\section{Zero-sum lattices and local separating degree}\label{sec:zero-sum} Let $G$ be a finite abelian group and let $\varnothing\ne S\subset G$. Write \[ \calB(S)=\left\{a\in\N_0^S:\sum_{s\in S}a_s s=0\right\} \] for the zero-sum sequence monoid. Its atoms are denoted by $\calA(S)$. For $d\ge1$, let $\calB(S)_{\le d}$ be the submonoid generated by atoms of length at most $d$. The integral relation lattice is \[ K(S)=\ker\left(\Z^S\longrightarrow\angles S\right),\qquad (a_s)\longmapsto\sum_s a_s s, \] and $\calL_{\le d}(S)$ is generated by all nonnegative zero-sum vectors of length at most $d$. Using all rows of length at most $d$ or only atoms of length at most $d$ is equivalent at the integral level. A reducible zero-sum sequence factors as a sum of shorter zero-sum sequences and adds no new generator beyond its atoms. We retain the row language in coordinate proofs because one autocorrelation monomial naturally gives one row; the atom language is the bridge to separating invariants. For reference, the objects used throughout are separated in Table~\ref{tab:objects}. In particular, $A_d(S)$ contains the finite target coordinates as well as the obstruction; it should not be called the ambiguity group. \begin{table}[ht] \centering \small \begin{tabularx}{\textwidth}{@{}p{3.2cm}YY@{}} \toprule Object & Definition & Role\\ \midrule $\calB(S)$ & Nonnegative zero-sum sequences & Monoid of Fourier monomial indices\\ $K(S)$ & Full integral kernel in $\Z^S$ & All additive relations on the support\\ $\calL_{\le d}(S)$ & Span of zero-sum rows through length $d$ & Relations visible to bounded autocorrelations\\ $A_d(S)$ & $\Z^S/\calL_{\le d}(S)$ & Universal ambient presentation\\ $K(S)/\calL_{\le d}(S)$ & Kernel of $A_d(S)\to\angles S$ & Integral obstruction lattice\\ $\Hom(K/\calL_{\le d},\C^\times)$ & Character dual of the obstruction & Complex phase ambiguity modulo translations\\ $\depth(S)$ & Least $d$ with $\calL_{\le d}=K$ & Local closing/separating degree\\ \bottomrule \end{tabularx} \caption{Notation and logical roles.}\label{tab:objects} \end{table} For a cancellative monoid $M$, write $\mathsf q(M)$ for its quotient group. The next proposition is the precise bridge between the lattice used here and the separating-atom filtration of Schefler--Zhao--Zhong \cite{ScheflerZhaoZhong2026}. \begin{proposition}[Group-completion dictionary]\label{prop:dictionary} For every $S$ and $d$, \[ \calL_{\le d}(S)=\mathsf q(\calB(S)_{\le d}), \qquad K(S)=\mathsf q(\calB(S)), \tag{2.1} \] where both quotient groups are viewed in $\Z^S$. Consequently \[ \depth(S)=\max\{|A|:A\in\calA_{\mathrm{sep}}(S)\}, \tag{2.2} \] the maximum length of a separating atom over $S$. \end{proposition} \begin{proof} Every nonnegative zero-sum vector of length at most $d$ factors into atoms, each of length at most $d$. Conversely, each atom of length at most $d$ is one of the rows generating $\calL_{\le d}(S)$. Taking integral differences gives the first identity in (2.1). The inclusion $\mathsf q(\calB(S))\subset K(S)$ is immediate. For the reverse inclusion, let $x=x^+-x^-\in K(S)$. The two nonnegative parts have the same group sum, say $h\in\angles S$. Since the finite group $\angles S$ is generated by $S$, there is a nonnegative vector $c\in\N_0^S$ with group sum $-h$: begin with an integral expression and add suitable multiples of element orders to make every coefficient nonnegative. Then $x^++c$ and $x^-+c$ lie in $\calB(S)$ and their difference is $x$. By Definition 2.5 of \cite{ScheflerZhaoZhong2026}, a separating atom is an atom not contained in the quotient group generated by shorter atoms. The same source proves that separating atoms generate the full quotient group. Thus the least truncation degree generating $K(S)$ is their maximum length, which is (2.2). \end{proof} The proposition does not require $S=-S$. This is useful because it separates the algebraic dictionary from Fourier rationality. When $S$ is the exact support of a rational-valued signal, Galois stability later implies $S=-S$, but no symmetry hypothesis is needed in Proposition~\ref{prop:dictionary}. Domokos's lattice criterion for diagonalizable actions states the same generation principle at the level of coordinate subrepresentations \cite[Theorem~1.3]{Domokos2023}. Its non-closed-field extension requires the criterion on every Galois-stable character subset of the regular representation \cite[Theorem~5.4]{Domokos2026}. We use that result as the general field-theoretic boundary, not as a source of the numerical profiles: our local invariant asks for the \emph{least} $d$ generating the group completion on each prescribed stable support. Sections~\ref{sec:boxes}-- \ref{sec:profiles} compute that truncation exactly for all $255$ supports of $C_{2pq}$. \begin{example}[The first two profile rows] For $S=\{0\}$, the vector $e_0$ is a length-one zero sum, so $K(S)=\calL_{\le1}(S)$ and $\depth(S)=1$. For the nonzero order-two orbit $S=\Orb2$, the map $\Z e\to C_2$ has kernel $2\Z e$ and its first nonnegative generator is the pair row $2e$. Hence $\depth(S)=2$. These are the mechanisms behind the first two entries of both profiles. \end{example} \subsection{Ambient and obstruction quotients} Define \[ \Adeg dS=\Z^S/\calL_{\le d}(S). \] There is an exact sequence \[ 0\longrightarrow K(S)/\calL_{\le d}(S) \longrightarrow\Adeg dS \longrightarrow\angles S\longrightarrow0. \tag{2.3} \] The middle group is convenient for presentations. The left group is the obstruction to closing at degree $d$. Its complex character group is the phase ambiguity modulo translations. These three objects will be kept distinct throughout. If $A_d(S)$ is finite, (2.3) gives the index identity \[ [K(S):\calL_{\le d}(S)] =\frac{|A_d(S)|}{|\angles S|}. \tag{2.3a} \] If $A_d(S)$ has a free summand, the obstruction does as well. The later proofs retain explicit presentations rather than using only (2.3a), because a presentation identifies both the obstruction generator and the row that eventually kills it. For later comparison, define the support-restricted orbit-stable maximum \[ \beta_{\mathrm{orb}}(G)= \max\{\depth(S):\varnothing\ne S\subset G,\ \alpha S=S \text{ for every }\alpha\in\Aut(G)\}. \tag{2.4} \] This is only working notation. It is not the ordinary separating Noether number $\beta_{\mathrm{sep}}(G)$ studied in \cite{ScheflerZhaoZhong2026,Huang2026}. Theorem~\ref{thm:profiles} states that (2.4) equals six for $C_{2pq}$ and gives the complete distribution below that maximum. The stability restriction is substantial. Without it one may choose small supports whose relation lattices close much later, but such supports cannot be exact Fourier supports of rational signals. Thus (2.4) is adapted to the rational Fourier problem and is not a new name for the unrestricted separating Noether number. \subsection{The zero-sum face picture} Let $\Delta_S$ be the coordinate face of the standard simplex supported on $S$, and let $\Lambda_G$ impose the integral congruence $\sum_s a_s s=0$. A zero-sum sequence of length $m$ supported in $S$ is exactly a lattice point of \[ m\Delta_S\cap\Lambda_G. \tag{2.5} \] Indeed, the simplex equation is $\sum_s a_s=m$, nonnegativity gives the face, and membership in $\Lambda_G$ is the zero-sum condition. This is the support-face identity used in the zero-sum-polytope theory of Han--Wang--Zhang--Zhang \cite{HanWangZhangZhang2026}. Our question is not to count the points in (2.5), but to ask at which dilation their integral span generates the full relation lattice. This is generation rather than enumeration. Ehrhart-type data count the zero-sum sequences at each length; the closing degree asks when their integral combinations span every relation. Many short rows may be redundant, while a single separating atom can delay closure. The two Skeleton arguments below replace large row lists by a small set of functional consequences whose span is proved uniformly. There is also an equivariance distinction. A setwise $\Aut(G)$-stable face need not consist of fixed lattice points: automorphisms may permute its coordinates. We classify generation on stable supports. We do not require each zero-sum sequence or multiplicity vector to be fixed by the action.