Conditioned Ramanujan reciprocity for odd zeta values

In this paper, we develop a certified analytic-computational form of the transformation. Mellin inversion and a complete residue calculation first recover Ramanujan’s identity. Solving that identity for the odd zeta value then reveals the exact extraction denominator. Under an exponential parameterization of the reciprocal pair, the denominator is nonsingular for odd index and vanishes at the symmetric point for even index. A geometric estimate for each omitted Lambert tail gives an explicit, computable error certificate. For even index, asymptotic minimization produces a universal leading parameter rule governed by a single transcendental constant. Five terms in each Lambert series certify at least 18 decimal places for the third and fifth zeta values and 21 decimal places for the seventh and ninth. The method supplies a reproducible evaluation framework and explains the parity geometry hidden in Ramanujan’s reciprocity law.