Abstract
We develop a spectral-zeta framework for quantum mechanics with the PT-symmetric potential V_ PT(x)=x^2K(ix)^ε (K,ε ∈ N) and the Hermitian potential V_ H(x)=x^2M (M ∈ N+1), based on the fusion relations of the A_2M-1 T-system. Using the ODE/IM correspondence, we construct exact sum rules (ESRs) and zeta generating formulas (ZGFs) for the spectral zeta functions (SZFs) ζ_n(s). In contrast to recursive T-Q relations, the ZGFs provide fixed-source, closed-form mappings between different fusion sectors. For Hermitian M=2, our ESRs reproduce exact WKB results, extending them systematically to PT sectors and (half-)integer M. Our analysis reveals a phenomenon of \textit{algebraic information loss}, distinct from analytic ambiguity. The structure is governed by a selection rule S_n, derived from the Chebyshev structure of fusion relations and Z_2M+2 Symanzik symmetry. For odd integer M, we identify a structural non-invertibility: mapping from \textit{odd} to \textit{even} fusion sectors causes exact coefficient cancellation due to phase interference, rendering the map non-invertible. This implies even-sector data carry strictly less information than odd-sector data, yielding a \textit{no-go} statement for inverse spectral reconstruction. Conversely, for even and half-integer M, all relevant sectors form an information-equivalent, mutually invertible family. Finally, we provide a spectral-zeta formulation of the massless Ai-Bender-Sarkar (ABS) conjecture. By connecting PT and Hermitian spectra via ZGFs, we establish a purely spectral-theoretic route to the conjectured relation, avoiding explicit analytic continuation.