Intermediate Growth and Rigidity in Primitive Circulant Spectral Algebras

At prime graph orders, the complete nontrivial and primitive Fourier spectra of a circulant coincide. We show that this agreement conceals a strong algebraic separation when the graph order varies. For any nonempty finite jump set S, let ASᵖʳⁱᵐ be the algebra of arithmetic functions obtained by evaluating symmetric power sums on the primitive spectrum, filtered by power-sum weight. Removing the common Euler-totient term gives a reduced, locally finite algebra BS, and we prove a filtration-preserving isomorphism ASᵖʳⁱᵐ BS[T₂]. Thus ASᵖʳⁱᵐ is non-Noetherian and has Krull dimension one. Its filtered dimensions satisfy exp(cS√D/log D)≤ ρSᵖʳⁱᵐ(D) ≤exp(CS√ D) for large D, and its associated-graded Hilbert series is nonrational. Despite agreement on every prime order, the primitive and full spectral evaluation algebras intersect only in Q. More strongly, no nonconstant primitive spectral function is eventually quasirational on residue classes. We give an exact finite-matrix rank formula and determine the quotient obtained by restricting to integers with large prime factors.