Signed Rankin–Selberg Systems for Ramanujan's Third-Order Mock-Theta Multiplier: Meromorphic Continuation, Principal-Point Separation, and Sign-Resolved Mean Squares

This paper develops a signed Rankin–Selberg theory for the positive- and negative-frequency Fourier channels of the unitary weight-$1/2$, three-dimensional multiplier associated with Ramanujan’s third-order mock-theta completion. The work is a standalone arithmetic sequel to: Boundary-Heat Pair Reconstruction for Ramanujan’s Third-Order Mock-Theta Completion: Sesquilinear Spectral Packets and a Mellin Regulator Obstruction, Version v0.2r1, DOI: 10.5281/zenodo.21768928. The earlier paper reconstructs a positive mock-theta coefficient-square observable as a boundary heat energy and a discrete sesquilinear spectral packet. The present paper passes from that positive-frequency boundary theory to signed positive/negative Fourier-pair Dirichlet series. For unitary spectral eigenforms $\widetilde u_j$ with Fourier coefficients $\rho_{j,a}(\nu)$, the signed coefficient-pair series are $$Z_{jk}^{\pm}(z) = \sum_{a=1}^{3} \sum_{\substack{\nu\in\mathbb Z+\alpha_a \\ \pm\nu>0}} \rho_{j,a}(\nu) \overline{\rho_{k,a}(\nu)} \vert{}\nu\vert{}^{-z}.$$ Two exact full-group Eisenstein periods are constructed. The first uses the invariant scalar pairing of the original weight-$1/2$ eigenforms, while the second uses their Maass-lowered weight-$-3/2$ forms. Both completed periods admit meromorphic continuation and satisfy the functional equation induced by the full modular Eisenstein series. Their canonical diagonal residues are $$\operatorname{Res}_{z=0} C_{jk}^{(0)}(z) = \frac{3}{\pi}\delta_{jk}, \qquad \operatorname{Res}_{z=0} C_{jk}^{(L)}(z) = \frac{3}{\pi}\mu_j\delta_{jk},$$ where $$\mu_j = t_j^2 + \frac{1}{16}.$$ The sign-sensitive action of the Maass lowering operator provides a second independent equation for the positive and negative Fourier channels. At the diagonal principal point, the associated archimedean determinant has the explicit positive-energy representation $$\Delta_{jj}(0) = \frac{1}{2} \left( G_{jj}^{+}(0)H_j^{-} + G_{jj}^{-}(0)H_j^{+} \right) > 0.$$ This unconditional positivity separates the two diagonal sign channels and determines their individual principal residues $$r_j^{\pm} = \operatorname{Res}_{z=0} Z_{jj}^{\pm}(z).$$ The general vector-valued real-weight Rankin–Selberg theory supplies an independent meromorphic continuation and signed mean-square input for the diagonal series. The two-period inversion gives a separate full-group derivation and, crucially, provides an explicit allocation of the canonical residue between the positive and negative channels in the concrete third-order multiplier normalization. With $$b_j^{\pm} = \operatorname{Res}_{s=1} \left( \zeta(2s)R_{\pm,j}(s) \right) = \zeta(2)r_j^{\pm},$$ the resulting sign-resolved logarithmic mean-square asymptotics are $$\sum_{a=1}^{3} \sum_{0<\pm\nu\le X} \vert{}\rho_{j,a}(\nu)\vert{}^2 = \zeta(2)r_j^{\pm}\log X + o(\log X).$$ In the lowest-weight case ($\mu_j=0$), the negative-frequency channel vanishes identically, so that $$Z_{jj}^{-}\equiv 0, \qquad r_j^{-}=0.$$ For off-diagonal pairs ($j\ne k$), individual sign separation is obtained conditionally when the corresponding archimedean determinant is not identically zero. A complete off-diagonal theory would require either explicit determinant control or the calculation of the full cusp–eigenline–sign scattering system. The paper does not claim an Euler product, an unconditional off-diagonal separation theorem for all spectral pairs, conductor-uniform cancellation estimates, or any implication for the Riemann hypothesis. Its principal contribution is a concrete sign-separation mechanism in the third-order mock-theta multiplier: two exact Eisenstein periods, a lowering-generated second channel equation, an unconditionally positive diagonal determinant, explicit sign-resolved residues, and correctly normalized signed mean-square asymptotics. This Zenodo record contains the manuscript PDF only.