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

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 σ > 0, the paper introduces the coefficient-square heat kernel Φ3,σ(t) = ∑ν>0 e-2σν A₃(ν)² e-tν². Its first main result identifies this kernel exactly with the heat energy of a positive-frequency boundary vector on the period-$24$ horocycle: Φ3,σ(t) = e-tD₊²/2 B3,σ _L²( T₂₄)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 cⱼ(Y)cₖ(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 Re(s) ≥ 0, positive self-adjoint for real s ≥ 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 ∫₀^∞ Aⱼₖ(σ)(s) \, σʷ⁻¹ \, dσ = 2¹⁻ʷ√π \, Gⱼₖ(w-12) Zⱼₖ(2s+w-12) in an explicit common region of absolute convergence. Here Gⱼₖ is an archimedean Whittaker-product Mellin transform, while Zⱼₖ 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 σ=σ(T)>0 can make the corresponding Mellin transforms converge uniformly to ξ(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 Zⱼₖ, 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.