Exact Certificates and Falsification Controls for Narain Spectra, Graph Codes, and Integral Lattice Splitting

The information retained by a toroidal charge spectrum, a graph-code enumerator, or a higher-genus Gaussian expression depends on both the observable and the equivalence relation. This paper reports exact computations and falsification controls while keeping classical results and the scope of each conclusion explicit. In a rational two-torus family, the first spin-labelled zero-mode discrepancy for b = 1/5 occurs at energy 3313/1325; an exact parity map relates the b and −b backgrounds and prevents this observation from establishing inequivalence of unoriented theories. Exhaustive classification of simple graphs through six vertices finds 26 local-complementation/permutation classes, 64 coarse signatures, and no cross-class overlap among 32,768 labelled six-vertex graphs, although the signatures themselves are not invariant under local complementation. A previously published seven-vertex pair has 756 differing coefficients in a ten-variable genus-two refinement, and the implemented nonseparating Gaussian sum admits a strictly signed interval enclosure; physical theta normalization remains an independent requirement. We give explicit sufficient non-splitting certificates for rational and parameterized Narain lattice families, a standard real-analytic genericity consequence, and a reduction of simultaneous two-form decomposition to classical positive-form components. These structural arguments use established Kneser-type methods and are not asserted to establish new lattice-decomposition algorithms. Additional controls delimit the interpretation of code-CFT examples, symmetry-marked traces, and internal theta-computation timings. The contribution is a bounded, checkable evidence synthesis rather than a new physical inequivalence theorem or a demonstrated advance over existing theta algorithms.