, AND THE STRUCTURE OF THE RIEMANN HYPOTHESIS |
| url | https://doi.org/10.5281/zenodo.18920297 |