\section{Introduction and exact results}\label{sec:intro}

Higher-order autocorrelations are translation invariants built from products of
a function along several shifted copies.  Their use in pattern recognition and
periodic reconstruction has a long history
\cite{AdlerKonheim1962,McLaughlinRaviv1968,ChazanWeiss1970,
KakaralaIverson1993}.  On a finite
cyclic group, the Fourier transform converts an order-$k$ autocorrelation into
products over zero-sum $k$-tuples.  This makes reconstruction below the
classical global order-six line of Gr\"unbaum--Moore
\cite{GrunbaumMoore1995} a problem about the integral generation of zero-sum
relations.

The answer depends sharply on the exact Fourier support.  For a support
$S\subset C_n$, let $\calL_{\le d}(S)$ be generated by the nonnegative
zero-sum multiplicity vectors of length at most $d$, and let $K(S)$ be the full
integral relation lattice.  We call
\[
 \depth(S)=\min\{d:\calL_{\le d}(S)=K(S)\}
\]
the local closing degree.  Section~\ref{sec:zero-sum} identifies it with the
maximum separating-atom length on $S$, using the established quotient-group
language of Schefler--Zhao--Zhong
\cite{ScheflerZhaoZhong2026}.  Section~\ref{sec:phase} explains why the same
lattice controls phase ratios invisible to autocorrelations.

We compute this object in two directions.  The first is a signal-level theorem
and complete ambiguity parametrization for the exact unit support of every
even cyclic group.  The second is a complete supportwise classification when
the group order is $2pq$, together with exact rational and integer thresholds
and explicit lower-bound pairs on every stable support of depth greater than
one.  The sole depth-one support is direct.  Their common row is the unit orbit.

\subsection{The orbit-stable classification}

Let $p<q$ be distinct odd primes.  The automorphism orbits in $C_{2pq}$ are the
eight element-order classes $\Orb d$ for $d\mid2pq$.  Hence a rational Fourier
support, which is stable under the Galois action, is a union of these eight
orbits and there are $2^8-1=255$ nonempty possibilities.

\begin{theorem}[Orbit-stable profiles]\label{thm:profiles}
For $n=2pq$, the numbers of nonempty $\Aut(C_n)$-stable supports of local
closing degree $1,\ldots,6$ are
\[
 (1,2,154,80,4,14)\qquad\text{if }p=3,
\]
and
\[
 (1,2,154,92,4,2)\qquad\text{if }p,q\ge5.
\]
Equivalently, these are the distributions of the maximum separating-atom
length on the stable supports.  In both regimes the orbit-stable local maximum
is six.
\end{theorem}

For $A\in\{\Q,\Z\}$, let $\depth_A(S)$ be the least $d\ge1$ such that any two
$A$-valued signals with exact Fourier support $S$ and equal autocorrelations
of orders $1,\ldots,d$ differ by a translation.  The phase-ratio bridge gives
the upper bounds $\depth_{\Z}(S)\le\depth_{\Q}(S)\le\depth(S)$.  The reverse
bounds require arithmetic realizations of the lattice obstructions and are
the content of the following theorem.

\begin{theorem}[Uniform rational and integer realization]
\label{thm:uniform-descent}
Let $n=2pq$, where $p<q$ are distinct odd primes.  For every nonempty
$\Aut(C_n)$-stable exact Fourier support $S$,
\[
 \depth_{\Z}(S)=\depth_{\Q}(S)=\depth(S).                  \tag{1.0}
\]
Thus the rational-signal and integer-signal reconstruction profiles are the
two profiles in Theorem~\ref{thm:profiles}.  Equivalence is by translation
only.
\end{theorem}

For a subset $A\subset C_n$, its analytical ordered $k$-deck is (3.1) with
$f=\mathbf1_A$.  Let $r_{\rm set}(C_n)$ be the least $d$ for which this
$d$-deck determines every subset up to translation.  Indicators are
idempotent, so zero shifts make the exact $d$-deck contain every lower deck;
this fact is special to the binary class.  Lemma~\ref{lem:pebody-deck-crosswalk}
shows, including for the empty set and sets of size below $d$, that this is
exactly Pebody's cumulative necklace-deck convention rather than a different
analytical reconstruction problem.

\begin{theorem}[Binary reconstruction in the $2pq$ family]
\label{thm:set-reconstruction}
If $p<q$ are distinct odd primes, then
\[
                     r_{\rm set}(C_{2pq})=4.                \tag{1.0a}
\]
The upper bound follows by combining the supportwise phase-lattice criterion
with a binary exclusion of every depth-five and depth-six support.  The lower
bound is the published even-order three-deck obstruction of
Keleti--Kolountzakis \cite{KeletiKolountzakis2006}.
\end{theorem}

Pebody determined the multiset reconstruction numbers of finite abelian
groups and obtained the general cyclic set bound six
\cite[Theorems~2.1--2.2 and Lemma~9.3]{Pebody2004}.  Combining his comparison of set and multiset
reconstruction with the multiset classification and the even-order lower
bound of Keleti--Kolountzakis already gives
\[
 r_{\rm set}(C_{2^a})=4\quad(a\ge4),\qquad
 r_{\rm set}(C_{2\ell^k})=4
 \quad(\ell>3\text{ prime},\ 2\ell^k>10).                 \tag{1.0b}
\]
Pebody later conjectured value four for every even cyclic order greater than
ten and reported that his method covered many even orders, including every
one below $240$ \cite{Pebody2007}.  Theorem
\ref{thm:set-reconstruction} adds the arithmetically disjoint squarefree
family $2pq$, with two distinct odd prime factors, inside that conjectural
line; no absolute priority or firstness assertion is made.

The proof is uniform in the primes.  It does not arise from interpolation of
finite atlases.  We first compute the cumulative degree-two, degree-three and
degree-four presentations of seven nonzero CRT boxes.  Exactly nineteen cubic
box-count patterns can occur.  Five propagation lemmas convert them into a
ten-clause criterion selecting exactly 77 of the 127 nonempty seven-box masks;
ten explicit defect families prove the converse.  At order four, fourteen
connector types reduce the generic branch to complete primary-coordinate
graphs and the $p=3$ branch to a fixed six-coordinate integral presentation.
The remaining masks are two depth-five bridges and one or seven depth-six
critical bases.  Adding the optional zero orbit doubles every base row and
gives the two displayed profiles.

The ordinary separating Noether number is a different invariant: it maximizes
over modules and supports, and its exact finite-abelian value belongs to the
separating-invariant literature \cite{Huang2026}.  Theorem~\ref{thm:profiles}
is a supportwise distribution restricted to the eight order orbits of one
cyclic group.

There is also a closer general framework.  Domokos characterizes separating
monomials by integral generation of zero-sum sequence monoids, first for
diagonalizable actions and then, over non-closed fields, by imposing the
criterion on every Galois-stable character subset
\cite{Domokos2023,Domokos2026}.  That criterion explains why the stable
supports considered here are the natural local objects.  It does not give the
least truncation degree for any prescribed support, the ten-clause Cubic
Skeleton, either arithmetic profile, or the exact unit-support quotient.
Theorem~\ref{thm:profiles} computes precisely this supportwise layer for the
regular representation of $C_{2pq}$.

\subsection{Exact unit supports}

Write $\units n=(\Z/n\Z)^\times$.  The second theorem begins with an integral
dichotomy and then performs the Galois descent needed for rational signals.

\begin{theorem}[Even unit-support dichotomy and signal classification]
\label{thm:unit-main}
Let $n$ be even.
\begin{enumerate}
\item The fourth-order quotient is
\[
 \Z^{\units n}/\calL_{\le4}(\units n)\cong
 \begin{cases}
 C_n,&3\nmid n,\\
 \Z\oplus C_{n/6},&6\mid n.
 \end{cases}                                                   \tag{1.1}
\]
If $6\mid n$, then $\calL_{\le6}(\units n)=K(\units n)$.
In every even modulus,
$\calL_{\le5}(\units n)=\calL_{\le4}(\units n)$.
\item If $3\nmid n$, every two rational-valued signals with exact Fourier
support $\units n$ and equal autocorrelations through order four differ by a
translation.
\item If $n=6m$, every pair with exact support $\units n$ and equal data
through order four has, after translating the second signal, the form
\[
 \widehat f(u)=\sigma_u(\alpha),\qquad
 \widehat g(u)=\sigma_u(z\alpha),                              \tag{1.2}
\]
where $\alpha\in\Q(\zeta_n)^\times$ and
$z\in\Q(\zeta_6)^\times$ satisfies $z\overline z=1$.  Conversely every pair
in \emph{(1.2)} agrees through order five, and hence through order four.
Normalized parameters are
translation-equivalent exactly modulo $\mu_6$, and equality at order six is
equivalent to $z^6=1$.
\end{enumerate}
\end{theorem}

The integral calculation uses a universal ambient group for the sifted set
\[
 T_{q,c}=\{t\in C_q:\gcd(1+ct,q)=1\}.
\]
It is a concrete use of the Freiman-homomorphism framework of Konyagin--Lev and
Grynkiewicz \cite{KonyaginLev2000,Grynkiewicz2013}.  The exceptional free
direction is not an arbitrary cyclotomic effect: it is the second admissible
class modulo three.  After translation and rational descent, the relative
phase lies in the quadratic subfield even though the common amplitude ranges
over the full cyclotomic field.

\begin{corollary}[The common unit row]\label{cor:unit-row}
For $n=2pq$, the unit support is the single orbit
$\units{2pq}=\Orb{2pq}$.  Its closing degree is
\[
 \depth(\units{2pq})=
 \begin{cases}
 6,&p=3,\\
 4,&p,q\ge5.
 \end{cases}
\]
Theorem~\ref{thm:profiles} and Theorem~\ref{thm:unit-main} therefore agree on
their literal common support.
\end{corollary}

\begin{corollary}[$\units{30}$]\label{cor:u30-intro}
For exact support $\units{30}$, formula \emph{(1.2)} is the complete rational
classification through both orders four and five.  The larger field
$\Q(\zeta_{30})$
enlarges the common amplitude $\alpha$, while the relative phase remains in
the norm-one torus of $\Q(\zeta_6)$.
\end{corollary}

This specializes to the question raised by Agulnick--Busick-Warner
\cite{AgulnickBusickWarner2026}; Section~\ref{sec:signals} gives the precise
comparison.  The Fourier monomial and Galois criteria used in passing from
lattices to signals are due to Casper--Orozco
\cite{CasperOrozco2026}.

\subsection{Claim levels and organization}

The distinction between the two headline conclusions is binding.  On the
unit support, all surviving lattice characters are classified and rationally
realized.  On all 255 supports, we determine the exact rational and integer
thresholds; whenever the depth exceeds one, we construct at least one
Galois-equivariant lower-bound ratio for the failure immediately below
closure.  The sole depth-one support is handled directly.  We do not classify
the complete rational ambiguity torus on every atlas row.

Sections~\ref{sec:zero-sum}--\ref{sec:signals} establish the common dictionary
and the unit-support theorem.  Sections~\ref{sec:boxes}--\ref{sec:profiles}
prove the $2pq$ lattice classification.  Section~\ref{sec:galois} constructs
the nontrivial supportwise rational and integer lower-bound pairs and proves
Theorem~\ref{thm:uniform-descent}.  Section~\ref{sec:set-reconstruction}
specializes the classifier to binary indicators, proves
Theorem~\ref{thm:set-reconstruction}, and records the oriented-deck
translation/reflection corollary.  Section~\ref{sec:interpretation} returns
to autocorrelation and literature boundaries.  The appendices record the full
connector and Boolean tables and separate computational assurance from proof.

\begin{proofboundary}
No absolute priority or firstness assertion is made.  The binary phase/deck
crosswalk is part of the Pebody--Gr\"unbaum--Moore--Simon lineage, and the
even three-deck lower bound is imported from Keleti--Kolountzakis.  The
generic critical
support used later is the sharpness family of Casper--Orozco; the global
sixth-order line and the ratio-character method have earlier sources.  The
general non-closed-field separating criterion is Domokos's.  The contribution
certified here is the displayed fixed-support theorem chain, its exact
supportwise classification, and the explicit Galois-equivariant realization
of every nontrivial lower-bound layer.
\end{proofboundary}