Ramanujan's Page-336 Formulas: Period Functions, Reciprocal Discrepancies, and Mellin–Laplace Descent

Two formulas on page 336 of Ramanujan’s lost notebook are invalid as ordinary scalar identities. One relates sine series at reciprocal arguments; the other juxtaposes a divergent series of complex powers with exponentially damped oscillations. Both involve divisor sums and a tangent factor. We give a common analytic reconstruction of these particular expressions, rather than corrected scalar equalities. Starting from the divisor Dirichlet series, we use a normalized Eisenstein–Lambert period function and its distributional boundary to distinguish the reciprocal sum supplied by the period from the reciprocal difference appearing in Ramanujan’s sine formula. An explicit identity reconstructs that historical difference, including its contact terms; Mellin contour displacement retains the crossed residues, and the complete discrepancy is shown to be nonzero. A jointly continued weighted Laplace family then transports this discrepancy to the meromorphically interpreted discrepancy of the companion formula, recovering its complex powers, homogeneous terms, and oscillatory phase for (0<\Re s<1), (s\neq 1/2). The tangent factor is obtained from an oriented local derivative of the signed Mellin multiplier.