For a finite cyclic group and an exact Fourier support $S$, we study the
lattice $\calL_{\le d}(S)$ generated by zero-sum relations of length at most
$d$ inside the full relation lattice $K(S)$.  Its closing degree is both a
local separating degree and the order at which every invisible phase ratio
becomes a translation character.  For the exact unit support of every even
cyclic group, we compute
\[
 \Z^{\units n}/\calL_{\le4}(\units n)\cong
 \begin{cases}
   \Z/n\Z,&3\nmid n,\\
   \Z\oplus\Z/(n/6)\Z,&6\mid n.
 \end{cases}
\]
Moreover $\calL_{\le5}(\units n)=\calL_{\le4}(\units n)$.  Thus fourth- and
fifth-order rational ambiguity coincide: fourth order reconstructs up to
translation when $3\nmid n$, while for $n=6m$ the complete surviving family
has normalized ratios parameterized by norm-one elements of $\Q(\zeta_6)$,
modulo the residual translation subgroup $\mu_6$; order six closes it.
For $n=2pq$ with distinct odd primes, we classify every nonempty
$\Aut(C_n)$-stable support.  The distributions of these 255 supports by
closing degree are
\[
 (1,2,154,80,4,14)\quad(p=3),\qquad
 (1,2,154,92,4,2)\quad(p,q\ge5).
\]
The proof uses seven local CRT presentations, cubic and quartic Skeletons,
and explicit depth-five and depth-six obstructions.  Galois-equivariant
lower-bound ratios show that the lattice, rational-signal and integer-signal
closing degrees coincide on every stable support.  Finally, for binary
indicators, a supportwise crosswalk and a uniform exclusion of all deep
supports give $r_{\rm set}(C_{2pq})=4$ under translations.  The value is also
four for the same oriented analytical deck when outputs are identified up to
translation or reflected translation.