محفوظ في:
| المؤلف الرئيسي: | |
|---|---|
| التنسيق: | Recurso digital |
| اللغة: | |
| منشور في: |
Zenodo
2026
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| الموضوعات: | |
| الوصول للمادة أونلاين: | https://doi.org/10.5281/zenodo.19520558 |
| الوسوم: |
إضافة وسم
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جدول المحتويات:
- <p class="p1">Research Note 51 in the "Geometry of the Critical Line" programme.</p> <p class="p1">This note establishes two results on restricted Weil positivity:</p> <p class="p1"><strong>Theorem A </strong>(restricted archimedean positivity): At the maximal prime-free support parameter a₂ = ½ log 2, the Weil functional is non-negative on the admissible class. For every G <span class="s1">∈</span> L²([−a₂, a₂]) satisfying Ĝ(±i/2) = 0, the archimedean Weil form W[G] ≥ 0. The proof constructs a codimension-one coercive estimate on the even Paley–Wiener space using Slepian concentration theory: the archimedean Toeplitz operator has exactly one negative eigenvalue, and the admissibility constraint projects out the corresponding eigendirection. The coercive gap is c_gap ≥ 0.063 > 0. An explicit min-max computation proves positive definiteness (not just semidefiniteness) on the admissible hyperplane. This coincides with the archimedean positivity theorem of Connes and Consani (2021, Selecta Math.); the proof here is independent, using Toeplitz decomposition and Slepian concentration rather than Sonin space compression.</p> <p class="p1"><strong>Theorem B </strong>(local prime-active extension): In the first prime-active window a <span class="s1">∈</span> [a₂, a₃) with a₃ = ½ log 3, the Weil functional reduces to a single Toeplitz form with modified symbol Ψ(t) = φ(t) − √2 log 2 cos(t log 2). There exists δ* > 0 such that restricted Weil positivity persists for all a <span class="s1">∈</span> [a₂, a₂ + δ*]. The proof is by contradiction: after transport to a fixed Hilbert space L²([−1,1]), Hilbert–Schmidt compactness of bounded-window Fourier restriction and the positive tail of Ψ exclude any sequence of negative admissible minimisers accumulating at a₂. The extension is nonquantitative: no explicit value for δ* is produced. Full positivity on Window I is not claimed. The Riemann Hypothesis is not claimed.</p>