Reconstructing Numerical Semigroups from Circulant Spectral Dimensions

The polynomial algebra generated by the spectral moments of a family of circulant graphs consists of all eventually polynomial sequences in the graph order; as an unfiltered algebra, it therefore carries little information about the jumps. We show that its natural power-sum weight filtration retains substantially finer arithmetic data. For a finite jump set S with (S)=1 and M=max S, the relevant invariant is the numerical semigroup ΣS= M,M-s,M+s:s∈ S, which arises from the lengths of closed walks winding around the cycle. We prove that the eventual global filtered dimension is MD+ D/2+1-M(M+1)-g(ΣS). Restricting graph orders to n≡ a q, for q M, replaces the genus term by the number of gaps of ΣS in the class -a q; the class $a=0$ gives the genus of the quotient semigroup ΣS/q. These formulas yield an inverse reconstruction theorem: the unrestricted dimensions determine M, and the dimensions in all classes modulo M at one common stable degree determine the Apéry set and hence ΣS. We give effective thresholds, exact examples, and a finite-rank algorithm valid in every degree. Reconstruction of the semigroup need not identify the original jump set.