Eid’s Covenant: An Exploratory Study on a Structural Approach to the Riemann Hypothesis

Eid’s Covenant: An Exploratory Study on a Structural Approach to the Riemann Hypothesis | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 9 April 2026 V1 Latest version Share on Eid’s Covenant: An Exploratory Study on a Structural Approach to the Riemann Hypothesis Author : SALEM EID 0009-0001-1788-9083 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.177575358.87971935/v1 124 views 29 downloads Contents Abstract Introduction Fundamental Definitions and Adelic Spaces The Adelic Hilbert Space The Hermitian Spectral Operator Spectral Linking and the Eigenstate-Zero Correspondence Cohomological Constraints on Zero Distribution Conclusion References Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This paper presents a novel structural framework exploring a potential proof of the Riemann Hypothesis by constructing a spectral representation over the adelic class space \(C_{\mathbb{A}}^{1}\). We define a self-adjoint Hamiltonian operator \(\mathcal{H}\) on the pure adelic Hilbert space \(L_{0}^{2}(C_{\mathbb{A}}^{1})\), aiming to establish a correspondence between the non-trivial zeros of the Riemann Zeta function and the real eigenvalues of \(\mathcal{H}\). We further investigate cohomological properties of the adelic configuration that may impose structural constraints on the distribution of zeros. This work is presented as an exploratory study proposing a novel approach and does not constitute a completed proof of the Riemann Hypothesis. Introduction The Riemann Hypothesis (RH) asserts that all non-trivial zeros of the Riemann Zeta function \(\zeta(s)\) lie on the critical line \(Re(s)=1/2\). Despite extensive numerical evidence, a structural explanation for this distribution remains elusive. In this exploratory study, we introduce Eid’s Covenant , a framework integrating adelic analysis, spectral theory, and algebraic geometry, aiming to relate the zeros of \(\zeta(s)\) to eigenvalues of a Hermitian operator defined on the adelic class space. We investigate how cohomological properties of this space could impose structural constraints that align zeros with the critical line. We emphasize that the present work is a proposal and conceptual exploration. The results should be interpreted as a suggested approach rather than a definitive proof of the Riemann Hypothesis. Our goal is to provide a rigorous mathematical framework that may inspire further detailed analysis and validation by the research community. Fundamental Definitions and Adelic Spaces The Adèle Ring and Idèle Group We define the Adèle ring \(\mathbb{A}\) over \(\mathbb{Q}\) as the restricted product: \begin{equation} \mathbb{A}=\mathbb{R}\times\prod^{\prime}_{p}\mathbb{Q}_{p}.\par\\ \end{equation} The Idèle group \(I_{\mathbb{A}}\) is the group of units \(\mathbb{A}^{\times}\). The adelic class space is given by \begin{equation} C_{\mathbb{A}}=I_{\mathbb{A}}/\mathbb{Q}^{\times}.\par\\ \end{equation} The Compact Adelic Class Space The fundamental domain for our analysis is the compact subgroup: \begin{equation} C_{\mathbb{A}}^{1}=\{x\in C_{\mathbb{A}}:|x|_{\mathbb{A}}=1\}.\par\\ \end{equation} This space is compact and carries a normalized Haar measure, allowing for a discrete spectral decomposition. The Adelic Hilbert Space We consider the Hilbert space \(L^{2}(C_{\mathbb{A}}^{1})\). To eliminate the pole of the Zeta function at \(s=1\), we define the Pure Subspace : \begin{equation} L_{0}^{2}(C_{\mathbb{A}}^{1})=\left\{f\in L^{2}(C_{\mathbb{A}}^{1}):\int_{C_{\mathbb{A}}^{1}}f(x)dx=0\right\}.\par\\ \end{equation} This space serves as the domain for our spectral operator. The Hermitian Spectral Operator Definition of the Operator We define the infinitesimal dilatation operator \(\mathcal{H}\) acting on \(L_{0}^{2}(C_{\mathbb{A}}^{1})\) as: \begin{equation} \mathcal{H}=i\left(x\frac{d}{dx}+\frac{1}{2}\right).\par\\ \end{equation} Hermiticity and Eigenvalues The operator \(\mathcal{H}\) is self-adjoint (Hermitian) on the dense domain of Schwartz-Bruhat functions \(\mathcal{S}(\mathbb{A})\subset L_{0}^{2}(C_{\mathbb{A}}^{1})\). Consequently, all eigenvalues \(E\in\text{Spec}(\mathcal{H})\) are strictly real (\(E\in\mathbb{R}\)). Spectral Linking and the Eigenstate-Zero Correspondence [Eigenstate-Zero Correspondence, Exploratory] Every non-trivial zero \(\rho\) of the Riemann Zeta function \(\zeta(s)\) may correspond to an eigenvalue \(E\) of the operator \(\mathcal{H}\) in the form: \begin{equation} \rho\approx\frac{1}{2}+iE.\par\\ \end{equation} Remark: At this stage, this correspondence is a proposed framework. A full rigorous bijection between zeros and eigenvalues requires further validation. Cohomological Constraints on Zero Distribution Vanishing of the Picard Group We consider the cohomological property of the adelic class space: \begin{equation} H^{1}(C_{\mathbb{A}}^{1},\mathcal{O}^{*})\cong 0.\par\\ \end{equation} Implications for Zeros The vanishing of \(H^{1}\) suggests structural rigidity of the space, which may restrict deformations corresponding to off-axis zeros. We present this as a conceptual observation rather than a formal prohibition: This rigidity indicates a potential mechanism favoring zeros on the critical line, but a complete proof of necessity remains to be established. Conclusion We have introduced Eid’s Covenant as an exploratory framework linking adelic spectral theory and cohomological rigidity to the distribution of non-trivial zeros of the Riemann Zeta function. While the proposed correspondence and constraints are mathematically suggestive, they should be interpreted as **conceptual proposals for further investigation**, rather than definitive proofs. Future work is needed to rigorously validate each step. References 1. Tate, J. T. (1950). Fourier Analysis in Number Fields and Hecke’s Zeta-Functions . 2. Connes, A. (1999). Trace formula in noncommutative geometry and the Riemann hypothesis . 3. Riemann, B. (1859). On the Number of Prime Numbers Less Than a Given Quantity . Information & Authors Information Version history V1 Version 1 09 April 2026 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords adelic analysis adelic hilbert space algebraic geometry riemann hypothesis salem eid spectral theory zeta function Authors Affiliations SALEM EID 0009-0001-1788-9083 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 124 views 29 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation SALEM EID. Eid’s Covenant: An Exploratory Study on a Structural Approach to the Riemann Hypothesis. Authorea . 09 April 2026. DOI: https://doi.org/10.22541/au.177575358.87971935/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. 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