Boundary-Heat Pair Reconstruction for Ramanujan's Third-Order Mock-Theta Completion: Sesquilinear Spectral Packets and a Mellin Regulator Obstruction

Abstract

This paper develops a boundary-heat and spectral reconstruction theory for a positive quadratic observable derived from the positive-frequency Fourier coefficients of Zwegers’ vector-valued completion of Ramanujan’s third-order mock theta functions. For a damping parameter $\sigma > 0$, the paper introduces the coefficient-square heat kernel $$\Phi_{3,\sigma}(t) = \sum_{\nu>0} e^{-2\sigma\nu} \lVert A_3(\nu)\rVert^2 e^{-t\nu^2}.$$ Its first main result identifies this kernel exactly with the heat energy of a positive-frequency boundary vector on the period-$24$ horocycle: $$\Phi_{3,\sigma}(t) = \left\Vert{} e^{-tD_+^2/2} B_{3,\sigma} \right\Vert{}_{L^2(\mathbb T_{24})}^2.$$ Because this observable is Hermitian quadratic rather than complex linear, a linear spectral reconstruction is structurally impossible. The appropriate reconstruction is instead a sesquilinear pair expansion with coefficients $c_j(Y)\overline{c_k(Y)}$. The translation monodromy of the third-order completion has no invariant vector. This removes the zero-frequency cusp channel and yields a compact-resolvent spectral setting. After an invariant cusp truncation, the boundary vector therefore admits a purely discrete spectral expansion. The resulting finite sesquilinear pair packets converge to the full boundary-heat kernel, with an explicit Sobolev-controlled truncation estimate. The paper also constructs a basis-independent boundary pair operator. It is compact for $\operatorname{Re}(s) \ge 0$, positive self-adjoint for real $s \ge 0$, and trace class under a conservative Sobolev threshold. After explicitly conjugating the holomorphic Petersson model to the unitary Maass–Whittaker model, its boundary matrix coefficients satisfy the normalized Whittaker–Mellin factorization $$\int_0^\infty \mathcal A_{jk}^{(\sigma)}(s) \, \sigma^{w-1} \, d\sigma = 2^{1-w}\sqrt{\pi} \, \mathcal G_{jk}\left(w-\frac12\right) \mathcal Z_{jk}\left(2s+w-\frac12\right)$$ in an explicit common region of absolute convergence. Here $\mathcal G_{jk}$ is an archimedean Whittaker-product Mellin transform, while $\mathcal Z_{jk}$ is a same-frequency metaplectic coefficient-pair Dirichlet series. Finally, the paper proves a structural obstruction to directly approximating the completed Riemann zeta function with this positive coefficient-square Mellin family. No damping schedule $\sigma=\sigma(T)>0$ can make the corresponding Mellin transforms converge uniformly to $\xi(s)$ on expanding critical-line windows. The obstruction follows from the forced limiting behavior of the damping parameter and the incompatibility between the pole contributed by the gamma factor at $s=0$ and the entire completed zeta function. The paper does not claim an Euler product, meromorphic continuation, functional equation, or off-diagonal cancellation theorem for the coefficient-pair series $\mathcal Z_{jk}$, and it does not claim a proof of the Riemann hypothesis. Its contribution is a concrete and quantitatively controlled passage from a positive mock-theta coefficient-square observable to a boundary heat energy, a discrete sesquilinear spectral reconstruction, and a correctly normalized Whittaker–Mellin pair structure, together with a rigorous obstruction to a direct completed-zeta regulator. This Zenodo record contains the manuscript PDF only.