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 \(S_n\setminus A_n\). 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.