This work develops a theta-geometric framework for studying the Hankel positivity problem associated with the Riemann Hypothesis.Starting from the centered Riemann xi-function, the framework exploits the even spectral symmetry to introduce a folded spectral coordinate and a meromorphic resolvent whose poles encode the squared centered zeros. Passing to reciprocal spectral variables leads to a canonical moment sequence whose Stieltjes moment property is characterized by the positivity of two associated families of Hankel matrices.The framework then connects this spectral-moment structure with the theta-function representation of the xi-function. A tilted theta measure is introduced, yielding an exact expectation representation for the folded resolvent and a differential hierarchy governing its derivatives. The resulting analysis leads to an adjoint differential operator and a two-variable kernel representation, together with an exact inhomogeneous differential identity in which boundary contributions are retained explicitly.The central outcome is an exact reduction of the spectral question to an all-orders Hankel positivity problem. The work distinguishes rigorously established identities and equivalences from numerical diagnostics and does not claim a proof of the Riemann Hypothesis. Instead, it provides a structured theta-geometric and moment-theoretic framework for investigating the remaining positivity problem.