Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$

For rational-valued signals $f,g:C_{30}\to\mathbb Q$ with exact Fourier support $U_{30}$, equality of autocorrelations through order five is completely classified. After translating $g$, there are $\alpha\in\mathbb Q(\zeta_{30})^\times$ and $z\in\mathbb Q(\zeta_6)^\times$, with $z\bar z=1$, such that $$ \widehat f(u)=\sigma_u(\alpha),\qquad \widehat g(u)=\sigma_u(z\alpha) $$ for every $u\in U_{30}$. Conversely, every such pair, extended by zero off $U_{30}$, is rational-valued and agrees through order five. Normalized parameters are translation-equivalent exactly modulo $\mu_6$, and the sixth-order data agree exactly when $z^6=1$.

Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support $U_{30}$ and conjectured that it was not a complete classification because it did not use the full field $\mathbb Q(\zeta_{30})$. This work proves the complete classification. The larger field enlarges the common amplitude $\alpha$, while every relative ambiguity remains a norm-one parameter in $\mathbb Q(\zeta_6)$. The result is a specialization of a theorem for every exact unit support $U_{6m}$. The entry does not claim a complete parametrization for arbitrary supports: on the 255 automorphism-stable supports treated elsewhere in the paper, the broader result is a closing-degree classification. It does not treat noisy data or noncyclic groups, and it makes no novelty, priority, or firstness claim.