A Proof of the Riemann Hypothesis for the Riemann Zeta Function: Weil Positivity via Semilocal Spectral Descent
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2026
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| _version_ | 1866901984465911808 |
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| author | Franchi Viceré, Christian |
| author_facet | Franchi Viceré, Christian |
| contents | <p>I prove the positivity of the Weil quadratic form Q_W ≥ 0 for all smooth compactly supported test functions satisfying the vanishing conditions, which by the Weil criterion implies that all non-trivial zeros of the Riemann zeta function lie on Re(s) = 1/2.</p><p>My key idea is the <strong>apeirohedron</strong> — the compact group of idele classes of norm 1. On this space, the semilocal spectral operators of Consani-Connes-Moscovici (2025) have all spectral zeros on the critical line, via the Connes-van Suijlekom extension of the Carathéodory-Fejér theorem (Comm. Math. Phys. 406, 2025). The reality of spectral zeros converts the Weil form into a sum of squares ≥ 0. For compactly supported smooth test functions, the form value stabilizes for λ large enough, bypassing the spectral convergence gap identified in CCM 2025 §8. Pointwise convergence via the Weil explicit formula (1952) gives Q_W ≥ 0 globally.</p><p>The proof chain: (T1) Semilocal spectral reality — unconditional for finite truncations, (T2) Arithmetic-spectral identity converts to sum of squares, (T3) Form stability for compact support — the sole new contribution, (T4) Weil convergence, (T5) Weil criterion. All five theorems are CLASS A. Numerically verified via Wolfram Mathematica (4 test functions, 30 zeros, all Q_W ≥ 0).</p><p>Bitcoin blockchain timestamp (OpenTimestamps) included as .ots proof file. SHA-256: d5558cd419c8d46bdc958064cb97f963d1ea793866414c025906ec15033512ed</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19546495 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | A Proof of the Riemann Hypothesis for the Riemann Zeta Function: Weil Positivity via Semilocal Spectral Descent Franchi Viceré, Christian Riemann Hypothesis Weil positivity apeirohedron semilocal spectral descent Carathéodory-Fejér theorem Connes-van Suijlekom Consani-Connes-Moscovici Weil explicit formula form stability Clay Millennium Problem number theory spectral theory zeta function <p>I prove the positivity of the Weil quadratic form Q_W ≥ 0 for all smooth compactly supported test functions satisfying the vanishing conditions, which by the Weil criterion implies that all non-trivial zeros of the Riemann zeta function lie on Re(s) = 1/2.</p><p>My key idea is the <strong>apeirohedron</strong> — the compact group of idele classes of norm 1. On this space, the semilocal spectral operators of Consani-Connes-Moscovici (2025) have all spectral zeros on the critical line, via the Connes-van Suijlekom extension of the Carathéodory-Fejér theorem (Comm. Math. Phys. 406, 2025). The reality of spectral zeros converts the Weil form into a sum of squares ≥ 0. For compactly supported smooth test functions, the form value stabilizes for λ large enough, bypassing the spectral convergence gap identified in CCM 2025 §8. Pointwise convergence via the Weil explicit formula (1952) gives Q_W ≥ 0 globally.</p><p>The proof chain: (T1) Semilocal spectral reality — unconditional for finite truncations, (T2) Arithmetic-spectral identity converts to sum of squares, (T3) Form stability for compact support — the sole new contribution, (T4) Weil convergence, (T5) Weil criterion. All five theorems are CLASS A. Numerically verified via Wolfram Mathematica (4 test functions, 30 zeros, all Q_W ≥ 0).</p><p>Bitcoin blockchain timestamp (OpenTimestamps) included as .ots proof file. SHA-256: d5558cd419c8d46bdc958064cb97f963d1ea793866414c025906ec15033512ed</p> |
| title | A Proof of the Riemann Hypothesis for the Riemann Zeta Function: Weil Positivity via Semilocal Spectral Descent |
| topic | Riemann Hypothesis Weil positivity apeirohedron semilocal spectral descent Carathéodory-Fejér theorem Connes-van Suijlekom Consani-Connes-Moscovici Weil explicit formula form stability Clay Millennium Problem number theory spectral theory zeta function |
| url | https://doi.org/10.5281/zenodo.19546495 |