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 uⱼ with Fourier coefficients ρj,a(ν), the signed coefficient-pair series are Zⱼₖ±(z) = ∑ₐ₌₁³ ∑_ν∈ Z+αₐ \\ ±ν>0 ρj,a(ν) ρk,a(ν) ν⁻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 Resz=0 Cⱼₖ⁽⁰⁾(z) = 3/πδⱼₖ, Resz=0 Cⱼₖ(L)(z) = 3/πμⱼδⱼₖ, where μⱼ = tⱼ² + 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 Δⱼⱼ(0) = 1/2 ( Gⱼⱼ⁺(0)Hⱼ⁻ + Gⱼⱼ⁻(0)Hⱼ⁺ ) > 0. This unconditional positivity separates the two diagonal sign channels and determines their individual principal residues rⱼ± = Resz=0 Zⱼⱼ±(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 bⱼ± = Resₛ₌₁ ( ζ(2s)R±,j(s) ) = ζ(2)rⱼ±, the resulting sign-resolved logarithmic mean-square asymptotics are ∑ₐ₌₁³ ∑0<±ν≤ X ρj,a(ν)² = ζ(2)rⱼ±log X + o(log X). In the lowest-weight case (μⱼ=0), the negative-frequency channel vanishes identically, so that Zⱼⱼ⁻≡ 0, rⱼ⁻=0. For off-diagonal pairs (j≠ 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.