THE SYMMETRY AROUND (1/2) AND THE EQUILIBRIUM OF THE STRUCTURE OF THE ZEROS OF THE ZETA FUNCTION

Description for Zenodo publication (English) TitleThe Symmetry Around (1/2) and the Equilibrium of the Structure of the Zeros of the Zeta Function AuthorCláudio Vicente da Silva Abstract / Description This article develops a unified mathematical construction that brings together arithmetic, geometric and analytic perspectives around the central point 1/2, establishing a structural analogy with the symmetry of the functional equation of the Riemann zeta function. A normalization procedure is first introduced for arithmetic pairs and for right-triangular geometric figures. The elimination of absolute scale is shown to preserve a universal equilibrium point at 1/2. The resulting complementarity relation x+x∗=1x + x^* = 1x+x∗=1 is then applied to the real parts β\\betaβ of the non-trivial zeros of the zeta function. The displacement d=β−1/2d = \\beta - 1/2d=β−1/2 and the symmetric position β∗=1−β\\beta^* = 1 - \\betaβ∗=1−β are defined, and the same equilibrium structure is recovered. The construction is extended to the analytic domain by means of the Riemann Xi function Ξ(t)=ξ(1/2+it)\\Xi(t) = \\xi(1/2 + it)Ξ(t)=ξ(1/2+it). A hypothetical zero lying off the critical line (d≠0d \\neq 0d=0) generates a non-real quartet ±γ±id\\pm\\gamma \\pm id±γ±id. The first Laguerre operator L1[f]=(f′)2−ff′′L_1[f] = (f')^2 - ff''L1[f]=(f′)2−ff′′ is applied to the real polynomial Q(t)Q(t)Q(t) associated with this isolated quartet, and it is proved rigorously that L1[Q](γ)<0L_1[Q](\\gamma) < 0L1[Q](γ)<0. The global function is decomposed as Ξ(t)=Q(t)G(t)\\Xi(t) = Q(t)G(t)Ξ(t)=Q(t)G(t). The corresponding identity for the Laguerre operator is established, and a global coupling condition is formulated. Global positivity of the Laguerre operators is shown to be necessary and sufficient for the reality of all zeros of Ξ(t)\\Xi(t)Ξ(t). Consequently, the Riemann Hypothesis is reduced to the statement that the global factor G(t)G(t)G(t) cannot compensate the local negative signature produced by any such quartet. Further sections examine the geometric interpretation of the critical line as a fibre bundle whose fibres are the individual normalized structures associated with each zero, the macroscopic sculpture generated by the Hadamard product of the zeros, and the formal equivalence between the global rigidity condition and the Riemann Hypothesis itself. The work maintains a clear distinction between local geometric constructions, analytic identities, and global structural statements, providing a coherent framework that links normalization, complementarity, displacement, Laguerre operators and the reality of the zeros of the Xi function. KeywordsRiemann zeta function; Riemann Hypothesis; Riemann Xi function; symmetry and normalization; Laguerre operators; Laguerre–Pólya class; functional equation; quartet of zeros; Hadamard decomposition; fibre structure of the critical line. LanguageEnglish (translated from the original Portuguese manuscript) Date19 September 2026Londrina, Paraná, Brazil