Exact Sum Rules and Zeta Generating Formulas from the ODE/IM correspondence

We develop a spectral-zeta framework for quantum mechanics with the $\mathcal{P}\mathcal{T}$-symmetric potential $V_{\mathcal{P}\mathcal{T}}(x)=x^{2K}(ix{)}^{\epsilon }$ $(K,\epsilon \in \mathbb{N})$ and the Hermitian potential $V_{\mathcal{H}}(x)=x^{2M}$ $(M\in \mathbb{N}+1)$, 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) ${\zeta }_n(s)$. 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 $\mathcal{P}\mathcal{T}$ sectors and (half-)integer $M$. Our analysis reveals a phenomenon of $\mathit{\text{algebraic}}\mathit{\text{information}}\mathit{\text{loss}}$, distinct from analytic ambiguity. The structure is governed by a selection rule ${\mathcal{S}}_n$, derived from the Chebyshev structure of fusion relations and ${\mathbb{Z}}_{2M+2}$ Symanzik symmetry. For odd integer $M$, we identify a structural non-invertibility: mapping from $\mathit{\text{odd}}$ to $\mathit{\text{even}}$ 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 $\mathit{\text{no-go}}$ 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 $\mathcal{P}\mathcal{T}$ and Hermitian spectra via ZGFs, we establish a purely spectral-theoretic route to the conjectured relation, avoiding explicit analytic continuation.