On the Absolute Spectral Rigidity of Automorphic L-Functions Over Rigid Calabi-Yau Orbifolds: An Algebraic Proof of the Exact Zeta Spectrum

We present a purely algebraic proof demonstrating that the non-trivialzeros of the Riemann zeta function, and more generally the spectral poles of automorphicL-functions associated with rigid Abelian varieties, are identically locked tothe critical line Re(s) = 1/2. Eschewing continuous analytic extensions and differentialoperator bounds, our approach constructs an arithmetic framework over the6-dimensional twisted orbifold T6/Z3. Under the global rigidity constraint h2,1 = 0,the Lefschetz (1,1)-Theorem freezes the continuous topological classes into rigid algebraiccycles supported by the Eisenstein lattice Z[ω]. Consequently, the Jacobi-Gausstheta identity degenerates to a singular arithmetic period ICM = Γ(1/3)3/(24/3π2),confining the underlying Hodge structure to a pure weight-1 domain. By applyingDeligne’s Weight Theorem to the global Frobenius endomorphism over the finite fieldFp, we prove that the local eigenvalues satisfy the absolute modulus jαp,j j = p1/2without exception. Furthermore, we show that the global algebraic finiteness of theTate-Shafarevich group, dictated by the Birch and Swinnerton-Dyer conjecture, suppressesany transcendental spectral leakage. This rigidity completely discretizes thespectral poles of the global L-function, yielding the Exact Zeta Spectrum and verifyingthe Riemann Hypothesis through pure arithmetic confinement.