The First Spin-Resolved Spectral Discrepancy in a Narain Two-Torus Family: Exact Thresholds, Symmetry Boundaries, and Stability

Two toroidal backgrounds can have identical unlabelled energy spectra while differing in the distribution of states over spin. Identifying the first energy at which this distinction becomes visible is a finer problem than establishing spectral inequivalence. We study a rational two-dimensional Narain metric with opposite antisymmetric backgrounds and obtain an exact, parameter-dependent first-separation threshold. For a real field parameter b, writing t = dist(b, ℤ), the first energy at which the joint energy–spin zero-mode multiplicities differ is (133 − 6t + 18t²)/53 when 0 < t < 1/2; at integer and half-integer parameters the complete refined spectra coincide. The proof combines an analytic finite-charge completeness bound, an exact quadratic-polynomial certificate for nonzero-spin states, and the standard integral B-shift symmetry. A separate exact calculation gives a uniform competing-state gap of 6/265 on 1/10 ≤ b ≤ 2/5. This gap, together with an exact identity for paired witness energies at a general positive-definite metric, yields a quantitative persistence theorem under small symmetric perturbations. We further identify the first nonvanishing Fourier–Laplace coefficient of the difference of zero-mode spin-refined theta sums, making the spectral distinction explicit at the partition-function level. The first detectable refined spectral difference varies continuously within open chambers yet disappears at their symmetry boundaries. The construction is specific to the displayed family and does not assert a new duality or a generic inverse-spectral classification.