Finite Spectral Reconstruction and Robust Non-Identifiability in Two-Dimensional Narain Compactifications

To what extent can the momentum–winding spectrum of a toroidal conformal field theory determine its background geometry? We study this inverse question for two-dimensional Narain compactifications, distinguishing labelled quadratic-form evaluations, unlabelled total-energy levels, and the joint energy–spin spectrum. Five specified charge-labelled energies reconstruct an arbitrary positive-definite metric and antisymmetric field in a fixed integral basis. For rectangular metrics in a bounded domain, the unlabelled spectrum below an explicit finite cutoff reconstructs both radii and the magnitude of the antisymmetric field. In contrast, we exhibit a rational pair of backgrounds whose complete unlabelled zero-mode energy spectra agree but whose spin-resolved spectra differ, giving an exact obstruction to orientation-preserving integral T-duality. An exhaustive, analytically complete charge bound certifies the witness; rational interval arithmetic extends it to a four-parameter neighbourhood. A local-finiteness argument further shows that a fixed spin-asymmetry witness persists on an open, dense, full-measure set. Finally, no background-independent finite cutoff of even spin-refined spectral data distinguishes all duality classes on unrestricted moduli space. The results identify which spectral information suffices for reconstruction, which information is lost by forgetting charge or spin, and where finite-window inference necessarily fails. They concern exact zero-mode data and do not assert a new worldsheet-parity mechanism or an experimental reconstruction procedure.