The Riemann Hypothesis remains one of the deepest open problems in mathematics because it connects the hidden geometry of complex zeros with the distribution of prime numbers. This paper develops a structured analytic framework that transfers the problem from its usual complex form into a real-variable setting where the desired zero structure can be studied through positivity. The central idea is to work with the completed zeta function after normalization and transformation, and to identify a precise positivity principle that would force the corresponding zeros into the correct location. The framework reduces the problem to a compact moment condition for the central logarithmic coefficients of the normalized completed function. This condition is then connected with complete Bernstein functions, Stieltjes functions, Hankel positivity, finite-difference inequalities, and an equivalent logarithmic integral representation. Each step is formulated as a rigorous implication, so the remaining difficulty is isolated in one concrete positivity theorem rather than hidden inside formal manipulation. The approach is designed to avoid the common weaknesses of many proposedarguments for the Riemann Hypothesis. It does not use the Euler product outside its valid region, does not infer the result from symmetry alone, does not replace the original zeta function with a modified object, and does not rely on numerical evidence as proof. Instead, it gives a connected chain from central derivative positivity to a Stieltjes logarithmic derivative, from there to a negative-real-axis zero structure, and finally back to the critical-line statement. The paper does not claim a completed proof of the Riemann Hypothesis. Its contribution is a clean reduction that identifies a single remaining positivity problem in a form suitable for rigorous verification. This gives a clear and testable route for future work, with finite conditions that can be studied through moment theory, operator theory, and the analytic theory of special functions.