A Degree-24 Obstruction to Hopf Structures on Symmetric-Group Supercharacter Spaces

Consider the supercharacter theory of S_n whose superclasses are the S_n-conjugacy classes contained in A_n, together with the single block S_n-A_n. We prove that its superclass-function spaces, summed over n with one-dimensional components in degrees zero and one, admit no connected graded Hopf algebra structure over a field of characteristic zero. The argument uses a known nonnegative Euler-product condition for Hopf Hilbert series. The first negative Euler exponent occurs in degree 24 and equals -1: lower degrees force 795 basis monomials, but the required dimension is 794. The two coarser natural supercharacter families fail the same test in degree 4. These obstructions exclude arbitrary graded Hopf operations, not only quotients of symmetric functions. We state the families explicitly and do not claim a classification of all symmetric-group supercharacter theories. Unrefereed preprint. The theorem concerns three explicit families and the ordinary degree-n grading in characteristic zero. No classification of all supercharacter theories, independent peer review, formal verification or absolute priority certificate is claimed. AI-assisted tools supported research, exact computation and writing; the author is responsible for the final text. Corpus identifier: AIM-REPRESENTATION_THEORY-0102.